Open Access

On a difference scheme of the second order of accuracy for elliptic-parabolic equations

Boundary Value Problems20122012:80

DOI: 10.1186/1687-2770-2012-80

Received: 13 March 2012

Accepted: 10 July 2012

Published: 27 July 2012

Abstract

The second order of accuracy difference scheme generated by Crank-Nicholson difference scheme for approximately solving multipoint nonlocal boundary value problem is considered. Well-posedness of this difference scheme in Hölder spaces is established. Furthermore, as applications, coercivity estimates in Hölder norms for approximate solutions of the multipoint nonlocal boundary value problems for mixed type equations are obtained. Moreover, the method is illustrated by numerical examples.

Keywords

difference scheme elliptic-parabolic equation well-posedness

1 Introduction

In recent years, more and more mathematicians have been studying nonlocal problems for ordinary differential equations and partial differential equations because of their existence in many applied problems included in applied sciences. Theory and numerical methods of solutions of the nonlocal boundary value problems for these partial differential equations were investigated by many researchers (see, e.g., [113] and the references therein). Several types of problems in fluid mechanics, other areas of physics, and mathematical biology led to partial differential equations of elliptic-parabolic type (see, [1418]). The purpose of this paper is to study the second order of accuracy difference schemes of elliptic-parabolic problem with nonlocal boundary value problems.

In [19], we established the well-posedness of multipoint nonlocal boundary value problem
{ d 2 u ( t ) d t 2 + A u ( t ) = g ( t ) ( 0 t 1 ) , d u ( t ) d t A u ( t ) = f ( t ) ( 1 t 0 ) , u ( 1 ) = i = 1 J α i u ( λ i ) + φ , 1 λ 1 < < λ J 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ1_HTML.gif
(1)
in a Hilbert space H with the self-adjoint positive definite operator A under assumption
i = 1 J | α i | 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ2_HTML.gif
(2)

The well-posedness of multipoint nonlocal boundary value problem (1) in Hölder spaces with a weight was established. In applications, coercivity inequalities for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.

In [20], we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem (1) under assumption (2).

In the present paper, we consider the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme
{ τ 2 ( u k + 1 2 u k + u k 1 ) + A u k = g k , g k = g ( t k ) , t k = k τ , 1 k N 1 , N τ = 1 , τ 1 ( u k u k 1 ) 1 2 ( A u k 1 + A u k ) = f k , f k = f ( t k 1 2 ) , t k 1 2 = ( k 1 2 ) τ , ( N 1 ) k 0 , u 2 4 u 1 + 3 u 0 = 3 u 0 + 4 u 1 u 2 , u N = k = 1 J α i ( u [ λ i τ ] + ( λ i [ λ i τ ] τ ) ( f [ λ i τ ] + A u [ λ i τ ] ) ) + φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ3_HTML.gif
(3)

for the approximate solution of boundary value problem (1) under assumption (2).

The well-posedness of difference scheme (3) in Hölder spaces is established. In applications, the stability, almost coercivity stability, coercivity stability estimates for solutions of the second order of accuracy difference scheme for elliptic-parabolic equations are obtained. Furthermore, the theoretical statements for the solution of the first and second order of accuracy schemes for one-dimensional elliptic-parabolic differential equation are supported by the results of a numerical example.

2 Main theorems

Let us give some auxiliary lemmas we need below. Throughout the paper, H is a Hilbert space and we denote B = 1 2 ( τ A + A ( 4 + τ 2 A ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq1_HTML.gif, where A is a self-adjoint positive definite operator. Then, it is clear that B is a self-adjoint positive definite operator and B δ 1 2 I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq2_HTML.gif, where δ > δ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq3_HTML.gif, and R = ( I + τ B ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq4_HTML.gif which is defined on the whole space H is a bounded operator. The following operators
P = ( I τ A 2 ) G , G = ( I + τ A 2 ) 1 , R = ( I + τ B ) 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equa_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ4_HTML.gif
(4)
exist and are bounded for a self-adjoint positive operator A. Here,
B = 1 2 ( τ A + A ( 4 + τ 2 A ) ) , K = ( I + 2 τ A + 5 4 ( τ A ) 2 ) 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equb_HTML.gif

and I is the identity operator.

