Open Access

Computing eigenvalues and Hermite interpolation for Dirac systems with eigenparameter in boundary conditions

Boundary Value Problems20132013:36

DOI: 10.1186/1687-2770-2013-36

Received: 8 November 2012

Accepted: 5 February 2013

Published: 21 February 2013

Abstract

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper we use the derivative sampling theorem ‘Hermite interpolations’ to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Using computable error bounds, we obtain eigenvalue enclosures. Examples with tables and illustrative figures are given. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc-method in Annaby and Tharwat (BIT Numer. Math. 47:699-713, 2007) and explain that the Hermite interpolations method gives remarkably better results.

MSC:34L16, 94A20, 65L15.

Keywords

Dirac systems eigenvalue problems with eigenparameter in the boundary conditions Hermite interpolations truncation error amplitude error sinc methods

1 Introduction

Let σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq1_HTML.gif and PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq2_HTML.gif be the Paley-Wiener space of all L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq3_HTML.gif-entire functions of exponential type σ. Assume that f ( t ) PW σ 2 PW 2 σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq4_HTML.gif. Then f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq5_HTML.gif can be reconstructed via the Hermite-type sampling series
f ( t ) = n = [ f ( n π σ ) S n 2 ( t ) + f ( n π σ ) sin ( σ t n π ) σ S n ( t ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ1_HTML.gif
(1.1)
where S n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq6_HTML.gif is the sequences of sinc functions
S n ( t ) : = { sin ( σ t n π ) ( σ t n π ) , t n π σ , 1 , t = n π σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ2_HTML.gif
(1.2)
Series (1.1) converges absolutely and uniformly on , cf. [14]. Sometimes, series (1.1) is called the derivative sampling theorem. Our task is to use formula (1.1) to compute eigenvalues of Dirac systems numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1.1), cf. [5]. Both types of errors normally appear in numerical techniques that use interpolation procedures. In the following we summarize these estimates. The truncation error associated with (1.1) is defined to be
R N ( f ) ( t ) : = f ( t ) f N ( t ) , N Z + , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ3_HTML.gif
(1.3)
where f N ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq7_HTML.gif is the truncated series
f N ( t ) = | n | N [ f ( n π σ ) S n 2 ( t ) + f ( n π σ ) sin ( σ t n π ) σ S n ( t ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ4_HTML.gif
(1.4)
It is proved in [5] that if f ( t ) PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq8_HTML.gif and f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq5_HTML.gif is sufficiently smooth in the sense that there exists k Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq9_HTML.gif such that t k f ( t ) L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq10_HTML.gif, then, for t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq11_HTML.gif, | t | < N π / σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq12_HTML.gif, we have
| R N ( f ) ( t ) | T N , k , σ ( t ) : = ξ k , σ E k | sin σ t | 2 3 ( N + 1 ) k ( 1 ( N π σ t ) 3 / 2 + 1 ( N π + σ t ) 3 / 2 ) + ξ k , σ ( σ E k + k E k 1 ) | sin σ t | 2 σ ( N + 1 ) k ( 1 N π σ t + 1 N π + σ t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ5_HTML.gif
(1.5)
where the constants E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq13_HTML.gif and ξ k , σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq14_HTML.gif are given by
E k : = | t k f ( t ) | 2 d t , ξ k , σ : = σ k + 1 / 2 π k + 1 1 4 k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ6_HTML.gif
(1.6)
The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be
A ( ε , f ) ( t ) = n = [ { f ( n π σ ) f ˜ ( n π σ ) } S n 2 ( t ) + { f ( n π σ ) f ˜ ( n π σ ) } sin ( σ t n π ) σ S n ( t ) ] , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ7_HTML.gif
(1.7)
where f ˜ ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq15_HTML.gif and f ˜ ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq16_HTML.gif are approximate samples of f ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq17_HTML.gif and f ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq18_HTML.gif, respectively. Let us assume that the differences ε n : = f ( n π σ ) f ˜ ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq19_HTML.gif, ε n : = f ( n π σ ) f ˜ ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq20_HTML.gif, n Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq21_HTML.gif, are bounded by a positive number ε, i.e., | ε n | , | ε n | ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq22_HTML.gif. If f ( t ) PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq23_HTML.gif satisfies the natural decay conditions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ8_HTML.gif
(1.8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ9_HTML.gif
(1.9)
0 < ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq24_HTML.gif, then for 0 < ε min { π / σ , σ / π , 1 / e } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq25_HTML.gif, we have, [5],
A ( ε , f ) 4 e 1 / 4 σ ( ω + 1 ) { 3 e ( 1 + σ ) + ( ( π / σ ) A + M f ) ρ ( ε ) + ( σ + 2 + log ( 2 ) ) M f } ε log ( 1 / ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ10_HTML.gif
(1.10)
where
A : = 3 σ π ( | f ( 0 ) | + M f ( σ π ) ω ) , ρ ( ε ) : = γ + 10 log ( 1 / ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ11_HTML.gif
(1.11)

and γ : = lim n [ k = 1 n 1 k log n ] 0.577216 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq26_HTML.gif is the Euler-Mascheroni constant.

The classical [6] sampling theorem of Whittaker, Kotel’nikov and Shannon (WKS) for f PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq27_HTML.gif is the series representation
f ( t ) = n = f ( n π σ ) S n ( t ) , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ12_HTML.gif
(1.12)
where the convergence is absolute and uniform on and it is uniform on compact sets of , cf. [68]. Series (1.12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, e.g., [915]. The use of (1.12) in numerical analysis is known as the sinc-method established by Stenger, cf. [1618]. In [10, 12], the authors applied (1.12) and the regularized sinc-method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [19, 20]. In [12] the Dirac system has an eigenparameter appearing in the boundary conditions. The aim of this paper is to investigate the possibilities of using Hermite interpolations rather than Lagrange interpolations, to compute the eigenvalues numerically. Notice that, due to Paley-Wiener’s theorem [21], f PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq28_HTML.gif if and only if there is g ( ) L 2 ( σ , σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq29_HTML.gif such that
f ( t ) = 1 2 π σ σ g ( x ) e i x t d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ13_HTML.gif
(1.13)
Therefore f ( t ) PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq30_HTML.gif, i.e., f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq31_HTML.gif also has an expansion of the form (1.12). However, f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq31_HTML.gif can be also obtained by the term-by-term differentiation formula of (1.12)
f ( t ) = n = f ( n π σ ) S n ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ14_HTML.gif
(1.14)

see [[6], p.52] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples f ( n π σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq32_HTML.gif will be used to compute both f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq5_HTML.gif and f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq31_HTML.gif according to (1.12) and (1.14), respectively.

