- Research
- Open access
- Published:
Approximation of eigenvalues of boundary value problems
Boundary Value Problems volume 2014, Article number: 51 (2014)
Abstract
In the present paper we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of discontinuous Dirac systems, which contain an eigenvalue parameter in one boundary condition, with transmission conditions at the point of discontinuity. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper showing that this method gives us better results.
MSC: 34L16, 94A20, 65L15.
1 Introduction
Consider the discontinuous Dirac system which consists of the system of differential equations
with boundary conditions
and transmission conditions
where ; ; the real-valued functions and are continuous in and , and have finite limits , ; , ; and .The aim of the present work is to compute the eigenvalues of (1.1)-(1.5) numerically by the sinc-Gaussian technique with errors analysis, truncation error and amplitude error.
Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. A fundamental result in information theory is the Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem [1–3]. It states that any , ,
can be reconstructed from its sampled values by the formula
where
Series (1.6) converges absolutely and uniformly on compact subsets of ℂ, and uniformly on ℝ, cf. [4]. Expansion (1.6) is used in several approximation problems which are known as sinc-methods; see, e.g., [5–8]. In particular the sinc-method is used to approximate eigenvalues of boundary value problems; see, for example, [9–12]. The sinc-method has a slow rate of decay at infinity, which is as slow as . There have been several attempts to improve the rate of decay. One of the interesting ways is to multiply the sinc-function in (1.6) by a kernel function; see, e.g., [13–15]. Let and . Assume that such that , then for we have the expansion [16]
The speed of convergence of the series in (1.8) is determined by the decay of . But the decay of an entire function of exponential type cannot be as fast as as for some positive c [16]. In [17], Qian has introduced the following regularized sampling formula. For , and , Qian defined the operator [17]
where , which is called the Gaussian function, and denotes the integer part of ; see also [18, 19]. Qian also derived the following error bound. If , and , then [17, 18]
In [16] Schmeisser and Stenger extended the operator (1.9) to the complex domain ℂ. For , and , they defined the operator [16]
where and . Note that the summation limits in (1.11) depend on the real part of λ. Schmeisser and Stenger [16] proved that if g is an entire function such that
where ϕ is a non-decreasing, non-negative function on and , then for , , , , we have
where
The amplitude error arises when the exact values of (1.11) are replaced by the approximations . We assume that are close to , i.e., there is sufficiently small such that
Let , and be fixed numbers. The authors in [20] proved that if (1.15) is held, then for , we have
where
Without eigenparameter appearing in any of boundary conditions, in [21] and [12] Tharwat et al. approximately computed the eigenvalues of the discontinuous Dirac system which is studied in the monographs of [22] by Hermite interpolations and regularized sinc-methods, respectively. In the regularized sinc-method, also the same in the Hermite interpolations method, the basic idea is as follows: The eigenvalues are characterized as the zeros of an analytic function which can be written in the form , where is a known part. The ingenuity of the approach is in trying to choose the function so that (unknown part) and can be approximated by the WKS sampling theorem if its values at some equally spaced points are known; see [9–12]. Recall that, in regularized sinc and Hermite interpolations methods, it is necessary that is an -function. In this paper we will use the sinc-Gaussian sampling formula (1.11) to compute eigenvalues of (1.1)-(1.5) numerically. As is expected, the new method reduced the error bounds remarkably (see the examples in Section 4). Also here, the basic idea is to write the function of eigenvalues as the sum of two terms, one known and the other unknown but an entire function of exponential type which satisfies (1.12). In other words, the unknown term is not necessarily an -function. Then we approximate the unknown part using (1.11) and obtain better results. We would like to mention that the papers in computing eigenvalues by the sinc-Gaussian method are few; see [20, 23–25]. In Sections 2, 3 we derive the sinc-Gaussian technique to compute the eigenvalues of (1.1)-(1.5) with error estimates. The last section involves some illustrative examples.
2 Preliminaries
In this section we derive approximate values of the eigenvalues of problem (1.1)-(1.5). Recall that problem (1.1)-(1.5) has a denumerable set of real and simple eigenvalues, cf. [26]; see also [22, 27–29]. Let
be the solution of (1.1) satisfying the following initial conditions:
In [26], Tharwat proved the existence and uniqueness of (2.2). Since satisfies (1.2), then the eigenvalues of problem (1.1)-(1.5) are the zeros of the function (see Lemma 2.4 of [[26], p.8])
Notice that both and are entire functions of λ, and satisfies the system of integral equations (cf. [26])
where , , and , , are the Volterra integral operators defined by
For convenience, we define the constants
Define and , , to be
Lemma 2.1 The functions and are entire in λ for any fixed and satisfy the growth condition
Proof Since , then from (2.4) and (2.5) we obtain
Using the inequalities and for leads for to
The above inequality can be reduced to
Similarly, we can prove that
Then from (2.12), (2.13) and Lemma 3.1 of [[28], p.204], we obtain (2.11). □
In a similar manner, we will prove the following lemma for and .