Lemma 2.1 For any g k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq5_HTML.gif, 1 k N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq6_HTML.gifand f k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq7_HTML.gif, N + 1 k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq8_HTML.gif, the solution of problem (3) exists and the following formulas hold:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ6_HTML.gif
(6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ7_HTML.gif
(7)
Proof Clearly, the solution formula of the problem
τ 1 ( u k u k 1 ) 1 2 ( A u k 1 + A u k ) = f k , ( N 1 ) k 0 , u 0 = γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ8_HTML.gif
(8)
is [22]:
u k = P k γ τ s = k + 1 0 P s k 1 G f s , N k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ9_HTML.gif
(9)

for any { f k } k = N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq9_HTML.gif and γ. Equation (9) and the fact that u 0 = γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq10_HTML.gif yield Equation (6).

The solution of the problem
{ τ 2 ( u k + 1 2 u k + u k 1 ) + A u k = g k , g k = g ( t k ) , t k = k τ , 1 k N 1 , u 0 = γ , u N = ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ10_HTML.gif
(10)
satisfies the following formula [21]:
u k = ( I R 2 N ) 1 { [ R k R 2 N k ] γ + [ R N k R N + k ] ψ [ R N k R N + k ] ( I + τ B ) ( 2 I + τ B ) 1 B 1 s = 1 N 1 [ R N s R N + s ] g s τ } + ( I + τ B ) ( 2 I + τ B ) 1 B 1 s = 1 N 1 [ R | k s | R k + s ] g s τ , 1 k N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ11_HTML.gif
(11)
Equation (5) follows from Equations (9) and (11), initial condition u 0 = γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq10_HTML.gif, and
ψ = k = 1 J α i ( u [ λ i τ ] + ( λ i [ λ i τ ] τ ) ( f [ λ i τ ] + A u [ λ i τ ] ) ) + φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equc_HTML.gif
Finally, let us obtain formula (7). Combining (5), (6), and the condition
u 2 4 u 1 + 3 u 0 = 3 u 0 + 4 u 1 u 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equd_HTML.gif
we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Eque_HTML.gif
From Equation (4), it follows that
u 0 = 1 2 T τ K G 2 × { ( 2 I τ 2 A ) { ( 2 + τ B ) R N × [ i = 1 n α i ( I + ( λ i [ λ i τ ] τ ) A ) ( τ s = [ λ i τ ] + 1 0 P s [ λ i τ ] G f s ) + ( λ i [ λ i τ ] τ ) f [ λ i τ ] + φ ] R N 1 B 1 s = 1 N 1 [ R N s R N + s ] g s τ + ( I R 2 N ) B 1 s = 1 N 1 R s 1 g s τ } + ( I R 2 N ) ( I + τ B ) ( τ B 1 g 1 4 G B 1 f 0 + P G B 1 f 0 + G B 1 f 1 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equf_HTML.gif

This finishes the proof of Lemma 2.1. □

Here, we study well-posedness of problem (3). First, we give some necessary estimates for P k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq11_HTML.gif, R k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq12_HTML.gif and T τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq13_HTML.gif.

Lemma 2.2 For a self-adjoint positive operator A the following estimates are satisfied[21, 22, 24]:
{ P k H H 1 , k τ A P k G 2 H H M ( δ ) , k τ B R k H H M ( δ ) , R k H H M ( δ ) ( 1 + δ τ ) k , ( I R 2 N ) 1 H H M ( δ ) , G H H 1 , P k e k τ A H H M ( δ ) τ k τ , R k e k τ A 1 2 H H M ( δ ) τ k τ , k 1 , δ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ12_HTML.gif
(12)

where M ( δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq14_HTML.gifis independent of τ.