Consider the Dirac system which consists of the system of differential equations
u 2 ( x ) r 1 ( x ) u 1 ( x ) = λ u 1 ( x ) , u 1 ( x ) + r 2 ( x ) u 2 ( x ) = λ u 2 ( x ) , x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ15_HTML.gif
(1.15)
and the boundary conditions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ16_HTML.gif
(1.16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ17_HTML.gif
(1.17)
where r 1 ( ) , r 2 ( ) L 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq33_HTML.gif and α i , β i , α i , β i R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq34_HTML.gif, i = 0 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq35_HTML.gif, satisfying
( ( α 1 , α 2 ) = ( 0 , 0 )  or  α 1 α 2 α 1 α 2 > 0 ) and ( ( β 1 , β 2 ) = ( 0 , 0 )  or  β 1 β 2 β 1 β 2 > 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ18_HTML.gif
(1.18)

The eigenvalue problem (1.15)-(1.17) will be denoted by Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif when ( α 1 , α 2 ) ( 0 , 0 ) ( β 1 , β 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq37_HTML.gif. It is a Dirac system when the eigenparameter λ appears linearly in both boundary conditions. The classical problem when α 1 = α 2 = β 1 = β 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq38_HTML.gif, which we denote by Γ ( r , α , β , 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq39_HTML.gif, is studied in the monographs of Levitan and Sargsjan [22, 23]. Annaby and Tharwat [24] used Hermite-type sampling series (1.1) to compute the eigenvalues of problem Γ ( r , α , β , 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq39_HTML.gif numerically. In [25], Kerimov proved that Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif has a denumerable set of real and simple eigenvalues with ±∞ as the limit points. Similar results are established in [26] for the problem when the eigenparameter appears in one condition, i.e., when α 1 = α 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq40_HTML.gif, ( β 1 , β 2 ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq41_HTML.gif or equivalently when ( α 1 , α 2 ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq42_HTML.gif and β 1 = β 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq43_HTML.gif, where also sampling theorems have been established. These problems will be denoted by Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif and Γ ( r , α , β , α , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq45_HTML.gif, respectively. The aim of the present work is to compute the eigenvalues of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif and Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif numerically by the Hermite interpolations with an error analysis. This method is based on sampling theorem, Hermite interpolations, but applied to regularized functions hence avoiding any (multiple) integration and keeping the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see [24]. In Sections 2 and 3, we derive the Hermite interpolation technique to compute the eigenvalues of Dirac systems with error estimates. We briefly derive some necessary asymptotics for Dirac systems’ spectral quantities. The last section contains three worked examples with comparisons accompanied by figures and numerics with the Lagrange interpolation method.