Lemma 2.2 The functions and are entire in λ for any fixed and satisfy the growth condition
Proof Since , then from (2.6) and (2.7) we obtain
Then from (2.4) and (2.5) and Lemma 2.1, we get
Similarly, we can prove that
□
3 The numerical scheme
In this section we derive the method of computing eigenvalues of problem (1.1)-(1.5) numerically. The basic idea of the scheme is to split into two parts a known part and an unknown one . Then we approximate using (1.11) to get the approximate and then compute the approximate zeros. We first split into two parts as follows:
where is the unknown part involving integral operators
and is the known part
Then, from Lemma 2.1 and Lemma 2.2, we have the following result.
Lemma 3.1 The function is entire in λ and the following estimate holds:
where
Proof From (3.2) we have
Using the inequalities and for , Lemma 2.1 and Lemma 2.2 imply (3.4). □
Thus is an entire function of exponential type . In the following we let since all eigenvalues are real. Now we approximate the function using the operator (1.11) where and and then, from (1.13), we obtain
where
The samples , cannot be computed explicitly in the general case. We approximate these samples numerically by solving the initial value problems defined by (1.1) and (2.2) to obtain the approximate values , , i.e., . Here we use the computer algebra system Mathematica to obtain approximate solutions with the required accuracy. However, a separate study for the effect of different numerical schemes and the computational costs would be interesting. Accordingly, we have the explicit expansion
Therefore we get (cf. (1.16))
Now let . From (3.6) and (3.9) we obtain
Let be an eigenvalue and be its desired approximation, i.e., and . From (3.10) we have . Define the curves
The curves , enclose the curve of for suitably large N. Hence the closure interval is determined by solving , which gives an interval
It is worthwhile to mention that the simplicity of the eigenvalues guarantees the existence of approximate eigenvalues, i.e., the for which . Next we estimate the error for the eigenvalue .
Theorem 3.2 Let be an eigenvalue of (1.1)-(1.5) and let be its approximation. Then, for , we have the following estimate:
where the interval is defined above.
Proof Replacing λ by in (3.10), we obtain
where we have used . Using the mean value theorem yields that for some ,
Since is simple and N is sufficiently large, then and we get (3.12). □
4 Numerical examples
This section includes two examples illustrating the sinc-Gaussian method. It is clearly seen that the sinc-Gaussian method gives remarkably better results. We indicate in these two 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 N and h by several choices. We would like to mention that Mathematica has been used to obtain the exact values for these examples where eigenvalues cannot be computed concretely. Mathematica is also used in rounding off the exact eigenvalues, which are square roots. Each example is presented via figures that accurately illustrate the procedure near some of the approximated eigenvalues. More explanations are given below.
Example 4.1 Consider the system
Here
, , and . Direct calculations give
and
As is clearly seen, the eigenvalues cannot be computed explicitly. The following three tables (Tables 1, 2, 3) indicate the application of our technique to this problem and the effect of ε. By exact we mean the zeros of computed by Mathematica.
Figures 1 and 2 illustrate the enclosure intervals dominating for , and , , respectively. The middle curve represents , while the upper and lower curves represent the curves of , , respectively. We notice that when , the two curves are almost identical. Similarly, Figures 3 and 4 illustrate the enclosure intervals dominating for , and , , respectively.
Example 4.2 In this example we consider the system
where
, , and . Direct calculations give
and
Tables 4, 5, give the exact eigenvalues and their approximate ones for different values of h, N, ε. In Table 6, we give the absolute error for different values of h and N.
Here Figures 5, 6, 7, 8 illustrate the enclosure intervals dominating and for , and , , respectively.
References
Kotel’nikov V: On the carrying capacity of the ‘ether’ and wire in telecommunications. 55. In Material for the First All-Union Conference on Questions of Communications. Izd. Red. Upr. Svyazi RKKA, Moscow; 1933:55-64. (Russian)
Shannon CE: Communications in the presence of noise. Proc. IRE 1949, 37: 10-21.
Whittaker ET: On the functions which are represented by the expansion of the interpolation theory. Proc. R. Soc. Edinb., Sect. A 1915, 35: 181-194.
Butzer PL, Schmeisser G, Stens RL: An introduction to sampling analysis. In Nonuniform Sampling: Theory and Practices. Edited by: Marvasti F. Kluwer Academic, New York; 2001:17-121.