From these estimates, it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ13_HTML.gif
(13)
Now, we study well-posedness of problem (3). Let F τ ( H ) = F ( [ a , b ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq15_HTML.gif be the linear space of mesh functions φ τ = { φ k } N ˜ N ˜ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq16_HTML.gif defined on [ a , b ] τ = { t k = k h , N ˜ k N ˜ ˜ , N ˜ τ = a , N ˜ ˜ τ = b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq17_HTML.gif with values in the Hilbert space H. Next, on F τ ( H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq18_HTML.gif we denote C ( [ a , b ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq19_HTML.gif, C 0 , 1 α ( [ 1 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq20_HTML.gif, C 0 , 1 α ( [ 1 , 0 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq21_HTML.gif, C 0 α ( [ 0 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq22_HTML.gif, C ˜ 0 , 1 α ( [ 1 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq23_HTML.gif, and C ˜ 0 α ( [ 1 , 0 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq24_HTML.gif, 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq25_HTML.gif Banach spaces with the norms
φ τ C ( [ a , b ] τ , H ) = max N a k N b φ k H , φ τ C 0 , 1 α ( [ 1 , 1 ] τ , H ) = φ τ C ( [ 1 , 1 ] τ , H ) + sup N k < k + r 0 φ k + r φ k E ( k ) α r α φ τ C 0 , 1 α ( [ 1 , 1 ] τ , H ) = + sup 1 k < k + r N 1 φ k + r φ k E ( ( k + r ) τ ) α ( N k ) α r α , φ τ C 0 α ( [ 1 , 0 ] τ , H ) = φ τ C ( [ 1 , 0 ] τ , H ) + sup N k < k + r 0 φ k + r φ k E ( k ) α r α , φ τ C 0 , 1 α ( [ 0 , 1 ] τ , H ) = φ τ C ( [ 0 , 1 ] τ , H ) φ τ C 0 , 1 α ( [ 0 , 1 ] τ , H ) = + sup 1 k < k + r N 1 φ k + r φ k E ( ( k + r ) τ ) α ( N k ) α r α , φ τ C ˜ 0 , 1 α ( [ 1 , 1 ] τ , H ) = φ τ C ( [ 1 , 1 ] τ , H ) + sup N k < k + 2 r 0 φ k + 2 r φ k E ( k ) α ( 2 r ) α φ τ C ˜ 0 , 1 α ( [ 1 , 1 ] τ , H ) = + sup 1 k < k + r N 1 φ k + r φ k E ( ( k + r ) τ ) α ( N k ) α r α , φ τ C ˜ 0 α ( [ 1 , 0 ] τ , H ) = φ τ C ( [ 1 , 0 ] τ , H ) + sup N k < k + 2 r 0 φ k + 2 r φ k E ( k ) α ( 2 r ) α . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equg_HTML.gif

Theorem 2.1 Nonlocal boundary value problem (3) is stable in C ( [ 1 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq26_HTML.gifnorm.

Proof By [21], we have
{ u k } 1 N 1 C ( [ 0 , 1 ] τ , H ) M [ g τ C ( [ 0 , 1 ] τ , H ) + u 0 H + u N H ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ14_HTML.gif
(14)

for the solution of boundary value problem (10).

By [22], we have
{ u k } N 0 C ( [ 1 , 0 ] τ , H ) M [ f τ C ( [ 1 , 0 ] τ , H ) + u 0 H ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ15_HTML.gif
(15)

for the solution of inverse Cauchy difference problem (8).

Then, the proof of Theorem 2.1 is based on stability inequalities (14), (15), and on estimates
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ16_HTML.gif
(16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ17_HTML.gif
(17)

for the solution of boundary value problem (3). Estimates (16) and (17) follow from formula (7) and estimates (12), (13). Theorem 2.1 is proved. □

Theorem 2.2 Assume that φ D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq27_HTML.gifand f 0 , f 1 , g 1 D ( I + τ B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq28_HTML.gif. Then, for the solution of difference problem (3), we have the following almost coercivity inequality:
{ τ 2 ( u k + 1 2 u k + u k 1 ) } 1 N 1 C ( [ 0 , 1 ] τ , H ) + { τ 1 ( u k u k 1 ) } N + 1 0 C ( [ 1 , 0 ] τ , H ) + { A u k } 1 N 1 C ( [ 0 , 1 ] τ , H ) + { 1 2 ( A u k + A u k 1 ) } N + 1 0 C ( [ 1 , 0 ] τ , H ) M ( δ ) [ min { ln 1 τ , 1 + | ln A H H | } [ f τ C ( [ 1 , 0 ] τ , H ) + g τ C ( [ 0 , 1 ] τ , H ) ] + A φ H + ( I + τ B ) f 0 H + ( I + τ B ) g 1 H + ( I + τ B ) f 1 H ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equh_HTML.gif

where M ( δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq14_HTML.gifis independent not only of f τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq29_HTML.gif, g τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq30_HTML.gif, φ but also of τ.

Proof By [24], we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ18_HTML.gif
(18)

for the solution of inverse Cauchy difference problem (8).