2 Treatment of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif

In this section we derive approximate values of the eigenvalues of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif. Recall that Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif has a denumerable set of real and simple eigenvalues, cf. [25]. Let φ ( , λ ) = ( φ 1 ( , λ ) , φ 2 ( , λ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq46_HTML.gif be a solution of (1.15) satisfying the following initial:
φ 1 ( 0 , λ ) = α 2 + λ α 2 , φ 2 ( 0 , λ ) = α 1 + λ α 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ19_HTML.gif
(2.1)
Here A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq47_HTML.gif denotes the transpose of a matrix A. Since φ ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq48_HTML.gif satisfies (1.16), then the eigenvalues of the problem Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif are the zeros of the function
Δ ( λ ) : = ( β 1 + λ β 1 ) φ 1 ( 1 , λ ) ( β 2 + λ β 2 ) φ 2 ( 1 , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ20_HTML.gif
(2.2)
Similarly to [[22], p.220], φ 1 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq49_HTML.gif and φ 2 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq50_HTML.gif satisfy the system of integral equations
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ21_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ22_HTML.gif
(2.4)
where T i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq51_HTML.gif and T ˜ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq52_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq53_HTML.gif, are the Volterra operators defined by
T i u ( x , λ ) : = 0 x sin λ ( x t ) r i ( t ) u ( t , λ ) d t , T ˜ i u ( x , λ ) : = 0 x cos λ ( x t ) r i ( t ) u ( t , λ ) d t , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ23_HTML.gif
(2.5)
For convenience, we define the constants
c 1 : = max { | α 1 | + | α 2 | , | α 1 | + | α 2 | } , c 2 : = 0 1 [ | r 1 ( t ) | + | r 2 ( t ) | ] d t , c 3 : = c 1 c 2 , c 4 : = c 3 exp ( c 2 ) , c 5 : = max { | β 1 | + | β 2 | , | β 1 | + | β 2 | } , c 6 : = c 4 c 5 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ24_HTML.gif
(2.6)
Define h 1 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq54_HTML.gif and h 2 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq55_HTML.gif to be
h 1 ( x , λ ) : = T 1 φ 1 ( x , λ ) + T ˜ 2 φ 2 ( x , λ ) , h 2 ( x , λ ) : = T ˜ 1 φ 1 ( x , λ ) + T 2 φ 2 ( x , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ25_HTML.gif
(2.7)
As in [12] we split Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq56_HTML.gif into two parts via
Δ ( λ ) : = G ( λ ) + S ( λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ26_HTML.gif
(2.8)
where G ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq57_HTML.gif is the known part
G ( λ ) : = ( β 1 + λ β 1 ) ( ( α 1 + λ α 1 ) sin λ + ( α 2 + λ α 2 ) cos λ ) ( β 2 + λ β 2 ) ( ( α 2 + λ α 2 ) sin λ + ( α 1 + λ α 1 ) cos λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ27_HTML.gif
(2.9)
and S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq58_HTML.gif is the unknown one
S ( λ ) : = ( β 1 + λ β 1 ) h 1 ( 1 , λ ) ( β 2 + λ β 2 ) h 2 ( 1 , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ28_HTML.gif
(2.10)
Then the function S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq58_HTML.gif is entire in λ for each x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq59_HTML.gif for which, cf. [12],
| S ( λ ) | c 6 ( 1 + | λ | ) 2 e | λ | , λ C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ29_HTML.gif
(2.11)
The analyticity of S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq58_HTML.gif as well as estimate (2.11) are not adequate to prove that S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq58_HTML.gif lies in a Paley-Wiener space. To solve this problem, we will multiply S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq58_HTML.gif by a regularization factor. Let θ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq60_HTML.gif and m Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq61_HTML.gif, m 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq62_HTML.gif, be fixed. Let F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif be the function
F θ , m ( λ ) : = ( sin θ λ θ λ ) m S ( λ ) , λ C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ30_HTML.gif
(2.12)
We choose θ sufficiently small for which | θ λ | < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq64_HTML.gif. More specifications on m, θ will be given latter on. Then F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif, see [12], is an entire function of λ which satisfies the estimate
| F θ , m ( λ ) | c 0 m c 6 ( 1 + | λ | ) 2 ( 1 + θ | λ | ) m e | λ | ( 1 + m θ ) , λ C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ31_HTML.gif
(2.13)
Moreover, λ m 3 F θ , m ( λ ) L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq65_HTML.gif and
E m 3 ( F θ , m ) = | λ m 3 F θ , m ( λ ) | 2 d λ 2 c 0 m c 6 ξ 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ32_HTML.gif
(2.14)
where
ξ 0 : = 1 θ 2 m 1 ( 3 + 2 m 2 6 θ + 6 θ 2 + 4 m θ 5 m 4 m 3 12 m 2 + 11 m 3 + 6 θ 3 ( θ + 2 m 5 ) ( 4 m 3 12 m 2 + 11 m 3 ) ( m 2 ) ( 2 m 5 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equa_HTML.gif
What we have just proved is that F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif belongs to the Paley-Wiener space PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq66_HTML.gif with σ = 1 + m θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq67_HTML.gif. Since F θ , m ( λ ) PW σ 2 PW 2 σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq68_HTML.gif, then we can reconstruct the functions F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif via the following sampling formula:
F θ , m ( λ ) = n = [ F θ , m ( n π σ ) S n 2 ( λ ) + F θ , m ( n π σ ) sin ( σ λ n π ) σ S n ( λ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ33_HTML.gif
(2.15)
Let N Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq69_HTML.gif, N > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq70_HTML.gif, and approximate F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif by its truncated series F θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq71_HTML.gif, where
F θ , m , N ( λ ) : = n = N N [ F θ , m ( n π σ ) S n 2 ( λ ) + F θ , m ( n π σ ) sin ( σ λ n π ) σ S n ( λ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ34_HTML.gif
(2.16)
Since all eigenvalues are real, then from now on we restrict ourselves to λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq72_HTML.gif. Since λ m 3 F θ , m ( λ ) L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq73_HTML.gif, the truncation error, cf. (1.5), is given for | λ | < N π σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq74_HTML.gif by
| F θ , m ( λ ) F θ , m , N ( λ ) | T N , m 3 , σ ( λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ35_HTML.gif
(2.17)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ36_HTML.gif
(2.18)
The samples { F θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq75_HTML.gif and { F θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq76_HTML.gif, in general, are not known explicitly. So, we approximate them by solving numerically 8 N + 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq77_HTML.gif initial value problems at the nodes { n π σ } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq78_HTML.gif. Let { F ˜ θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq79_HTML.gif and { F ˜ θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq80_HTML.gif be the approximations of the samples of { F θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq81_HTML.gif and { F θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq82_HTML.gif, respectively. Now we define F ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq83_HTML.gif, which approximates F θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq71_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ37_HTML.gif
(2.19)
Using standard methods for solving initial problems, we may assume that for | n | < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq84_HTML.gif,
| F θ , m ( n π σ ) F ˜ θ , m ( n π σ ) | < ε , | F θ , m ( n π σ ) F ˜ θ , m ( n π σ ) | < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ38_HTML.gif
(2.20)
for a sufficiently small ε. From (2.13) we can see that F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq63_HTML.gif satisfies the condition (1.9) when m 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq62_HTML.gif and therefore whenever 0 < ε min { π / σ , σ / π , 1 / e } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq85_HTML.gif, we have
| F θ , m , N ( λ ) F ˜ θ , m , N ( λ ) | A ( ε ) , λ R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ39_HTML.gif
(2.21)
where there is a positive constant M F θ , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq86_HTML.gif for which, cf. (1.10),
A ( ε ) : = 2 e 1 / 4 σ { 3 e ( 1 + σ ) + ( π σ A + M F θ , m ) ρ ( ε ) + ( σ + 2 + log ( 2 ) ) M F θ , m } ε log ( 1 / ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ40_HTML.gif
(2.22)
Here
A : = 3 σ π ( | F θ , m ( 0 ) | + σ π M F θ , m ) , ρ ( ε ) : = γ + 10 log ( 1 / ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equb_HTML.gif
In the following, we use the technique of [27], where only the truncation error analysis is considered, to determine enclosure intervals for the eigenvalues; see also [24, 28]. Let λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq87_HTML.gif be an eigenvalue with | θ λ | < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq88_HTML.gif, that is,
Δ ( λ ) = G ( λ ) + ( sin θ λ θ λ ) m F θ , m ( λ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equc_HTML.gif
Then it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equd_HTML.gif
and so
| G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) | | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Eque_HTML.gif
Since G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq89_HTML.gif is given and | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq90_HTML.gif has computable upper bound, we can define an enclosure for λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif by solving the following system of inequalities:
| sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ41_HTML.gif
(2.23)
Its solution is an interval containing λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif, and over which the graph
G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equf_HTML.gif
is squeezed between the graphs
| sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ42_HTML.gif
(2.24)
and
| sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ43_HTML.gif
(2.25)
Using the fact that
F ˜ θ , m , N ( λ ) F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equg_HTML.gif
uniformly over any compact set, and since λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif is a simple root, we obtain, for large N and sufficiently small ε,
λ ( G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equh_HTML.gif
in a neighborhood of λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif. Hence the graph of G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq92_HTML.gif intersects the graphs | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq93_HTML.gif and | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq94_HTML.gif at two points with abscissae a ( λ , N , ε ) a + ( λ , N , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq95_HTML.gif and the solution of the system of inequalities (2.23) is the interval
I ε , N : = [ a ( λ , N , ε ) , a + ( λ , N , ε ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equi_HTML.gif

and in particular λ I ε , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq96_HTML.gif. Summarizing the above discussion, we arrive at the following lemma which is similar to that of [27] for Sturm-Liouville problems.