Kowalski M, Sikorski K, Stenger F: Selected Topics in Approximation and Computation. Oxford University Press, London; 1995.
Lund J, Bowers K: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia; 1992.
Stenger F: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 1981, 23: 156-224.
Stenger F: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York; 1993.
Boumenir A: Higher approximation of eigenvalues by the sampling method. BIT Numer. Math. 2000, 40: 215-225. 10.1023/A:1022334806027
Boumenir A: Sampling and eigenvalues of non-self-adjoint Sturm-Liouville problems. SIAM J. Sci. Comput. 2001, 23: 219-229. 10.1137/S1064827500374078
Tharwat MM, Bhrawy AH, Yildirim A: Numerical computation of eigenvalues of discontinuous Sturm-Liouville problems with parameter dependent boundary conditions using sinc method. Numer. Algorithms 2013, 63: 27-48. 10.1007/s11075-012-9609-3
Tharwat MM, Bhrawy AH, Yildirim A: Numerical computation of eigenvalues of discontinuous Dirac system using sinc method with error analysis. Int. J. Comput. Math. 2012, 89: 2061-2080. 10.1080/00207160.2012.700112
Butzer PL, Stens RL: A modification of the Whittaker-Kotel’nikov-Shannon sampling series. Aequ. Math. 1985, 28: 305-311. 10.1007/BF02189424
Gervais R, Rahman QI, Schmeisser G: A bandlimited function simulating a duration-limited one. In Approximation Theory and Functional Analysis. Edited by: Butzer PL, Stens RL. Birkhäuser, Basel; 1984:355-362.
Stens RL: Sampling by generalized kernels. In Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Edited by: Higgins JR, Stens RL. Oxford University Press, Oxford; 1999:130-157.
Schmeisser G, Stenger F: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 2007, 6: 199-221.
Qian L: On the regularized Whittaker-Kotel’nikov-Shannon sampling formula. Proc. Am. Math. Soc. 2002, 131: 1169-1176.
Qian L, Creamer DB: A modification of the sampling series with a Gaussian multiplier. Sampl. Theory Signal Image Process. 2006, 5: 1-20.
Qian L, Creamer DB: Localized sampling in the presence of noise. Appl. Math. Lett. 2006, 19: 351-355. 10.1016/j.aml.2005.05.013
Annaby MH, Asharabi RM: Computing eigenvalues of boundary-value problems using sinc-Gaussian method. Sampl. Theory Signal Image Process. 2008, 7: 293-312.
Tharwat MM, Bhrawy AH: Computation of eigenvalues of discontinuous Dirac system using Hermite interpolation technique. Adv. Differ. Equ. 2012. 10.1186/1687-1847-2012-59
Tharwat MM, Yildirim A, Bhrawy AH: Sampling of discontinuous Dirac systems. Numer. Funct. Anal. Optim. 2013, 34: 323-348. 10.1080/01630563.2012.693565
Bhrawy AH, Tharwat MM, Al-Fhaid A: Numerical algorithms for computing eigenvalues of discontinuous Dirac system using sinc-Gaussian method. Abstr. Appl. Anal. 2012. 10.1155/2012/925134
Annaby MH, Tharwat MM: A sinc-Gaussian technique for computing eigenvalues of second-order linear pencils. Appl. Numer. Math. 2013, 63: 129-137.
Tharwat MM, Bhrawy AH, Alofi AS: Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions. Bound. Value Probl. 2013. 10.1186/1687-2770-2013-132
Tharwat MM: On sampling theories and discontinuous Dirac systems with eigenparameter in the boundary conditions. Bound. Value Probl. 2013. 10.1186/1687-2770-2013-65
Levitan BM, Sargsjan IS Translation of Mathematical Monographs 39. In Introduction to Spectral Theory: Self Adjoint Ordinary Differential Operators. Am. Math. Soc., Providence; 1975.
Levitan BM, Sargsjan IS: Sturm-Liouville and Dirac Operators. Kluwer Academic, Dordrecht; 1991.
Tharwat MM: Discontinuous Sturm-Liouville problems and associated sampling theories. Abstr. Appl. Anal. 2011. 10.1155/2011/610232
Acknowledgements
This research was supported by a grant from the Institute of Scientific Research at Umm Al-Qura University, Saudi Arabia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have equal contributions to each part of this article. All the authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tharwat, M.M., Al-Harbi, S.M. Approximation of eigenvalues of boundary value problems. Bound Value Probl 2014, 51 (2014). https://doi.org/10.1186/1687-2770-2014-51
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2014-51