By [21], we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ19_HTML.gif
(19)
for the solution of boundary value problem (10). Then, the proof of Theorem 2.2 is based on almost coercivity inequalities (18), (19), and on the estimates
A u 0 H M ( δ ) [ A φ H + ( I + τ B ) f 0 H + min { ln 1 τ , 1 + | ln A H H | } [ f τ C ( [ 1 , 0 ] τ , H ) + g τ C ( [ 0 , 1 ] τ , H ) ] ] , A u N H M ( δ ) [ [ A φ H + ( I + τ B ) f 0 H ] + min { ln 1 τ , 1 + | ln A H H | } [ f τ C ( [ 1 , 0 ] τ , H ) + g τ C ( [ 0 , 1 ] τ , H ) ] ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equi_HTML.gif

for the solution of boundary value problem (3). The proof of these estimates follows the scheme of papers [21, 24] and relies on both formula (7) and estimates (12), (13). This finalizes the proof of Theorem 2.2. □

Theorem 2.3 Let the assumptions of Theorem 2.2 be satisfied. Then, boundary value problem (3) is well-posed in Hölder spaces C 0 , 1 α ( [ 1 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq31_HTML.gif, C ˜ 0 , 1 α ( [ 1 , 1 ] τ , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq23_HTML.gifand the following coercivity inequalities hold:
{ τ 2 ( u k + 1 2 u k + u k 1 ) } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , H ) + { τ 1 ( u k u k 1 ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , H ) + { A u k } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , H ) + { 1 2 ( A u k + A u k 1 ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , H ) M ( δ ) [ 1 α ( 1 α ) [ f τ C 0 α ( [ 1 , 0 ] τ , H ) + g τ C 0 , 1 α ( [ 0 , 1 ] τ , H ) ] + A φ H + ( I + τ B ) f 0 H + ( I + τ B ) g 1 H + ( I + τ B ) f 1 H ] , { τ 2 ( u k + 1 2 u k + u k 1 ) } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , H ) + { τ 1 ( u k u k 1 ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , H ) + { A u k } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , H ) + { 1 2 ( A u k + A u k 1 ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , H ) M ( δ ) [ 1 α ( 1 α ) [ f τ C ˜ 0 α ( [ 1 , 0 ] τ , H ) + g τ C 0 , 1 α ( [ 0 , 1 ] τ , H ) ] + A φ H + ( I + τ B ) f 0 H + ( I + τ B ) g 1 H + ( I + τ B ) f 1 H ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equj_HTML.gif

where M is independent of f τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq29_HTML.gif, g τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq32_HTML.gif, φ, τ and α.

Proof By [24],
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ20_HTML.gif
(20)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ21_HTML.gif
(21)

for the solution of inverse Cauchy difference problem (8).

By [21], we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ22_HTML.gif
(22)
for the solution of boundary value problem (10). Then, the proof of Theorem 2.3 is based on coercivity inequalities (20)-(22) and estimates
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equk_HTML.gif

for the solution of difference scheme (3). The proof of these estimates follows the scheme of the papers [21, 24] and relies on both formula (7) and estimates (12), (13). This is the end of the proof of Theorem 2.3. □

3 Application

Now, the application of the abstract result is considered. In [ 1 , 1 ] × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq33_HTML.gif, let us consider the boundary value problem for multi-dimensional elliptic-parabolic equation
{ u t t r = 1 n ( a r ( x ) u x r ) x r = g ( t , x ) , 0 < t < 1 , x Ω , u t + r = 1 n ( a r ( x ) u x r ) x r = f ( t , x ) , 1 < t < 0 , x Ω , u ( t , x ) = 0 , x S , 1 t 1 ; u ( 1 , x ) = i = 1 J α i u ( λ i , x ) + φ ( x ) , i = 1 J | α i | 1 , 1 λ 1 < λ 2 < < λ i < < λ J 0 , u ( 0 + , x ) = u ( 0 , x ) , u t ( 0 + , x ) = u t ( 0 , x ) , x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ23_HTML.gif
(23)

where a r ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq34_HTML.gif ( x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq35_HTML.gif), φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq36_HTML.gif ( φ ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq37_HTML.gif, x S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq38_HTML.gif), g ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq39_HTML.gif ( t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq40_HTML.gif, x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq41_HTML.gif), and f ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq42_HTML.gif ( t ( 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq43_HTML.gif, x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq41_HTML.gif) are given smooth functions. Here, Ω is the unit open cube in the n-dimensional Euclidean space R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq44_HTML.gif ( 0 < x k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq45_HTML.gif, 1 k n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq46_HTML.gif) with boundary S, Ω ¯ = Ω S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq47_HTML.gif, and a r ( x ) a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq48_HTML.gif.