Lemma 2.1 For any eigenvalue λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq87_HTML.gif, we can find N 0 Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq97_HTML.gif and sufficiently small ε such that λ I ε , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq96_HTML.gif for N > N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq98_HTML.gif. Moreover,
[ a ( λ , N , ε ) , a + ( λ , N , ε ) ] { λ } as N and ε 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ44_HTML.gif
(2.26)
Proof Since all eigenvalues of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif are simple, then for large N and sufficiently small ε, we have λ ( G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq99_HTML.gif in a neighborhood of λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif. Choose N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq100_HTML.gif such that
G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N 0 ( λ ) = ± | sin θ λ θ λ | m ( T N 0 , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equj_HTML.gif
has two distinct solutions which we denote by a ( λ , N 0 , ε ) a + ( λ , N 0 , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq101_HTML.gif. The decay of T N , m 3 , σ ( λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq102_HTML.gif as N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq103_HTML.gif and A ( ε ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq104_HTML.gif as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq105_HTML.gif will ensure the existence of the solutions a ( λ , N , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq106_HTML.gif and a + ( λ , N , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq107_HTML.gif as N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq108_HTML.gif and ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq109_HTML.gif. For the second point, we recall that F ˜ θ , m , N ( λ ) F θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq110_HTML.gif as N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq108_HTML.gif and as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq105_HTML.gif. Hence, by taking the limit, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equk_HTML.gif

that is, Δ ( a + ) = Δ ( a ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq111_HTML.gif. This leads us to conclude that a + = a = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq112_HTML.gif since λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif is a simple root.

Let Δ ˜ N ( λ ) : = G ( λ ) + ( sin θ λ θ λ ) m F ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq113_HTML.gif. Then (2.17) and (2.21) imply
| Δ ( λ ) Δ ˜ N ( λ ) | | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) , | λ | < N π σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ45_HTML.gif
(2.27)
Therefore θ, m must be chosen so that for | λ | < N π σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq74_HTML.gif
m 4 , | θ λ | < π . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equl_HTML.gif

Let λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif be an eigenvalue and λ N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq114_HTML.gif be its approximation. Thus Δ ( λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq115_HTML.gif and Δ ˜ N ( λ N ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq116_HTML.gif. From (2.27) we have | Δ ˜ N ( λ ) | | sin θ λ θ λ | m ( T N , m 3 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq117_HTML.gif. Now we estimate the error | λ λ N | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq118_HTML.gif for an eigenvalue λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif. □

Theorem 2.2 Let λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif be an eigenvalue of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif. For sufficient large N, we have the following estimate:
| λ λ N | < | sin θ λ N θ λ N | m T N , m 3 , σ ( λ N ) + A ( ε ) inf ζ I ε , N | Δ ( ζ ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ46_HTML.gif
(2.28)

Moreover, | λ λ N | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq119_HTML.gif when N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq108_HTML.gif and ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq105_HTML.gif.

Proof Since Δ ( λ N ) Δ ˜ N ( λ N ) = Δ ( λ N ) Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq120_HTML.gif, then from (2.27) and after replacing λ by λ N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq114_HTML.gif, we obtain
| Δ ( λ N ) Δ ( λ ) | | sin θ λ N θ λ N | m ( T N , m 3 , σ ( λ N ) + A ( ε ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ47_HTML.gif
(2.29)
Using the mean value theorem yields that for some ζ J ε , N : = [ min ( λ , λ N ) , max ( λ , λ N ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq121_HTML.gif,
| ( λ λ N ) Δ ( ζ ) | | sin θ λ N θ λ N | m ( T N , m 3 , σ ( λ N ) + A ( ε ) ) , ζ J ε , N I ε , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ48_HTML.gif
(2.30)

Since the eigenvalues are simple, then for sufficiently large N inf ζ I ε , N | Δ ( ζ ) | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq122_HTML.gif and we get (2.28). The rest of the proof follows from the fact that Δ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq123_HTML.gif converges uniformly to Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq56_HTML.gif in and A ( ε ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq104_HTML.gif when ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq105_HTML.gif. □

3 The case of Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif

This section includes briefly a treatment similar to that of the previous section for the eigenvalue problem Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif introduced in Section 1 above. Notice that the condition (1.18) implies that the analysis of problem Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif is not included in that of Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif. Let ψ ( , λ ) = ( ψ 1 ( , λ ) , ψ 2 ( , λ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq124_HTML.gif be a solution of (1.15) satisfying the following initial:
ψ 1 ( 0 , λ ) = α 2 , ψ 2 ( 0 , λ ) = α 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ49_HTML.gif
(3.1)
Therefore, the eigenvalues of the problem in question are the zeros of the function
Ω ( λ ) : = ( β 1 + λ β 1 ) ψ 1 ( 1 , λ ) ( β 2 + λ β 2 ) ψ 2 ( 1 , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ50_HTML.gif
(3.2)
Similarly to [[22], p.220], ψ ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq125_HTML.gif satisfies the system of integral equations
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ51_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ52_HTML.gif
(3.4)
where T i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq51_HTML.gif and T ˜ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq52_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq53_HTML.gif, are the Volterra operators defined in (2.5) above. Define g 1 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq126_HTML.gif and g 2 ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq127_HTML.gif to be
g 1 ( x , λ ) : = T 1 ψ 1 ( x , λ ) + T ˜ 2 ψ 2 ( x , λ ) , g 2 ( x , λ ) : = T ˜ 1 ψ 1 ( x , λ ) + T 2 ψ 2 ( x , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ53_HTML.gif
(3.5)
As in [12] we split Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq128_HTML.gif into
Ω ( λ ) : = K ( λ ) + U ( λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ54_HTML.gif
(3.6)
where K ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq129_HTML.gif is the known part
K ( λ ) : = ( β 1 + λ β 1 ) ( α 2 cos λ α 1 sin λ ) ( β 2 + λ β 2 ) ( α 1 cos λ + α 2 sin λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ55_HTML.gif
(3.7)
and U ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq130_HTML.gif is the unknown one
U ( λ ) : = ( β 1 + λ β 1 ) g 1 ( 1 , λ ) ( β 2 + λ β 2 ) g 2 ( 1 , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ56_HTML.gif
(3.8)
Then U ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq130_HTML.gif is entire in λ for each x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq131_HTML.gif for which, see [12],
| U ( λ ) | c 6 ( 1 + | λ | ) e | λ | , λ C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ57_HTML.gif
(3.9)
Define R m , θ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq132_HTML.gif to be
R m , θ ( λ ) = ( sin θ λ θ λ ) m U ( λ ) , λ C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ58_HTML.gif
(3.10)
where θ is sufficiently small, for which | θ λ | < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq64_HTML.gif and m are as in the previous section, but m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq133_HTML.gif. Hence
| R m , θ ( λ ) | c 0 m c 6 ( 1 + | λ | ) ( 1 + θ | λ | ) m e | λ | ( 1 + m θ ) , λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ59_HTML.gif
(3.11)
and λ m 2 R m , θ ( λ ) L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq134_HTML.gif with
E m 2 ( R m , θ ) = | λ m 2 R m , θ ( λ ) | 2 d λ c 0 m c 6 ω 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ60_HTML.gif
(3.12)
where
ω 0 : = 2 ( 3 5 m + 2 m 2 3 θ + 2 m θ + θ 2 ) θ 2 m 1 ( 3 + 11 m 12 m 2 + 4 m 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equm_HTML.gif
Thus, R m , θ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq132_HTML.gif belongs to the Paley-Wiener space PW σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq66_HTML.gif with σ = 1 + m θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq67_HTML.gif. Since R θ , m ( λ ) PW σ 2 PW 2 σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq135_HTML.gif, then we can reconstruct the functions R θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq136_HTML.gif via the following sampling formula:
R θ , m ( λ ) = n = [ R θ , m ( n π σ ) S n 2 ( λ ) + R θ , m ( n π σ ) sin ( σ λ n π ) σ S n ( λ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ61_HTML.gif
(3.13)
Let N Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq69_HTML.gif, N > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq70_HTML.gif, and approximate R θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq136_HTML.gif by its truncated series R θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq137_HTML.gif, where
R θ , m , N ( λ ) : = n = N N [ R θ , m ( n π σ ) S n 2 ( λ ) + R θ , m ( n π σ ) sin ( σ λ n π ) σ S n ( λ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ62_HTML.gif
(3.14)
Since all eigenvalues are real, then from now on we restrict ourselves to λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq72_HTML.gif. Since λ m 2 R θ , m ( λ ) L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq138_HTML.gif, the truncation error, cf. (1.5), is given for | λ | < N π σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq74_HTML.gif by
| R θ , m ( λ ) R θ , m , N ( λ ) | T N , m 2 , σ ( λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ63_HTML.gif
(3.15)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ64_HTML.gif
(3.16)
The samples { R θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq139_HTML.gif and { R θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq140_HTML.gif, in general, are not known explicitly. So, we approximate them by solving numerically 8 N + 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq77_HTML.gif initial value problems at the nodes { n π σ } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq78_HTML.gif. Let { R ˜ θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq141_HTML.gif and { R ˜ θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq142_HTML.gif be the approximations of the samples of { R θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq143_HTML.gif and { R θ , m ( n π σ ) } n = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq144_HTML.gif, respectively. Now we define R ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq145_HTML.gif, which approximates R θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq137_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ65_HTML.gif
(3.17)
Using standard methods for solving initial problems, we may assume that for | n | < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq84_HTML.gif,
| R θ , m ( n π σ ) R ˜ θ , m ( n π σ ) | < ε , | R θ , m ( n π σ ) R ˜ θ , m ( n π σ ) | < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ66_HTML.gif
(3.18)
for a sufficiently small ε. From (2.13) we can see that R θ , m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq136_HTML.gif satisfies the condition (1.9) when m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq133_HTML.gif and therefore whenever 0 < ε min { π / σ , σ / π , 1 / e } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq85_HTML.gif, we have
| R θ , m , N ( λ ) R ˜ θ , m , N ( λ ) | A ( ε ) , λ R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ67_HTML.gif
(3.19)
where there is a positive constant M R θ , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq146_HTML.gif for which, cf. (1.10),
A ( ε ) : = 2 e 1 / 4 σ { 3 e ( 1 + σ ) + ( π σ A + M R θ , m ) ρ ( ε ) + ( σ + 2 + log ( 2 ) ) M R θ , m } ε log ( 1 / ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ68_HTML.gif
(3.20)
Here
A : = 3 σ π ( | R θ , m ( 0 ) | + σ π M R θ , m ) , ρ ( ε ) : = γ + 10 log ( 1 / ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equn_HTML.gif

As in the above section, we have the following lemma.

Lemma 3.1 For any eigenvalue λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq87_HTML.gif of the problem Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif, we can find N 0 Z + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq97_HTML.gif and sufficiently small ε such that λ I ε , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq147_HTML.gif for N > N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq98_HTML.gif, where
I ε , N : = [ b ( λ , N , ε ) , b + ( λ , N , ε ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equo_HTML.gif
b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq148_HTML.gif, b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq149_HTML.gif are the solutions of the inequalities
| sin θ λ θ λ | m ( T N , m 2 , σ ( λ ) + A ( ε ) ) Ω ˜ N ( λ ) | sin θ λ θ λ | m ( T N , m 2 , σ ( λ ) + A ( ε ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ69_HTML.gif
(3.21)
Moreover,
[ b ( λ , N , ε ) , b + ( λ , N , ε ) ] { λ } as N and ε 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ70_HTML.gif
(3.22)
Let Ω ˜ N ( λ ) : = K ( λ ) + ( sin θ λ θ λ ) m R ˜ θ , m , N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq150_HTML.gif. Then (3.15) and (3.19) imply
| Ω ( λ ) Ω ˜ N ( λ ) | | sin θ λ θ λ | m ( T N , m 2 , σ ( λ ) + A ( ε ) ) , | λ | < N π σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ71_HTML.gif
(3.23)
Therefore, θ, m must be chosen so that for | λ | < N π σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq74_HTML.gif,
m 3 , | θ λ | < π . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equp_HTML.gif