The discretization of problem (23) is carried out in two steps. In the first step, the grid sets
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equl_HTML.gif
are defined. To the differential operator A generated by problem (23), we assign the difference operator A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq49_HTML.gif by formula
A h x u h = r = 1 n ( a r ( x ) u x ¯ r h ) x r , m r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ24_HTML.gif
(24)
acting in the space of grid functions u h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq50_HTML.gif, satisfying the conditions u h ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq51_HTML.gif for all x S h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq52_HTML.gif. With the help of A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq49_HTML.gif, we arrive at the nonlocal boundary value problem
{ d 2 u h ( t , x ) d t 2 + A h x u h ( t , x ) = g h ( t , x ) , 0 < t < 1 , x Ω h , d u h ( t , x ) d t A h x u h ( t , x ) = f h ( t , x ) , 1 < t < 0 , x Ω h , u h ( 1 , x ) = u h ( 1 , x ) + φ h ( x ) , x Ω ˜ h , u h ( 0 + , x ) = u h ( 0 , x ) , d u h ( 0 + , x ) d t = d u h ( 0 , x ) d t , x Ω ˜ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ25_HTML.gif
(25)

for an infinite system of ordinary differential equations (see [21]).

Secondly, problem (25) is replaced by difference scheme (3), so that the following second order of accuracy difference scheme
{ u k + 1 h ( x ) 2 u k h ( x ) + u k 1 h ( x ) τ 2 + A h x u k h ( x ) = g k h ( x ) , g k h ( x ) = g h ( t k , x ) , t k = k τ , 1 k N 1 , N τ = 1 , x Ω h , u k h ( x ) u k 1 h ( x ) τ A h x 2 ( u k h ( x ) + u k 1 h ( x ) ) = f k h ( x ) , f k h ( x ) = f h ( t k 1 2 , x ) , t k 1 2 = ( k 1 2 ) τ , N + 1 k 0 , x Ω h , u 2 h ( x ) + 4 u 1 h ( x ) 3 u 0 h ( x ) = 3 u 0 h ( x ) 4 u 1 h ( x ) + u 2 h ( x ) , x Ω ˜ h , u N h ( x ) = k = 1 J α i ( u h [ λ i τ ] ( x ) + ( λ k [ λ i τ ] τ ) ( f h [ λ i τ ] + A h x u h [ λ i τ ] ( x ) ) ) + φ h ( x ) , x Ω ˜ h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equ26_HTML.gif
(26)

is obtained (see [21], [22]).

To formulate the results, we introduce the spaces L 2 h = L 2 ( Ω ¯ h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq53_HTML.gif, W 2 h 1 = W 2 1 ( Ω ¯ h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq54_HTML.gif, and W 2 h 2 = W 2 2 ( Ω ¯ h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq55_HTML.gif of the grid functions φ h ( x ) = { φ ( h 1 m 1 , , h n m n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq56_HTML.gif defined on Ω ¯ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq57_HTML.gif, equipped with the norms
φ h L 2 h = ( x Ω ¯ h | φ h ( x ) | 2 h 1 h n ) 1 / 2 , φ h W 2 h 1 = φ h L 2 h + ( x Ω ¯ h r = 1 n | ( φ h ) x r | 2 h 1 h n ) 1 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equm_HTML.gif
and
φ h W 2 h 2 = φ h L 2 h + ( x Ω ¯ h r = 1 n | ( φ h ) x r | 2 h 1 h n ) 1 / 2 + ( x Ω ¯ h r = 1 n | ( φ h ) x r x ¯ r , m r | 2 h 1 h n ) 1 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equn_HTML.gif
Theorem 3.1 Let τ and | h | = h 1 2 + + h n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq58_HTML.gifbe sufficiently small positive numbers. Then, solutions of difference scheme (26) satisfy the following stability and almost coercivity estimates:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equo_HTML.gif

where M is independent not only of τ, h, φ h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq59_HTML.gifbut also of f k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq60_HTML.gif, N + 1 k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq8_HTML.gifand g k h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq61_HTML.gif, 1 k N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq6_HTML.gif.

The proof of Theorem 3.1 is based on Theorem 2.1, Theorem 2.2, the symmetry properties of the difference operator A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq49_HTML.gif defined by formula (24) in L 2 h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq62_HTML.gif, the estimate
min { ln 1 τ , 1 + | ln A h x L 2 h L 2 h | } M ln 1 τ + | h | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equp_HTML.gif

and the following theorem in L 2 h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq62_HTML.gif:

Theorem 3.2 For the solution of the elliptic difference problem
A h x u h ( x ) = ω h ( x ) , x Ω h , u h ( x ) = 0 , x S h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equq_HTML.gif
the following coercivity inequality holds[23]:
r = 1 n ( u h ) x ¯ r x r , m r L 2 h M ω h L 2 h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equr_HTML.gif

Here M is independent of h and ω h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq63_HTML.gif.