Let λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif be an eigenvalue and λ N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq114_HTML.gif be its approximation. Thus Ω ( λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq151_HTML.gif and Ω ˜ N ( λ N ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq152_HTML.gif. From (3.23) we have | Ω ˜ N ( λ ) | | sin θ λ θ λ | m ( T N , m 2 , σ ( λ ) + A ( ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq153_HTML.gif. Now we estimate the error | λ λ N | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq118_HTML.gif for an eigenvalue λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif. Finally, we have the following estimate.

Theorem 3.2 Let λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq91_HTML.gif be an eigenvalue of the problem Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif. For sufficient large N, we have the following estimate:
| λ λ N | < | sin θ λ N θ λ N | m T N , m 2 , σ ( λ N ) + A ( ε ) inf ζ I ε , N | Ω ( ζ ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ72_HTML.gif
(3.24)

Moreover, | λ λ N | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq119_HTML.gif when N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq108_HTML.gif and ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq105_HTML.gif.

In the following section, we have taken θ = 1 / ( N m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq154_HTML.gif, where σ = 1 + m θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq67_HTML.gif, in order to avoid the first singularity of ( sin θ λ N θ λ N ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq155_HTML.gif.

4 Examples

This section includes three detailed worked examples illustrating the above technique accompanied by comparison with the sinc-method derived in [12]. It is clearly seen that the Hermite interpolations method gives remarkably better results. The first two examples are computed in [12] with the classical sinc-method where r 1 ( x ) = r 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq156_HTML.gif. But in the last example, where eigenvalues cannot be computed concretely, r 1 ( x ) r 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq157_HTML.gif. By E S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq158_HTML.gif and E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq159_HTML.gif we mean the absolute errors associated with the results of the classical sinc-method and our new method (Hermite interpolations), respectively. We indicate in these examples the effect of the amplitude error in the method by determining enclosure intervals for different values of ε. We also indicate the effect of the parameters m and θ by several choices. Each example is exhibited via figures that accurately illustrate the procedure near to some of the approximated eigenvalues. More explanations are given below. Recall that a ± ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq160_HTML.gif and b ± ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq161_HTML.gif are defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ73_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ74_HTML.gif
(4.2)
respectively. Recall also that the enclosure intervals I ε , N : = [ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq162_HTML.gif and I ε , N : = [ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq163_HTML.gif are determined by solving
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ75_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ76_HTML.gif
(4.4)

respectively. We would like to mention that Mathematica has been used to obtain the exact values for the three examples where eigenvalues cannot be computed concretely. Mathematica is also used in rounding the exact eigenvalues, which are square roots.

Example 1

The boundary value problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ77_HTML.gif
(4.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ78_HTML.gif
(4.6)
is a special case of the problem Γ ( r , α , β , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq36_HTML.gif when r 1 ( x ) = r 2 ( x ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq164_HTML.gif, α 1 = α 2 = β 1 = β 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq165_HTML.gif, α 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq166_HTML.gif and α 2 = β 1 = β 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq167_HTML.gif. Here the characteristic function is
Δ ( λ ) : = 2 λ cos ( 1 3 + λ ) ( λ 2 1 ) sin ( 1 3 + λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ79_HTML.gif
(4.7)
The function G ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq57_HTML.gif will be
G ( λ ) : = 2 λ cos λ + ( 1 λ 2 ) sin λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ80_HTML.gif
(4.8)
As is clearly seen, eigenvalues cannot be computed explicitly. Five tables indicate the application of our technique to this problem and the effect of ε, θ and m (Tables 1, 2, 3, 4 and 5). By exact, we mean the zeros of Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq56_HTML.gif computed by Mathematica.
Table 1

N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq169_HTML.gif , θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq170_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Sinc λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Hermite λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

E S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq173_HTML.gif

E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq174_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−1.505786875767961

−1.5057868758327264

−1.5057868758327246

6.47653 × 10−11

1.77636 × 10−15

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.11141619186432938

−0.11141619146375636

−0.11141619146375908

4.00573 × 10−10

2.72005 × 10−15

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.1223201536675476

1.1223201551741047

1.1223201551741295

1.50656 × 10−9

2.4869 × 10−14

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.3830704087110752

3.383070408212596

3.3830704082125935

4.98479 × 10−10

2.66454 × 10−15

Table 2

N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq179_HTML.gif , θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq180_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Sinc λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Hermite λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−1.5057868758327237144550336

−1.5057868758327218561623117

−1.5057868758327237144491654

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.1114161914637569327965574

−0.1114161914637563667829627

−0.1114161914637563668056111

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.1223201551741129577354075

1.1223201551741041543767735

1.1223201551741041544693398

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.3830704082126126090125379

3.3830704082125963004202471

3.3830704082125963003644934

Table 3

Absolute error | λ k λ k , N | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq181_HTML.gif for N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq179_HTML.gif , θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq180_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq182_HTML.gif

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq183_HTML.gif

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq184_HTML.gif

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq185_HTML.gif

E S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq158_HTML.gif

1.858 × 10−15

5.660 × 10−16

8.803 × 10−15

1.630 × 10−14

E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq159_HTML.gif

5.868 × 10−21

2.265 × 10−20

9.257 × 10−20

5.575 × 10−20

Table 4

For N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq169_HTML.gif and θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq170_HTML.gif , the exact solutions λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq186_HTML.gif are all inside the interval [ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq187_HTML.gif for different values of ε

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

[ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq188_HTML.gif, ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq189_HTML.gif

[ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq188_HTML.gif, ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq190_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−1.5057868758327264

[−1.650349,−1.220683]

[−1.508403,−1.502664]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.11141619146375636

[−0.179803,−0.071447]

[−0.130019,−0.100199]

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.1223201551741047

[0.429491,1.314588]

[0.884579,1.206467]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.383070408212596

[3.314309,3.464923]

[3.369349,3.400197]

E 7 ( F θ , m ) = 3.05294 × 10 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq191_HTML.gif, E 6 ( F θ , m ) = 2.53419 × 10 9 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq192_HTML.gif, ω = 1, M F θ , m = 3.56048 × 10 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq193_HTML.gif.

Table 5

With N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq179_HTML.gif and θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq180_HTML.gif , λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq186_HTML.gif are all inside the interval [ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq187_HTML.gif for different values of ε

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

[ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq188_HTML.gif, ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq189_HTML.gif

[ a , a + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq188_HTML.gif, ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq190_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−1.5057868758327218561623117

[−1.652755,−1.334613]

[−1.505894,−1.505678]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.1114161914637563667829627

[−0.331996,0.121731]

[−0.111834,−0.111003]

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.1223201551741041543767735

[0.923906,1.285003]

[1.120633,1.124014]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.3830704082125963004202471

[3.241846,3.533914]

[3.382059,3.384093]

E 12 ( F θ , m ) = 1.61064 × 10 13 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq194_HTML.gif, E 11 ( F θ , m ) = 1.71043 × 10 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq195_HTML.gif, ω = 1, M F θ , m = 3.98665 × 10 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq196_HTML.gif.

Figures 1 and 2 illustrate the comparison between Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq56_HTML.gif and Δ ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq197_HTML.gif for different values of m and θ. Figures 3 and 4, for N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif, m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq169_HTML.gif and θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq170_HTML.gif, illustrate the enclosure intervals for ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq198_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq199_HTML.gif, respectively. Also, Figures 5 and 6 illustrate the enclosure intervals for ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq198_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq199_HTML.gif, respectively, but for m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq179_HTML.gif, θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq180_HTML.gif.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig1_HTML.jpg
Figure 1

Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , Δ ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq201_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq203_HTML.gif and θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq204_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig2_HTML.jpg
Figure 2

Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , Δ ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq201_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq205_HTML.gif and θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq206_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig3_HTML.jpg
Figure 3

a + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq207_HTML.gif , Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq208_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq203_HTML.gif , θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq204_HTML.gif and ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq209_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig4_HTML.jpg
Figure 4

a + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq207_HTML.gif , Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq208_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq203_HTML.gif , θ = 1 / 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq204_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq210_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig5_HTML.jpg
Figure 5

a + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq207_HTML.gif , Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq208_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq205_HTML.gif , θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq206_HTML.gif and ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq209_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig6_HTML.jpg
Figure 6

a + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq207_HTML.gif , Δ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq200_HTML.gif , a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq208_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq205_HTML.gif , θ = 1 / 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq206_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq210_HTML.gif .

Example 2

The Dirac system
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ81_HTML.gif
(4.9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ82_HTML.gif
(4.10)
is a special case of the problem treated in the previous section with r 1 ( x ) = r 2 ( x ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq164_HTML.gif, α 1 = β 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq211_HTML.gif, α 2 = β 1 = β 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq212_HTML.gif and β 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq213_HTML.gif. The characteristic function is
Ω ( λ ) : = cos ( 1 3 + λ ) λ sin ( 1 3 + λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ83_HTML.gif
(4.11)
The function K ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq129_HTML.gif will be
K ( λ ) : = cos λ λ sin λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ84_HTML.gif
(4.12)
As in the previous example, Figures 7, 8, 9, 10, 11 and 12 illustrate the results of Tables 6, 7, 8, 9 and 10. Figures 7 and 8 illustrate the comparison between Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq128_HTML.gif and Ω ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq214_HTML.gif for different values of m and θ. Figures 9 and 10, for N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif, m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq215_HTML.gif and θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq216_HTML.gif, illustrate the enclosure intervals for ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq198_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq199_HTML.gif, respectively. Also, Figures 11 and 12 illustrate the enclosure intervals for ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq198_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq199_HTML.gif, respectively, but for m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq217_HTML.gif, θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq218_HTML.gif.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig7_HTML.jpg
Figure 7

Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , Ω ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq220_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq221_HTML.gif and θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq222_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig8_HTML.jpg
Figure 8

Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , Ω ˜ N ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq220_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq223_HTML.gif and θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq224_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig9_HTML.jpg
Figure 9

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq221_HTML.gif , θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq222_HTML.gif and ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq209_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig10_HTML.jpg
Figure 10

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq221_HTML.gif , θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq222_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq210_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig11_HTML.jpg
Figure 11

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq223_HTML.gif , θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq224_HTML.gif and ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq209_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig12_HTML.jpg
Figure 12

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq223_HTML.gif , θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq224_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq210_HTML.gif .

Table 6

N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq215_HTML.gif , θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq216_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Sinc λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Hermite λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

E S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq173_HTML.gif

E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq174_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.7364320716761927

−3.736432198331617

−3.7364321983463715

1.26655 × 10−7

1.47544 × 10−11

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−1.0801974353048152

−1.0801976714797825

−1.0801976714531203

2.36175 × 10−7

2.66622 × 10−11

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

0.6565189567613093

0.6565187872152198

0.6565187872029187

1.69546 × 10−7

1.23012 × 10−11

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.118561532798614

3.1185614501648167

3.1185614501681216

4.98479 × 10−8

3.30491 × 10−12

Table 7

N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq217_HTML.gif , θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq218_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Sinc λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Hermite λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.736432198332202082929465

−3.736432198331617091212013

−3.736432198331617091189782

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−1.080197671476027921290673

−1.080197671479782493157863

−1.080197671479782493947136

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

0.6565187872242083579354743

0.6565187872152199183983102

0.6565187872152199230592640

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.118561450158043898832776

3.118561450164816849643922

3.118561450164816845810261

Table 8

Absolute error | λ k λ k , N | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq181_HTML.gif for N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq217_HTML.gif , θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq218_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq182_HTML.gif