Theorem 3.3 Let τ and | h | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq64_HTML.gifbe sufficiently small positive numbers. Then, the solutions of difference scheme (26) satisfy the following coercivity stability estimates:
{ τ 2 ( u k + 1 h 2 u k h + u k 1 h ) } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) + { τ 1 ( u k h u k 1 h ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , L 2 h ) + { u k h } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , W 2 h 2 ) + { u k h + u k 1 h 2 } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , W 2 h 2 ) M [ φ h W 2 h 2 + τ f 0 h W 2 h 1 + τ f 1 h W 2 h 1 + τ g 1 h W 2 h 1 + 1 α ( 1 α ) [ { f k h } N + 1 1 C 0 α ( [ 1 , 0 ] τ , L 2 h ) + { g k h } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) ] ] , { τ 2 ( u k + 1 h 2 u k h + u k 1 h ) } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) + { u k h + u k 1 h 2 } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , W 2 h 2 ) + { τ 1 ( u k h u k 1 h ) } N + 1 0 C ˜ 0 α ( [ 1 , 0 ] τ , L 2 h ) + { u k h } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , W 2 h 2 ) M [ φ h W 2 h 2 + τ f 0 h W 2 h 1 + τ f 1 h W 2 h 1 + τ g 1 h W 2 h 1 + 1 α ( 1 α ) [ { f k h } N + 1 1 C ˜ 0 α ( [ 1 , 0 ] τ , L 2 h ) + { g k h } 1 N 1 C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) ] ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equs_HTML.gif

Here, M is independent not only of τ, h, φ h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq59_HTML.gifbut also of f k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq60_HTML.gif, N + 1 k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq8_HTML.gifand g k h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq61_HTML.gif, 1 k N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq6_HTML.gif.

The proof of Theorem 3.3 is based on the abstract Theorem 2.3, Theorem 3.2, and the symmetry properties of the difference operator A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq49_HTML.gif defined by formula (24).

4 Numerical Analysis

The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments of the nonlocal boundary value problem
{ u t + u x ( ( 1 + x ) u x ) = f ( t , x ) , f ( t , x ) = ( 2 e t + 1 t ) sin x + ( e t + t ) ( cos x x sin x ) , 1 < t 0 , 0 < x < π , 2 u t 2 + u x ( ( 1 + x ) u x ) = g ( t , x ) , g ( t , x ) = t sin x + ( e t + t ) ( cos x x sin x ) , 0 < t < 1 , 0 < x < π , u ( 1 , x ) = 1 2 u ( 1 , x ) + 1 2 u ( 1 2 , x ) + φ ( x ) , φ ( x ) = ( e 1 e 2 1 2 e 1 2 + 7 4 ) sin x , 0 x π , u ( t , 0 ) = u ( t , π ) = 0 , 1 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equt_HTML.gif
for the elliptic-parabolic equation. The exact solution of this problem is
u ( t , x ) = ( e t + t ) sin x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equu_HTML.gif
For the comparison, the errors computed by the following formula
E M N = max N k N 1 n M 1 | u ( t k , x n ) u n k | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_Equv_HTML.gif
are recorded for different values of N and M, where u ( t k , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq65_HTML.gif represents the exact solution and u n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq66_HTML.gif represents the numerical solution at ( t k , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq67_HTML.gif. The results are shown in Table 1 for N = M = 30 , 60 and 90 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq68_HTML.gif respectively.
Table 1

Error analysis for the solution u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-80/MediaObjects/13661_2012_Article_177_IEq69_HTML.gif

Method

N = M = 30

N = M = 60

N = M = 90

1st order of accuracy d. s.

0.042169

0.021639

0.014546

2nd order of accuracy d. s.

0.000908

0.000227

0.000101

Therefore, the results indicate that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.

Declarations

Acknowledgement

The authors are very grateful to Prof. P. E. Sobolevskii (Jerusalem, Israel) for valuable comments to the improvement of this article.

Authors’ Affiliations

(1)
Department of Mathematics, Fatih University
(2)
Department of Mathematics, ITTU

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.