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq183_HTML.gif

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq184_HTML.gif

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq185_HTML.gif

E S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq158_HTML.gif

5.849 × 10−13

3.755 × 10−12

8.988 × 10−12

6.773 × 10−12

E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq159_HTML.gif

2.223 × 10−20

7.893 × 10−19

4.661 × 10−18

3.834 × 10−18

Table 9

For N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq215_HTML.gif and θ = 1 / 14 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq216_HTML.gif , the exact solutions λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq186_HTML.gif are all inside the interval [ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq227_HTML.gif for different values of ε

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq189_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq190_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.736432198331617091212013

[−3.881037,−3.476447]

[−3.836682,−3.557513]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−1.080197671479782493157863

[−1.435432,−0.665868]

[−1.365324,−0.760935]

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

0.6565187872152199183983102

[0.410872,1.116247]

[0.492155,1.004381]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.118561450164816849643922

[2.884061,3.390359]

[2.940901,3.331955]

E 4 ( R θ , m ) = 2.9056 × 10 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq229_HTML.gif, E 3 ( R θ , m ) = 2.29859 × 10 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq230_HTML.gif, ω = 1, M R θ , m = 98845.4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq231_HTML.gif.

Table 10

With N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq217_HTML.gif and θ = 1 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq218_HTML.gif , λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq186_HTML.gif are all inside the interval [ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq227_HTML.gif for different values of ε

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq189_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq190_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.736432198331617

[−4.1011429,−3.3717065]

[−3.7364598,−3.7364045]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−1.0801976714797825

[−1.5078873,−0.4433678]

[−1.0808585,−1.07952734]

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

0.6565187872152198

[0.0168549,1.1086918]

[0.6528005,0.6602210]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.1185614501648167

[2.7401391,3.1185614]

[3.1157222,3.1214041]

E 10 ( R θ , m ) = 6.2724 × 10 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq232_HTML.gif, E 9 ( R θ , m ) = 8.21004 × 10 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq233_HTML.gif, ω = 1, M R θ , m = 501421 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq234_HTML.gif.

Example 3

The boundary value problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ85_HTML.gif
(4.13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ86_HTML.gif
(4.14)
is a special case of the problem Γ ( r , α , β , 0 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq44_HTML.gif when r 1 ( x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq235_HTML.gif, r 2 ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq236_HTML.gif, α 2 = β 1 = β 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq237_HTML.gif and α 1 = β 1 = β 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq238_HTML.gif. Here the characteristic function is
Ω ( λ ) : = 1 / ( AiryAiPrime [ λ ( 1 λ ) 1 / 3 ] AiryBi [ λ ( 1 λ ) 1 / 3 ] AiryAi [ λ ( 1 λ ) 1 / 3 ] AiryBiPrime [ λ ( 1 λ ) 1 / 3 ] ) × [ λ ( 1 λ ) 2 / 3 AiryAi [ ( 1 + λ ) ( 1 λ ) 1 / 3 ] AiryBi [ λ ( 1 λ ) 1 / 3 ] + AiryAiPrime [ ( λ + 1 ) ( 1 λ ) 1 / 3 ] AiryBi [ λ ( 1 λ ) 1 / 3 ] AiryAiPrime [ λ ( 1 λ ) 1 / 3 ] ( λ ( 1 λ ) 2 / 3 AiryBi [ ( λ + 1 ) ( 1 λ ) 1 / 3 ] + AiryBiPrime [ ( λ + 1 ) ( 1 λ ) 1 / 3 ] ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ87_HTML.gif
(4.15)
where AiryAi [ z ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq239_HTML.gif and AiryBi [ z ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq240_HTML.gif are Airy functions Ai ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq241_HTML.gif and Bi ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq242_HTML.gif, respectively, and AiryAiPrime [ z ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq243_HTML.gif and AiryBiPrime [ z ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq244_HTML.gif are derivatives of Airy functions. The function K ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq129_HTML.gif will be
K ( λ ) : = cos λ λ sin λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Equ88_HTML.gif
(4.16)
Figures 13, 14 and Tables 11, 12 illustrate the applications of the method to this problem.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig13_HTML.jpg
Figure 13

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq245_HTML.gif , θ = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq246_HTML.gif and ε = 10 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq247_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_Fig14_HTML.jpg
Figure 14

b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq225_HTML.gif , Ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq219_HTML.gif , b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq226_HTML.gif with N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq202_HTML.gif , m = 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq245_HTML.gif , θ = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq246_HTML.gif and ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq210_HTML.gif .

Table 11

N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq248_HTML.gif , θ = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq249_HTML.gif

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

λ k , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq172_HTML.gif

E H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq174_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.1976270593385675784857858037

−3.1976270593385675784857498452

3.596 × 10−23

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.64351783872891518984316280760

−0.64351783872891518984316309998

2.924 × 10−25

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.4487204290456776077365351429

1.4487204290456776077365176362

1.751 × 10−23

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.8015200831700579923508826075

3.8015200831700579923509045951

2.199 × 10−23

Table 12

With N = 20 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq168_HTML.gif , m = 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq248_HTML.gif and θ = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq249_HTML.gif , λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq186_HTML.gif are all inside the interval [ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq227_HTML.gif for different values of ε

λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

Exact λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq171_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq189_HTML.gif

[ b , b + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq228_HTML.gif, ε = 10 15 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq190_HTML.gif

λ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq175_HTML.gif

−3.1976270593385675784857858037

[−3.30255437,−3.11013060]

[−3.19791869,−3.19733846]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq176_HTML.gif

−0.64351783872891518984316280760

[−0.67219637,−0.61489406]

[−0.64356572,−0.64346999]

λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq177_HTML.gif

1.4487204290456776077365351429

[1.40795687,1.49107473]

[1.44812338,1.44932224]

λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq178_HTML.gif

3.8015200831700579923508826075

[3.60636554,4.19453907]

[3.80103975,3.80200804]

E 14 ( R θ , m ) = 2.16956 × 10 13 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq250_HTML.gif, E 13 ( R θ , m ) = 5.61116 × 10 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq251_HTML.gif, ω = 1, M R θ , m = 3.15557 × 10 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-36/MediaObjects/13661_2012_Article_289_IEq252_HTML.gif.

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, Faculty of Science, Beni-Suef University

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© Tharwat; licensee Springer. 2013

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