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Volume 2012 |Article ID 276080 | https://doi.org/10.1155/2012/276080

Abdullah Said Erdogan, Hulya Uygun, "A Note on the Inverse Problem for a Fractional Parabolic Equation", Abstract and Applied Analysis, vol. 2012, Article ID 276080, 26 pages, 2012. https://doi.org/10.1155/2012/276080

A Note on the Inverse Problem for a Fractional Parabolic Equation

Academic Editor: Ravshan Ashurov
Received15 May 2012
Accepted08 Jul 2012
Published10 Sep 2012

Abstract

For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.

1. Introduction

Inverse problems arise in many fields of science and engineering such as ion transport problems, chromatography, and heat determination problems with an unknown internal energy source. Different typed of inverse problems have been investigated, and the main results obtained in this field of research were given by many researchers (see [1โ€“10]). More than three centuries the theory of fractional derivatives developed mainly as a pure theoretical field of mathematics. Fractional integrals and derivatives appear in the theory of control of dynamical systems, when the controlled system or/and the controller is described by a fractional differential equation (see [11]). Recently, many application areas such as bioengineering applications, image and signal processing are also related to fractional calculus. Methods of solutions of problems and theory of fractional calculus have been studied by many researchers [11โ€“28]. Among them finite difference method is used for solving several fractional differential equations (see [20, 22, 23, 27] and the references therein).

1.1. Statement of the Problem

Many scientists and researchers are trying to enhance mathematical models of real-life cases for investigating and understanding the behavior of them. Therefore, some phenomena have been modeled and investigated as fractional inverse problems (see [29โ€“33] and the references therein). In this paper, we consider the fractional parabolic inverse problem with the Dirichlet condition ๐œ•๐‘ข(๐‘ก,๐‘ฅ)๐œ•๐œ•๐‘กโˆ’๐‘Ž2๐‘ข(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ2โˆ’๐ท๐‘ก1/2๐‘ข๎€ท๐‘ข(๐‘ก,๐‘ฅ)+๐œŽ๐‘ข(๐‘ก,๐‘ฅ)=๐‘(๐‘ก)๐‘ž(๐‘ฅ)+๐‘“(๐‘ก,๐‘ฅ),0<๐‘ฅ<๐œ‹,0<๐‘กโ‰ค๐‘‡,๐‘ข(๐‘ก,0)=๐‘ข(๐‘ก,๐œ‹)=0,0โ‰ค๐‘กโ‰ค๐‘‡,๐‘ข(0,๐‘ฅ)=๐œ‘(๐‘ฅ),0โ‰ค๐‘ฅโ‰ค๐œ‹,๐‘ก,๐‘ฅโˆ—๎€ธ=๐œŒ(๐‘ก),0<๐‘ฅโˆ—<๐œ‹,(1.1) where ๐‘ข(๐‘ก,๐‘ฅ) and ๐‘(๐‘ก) are unknown functions, ๐‘Ž(๐‘ฅ)โ‰ฅ๐‘Ž>0, and ๐œŽ>0 is a sufficiently large number. Here, ๐ท๐‘ก1/2=๐ท1/20+ is the standard Riemann-Liouvilleโ€™s derivative of order 1/2.

Theorems on the stability of problem (1.1) are analyzed by assuming that ๐‘ž(๐‘ฅ) is a sufficiently smooth function, ๐‘ž(0)=๐‘ž(๐œ‹)=0 and ๐‘ž(๐‘ฅโˆ—)โ‰ 0.

2. Main Results

In this section, stability estimates for the solution of (1.1) are investigated. For the mathematical substantiation, we introduce the Banach space โˆ˜๐ถ๐›ผ[0,๐œ‹],๐›ผโˆˆ(0,1), of all continuous functions ๐œ™(๐‘ฅ) defined on [0,๐œ‹] with ๐œ™(0)=๐œ™(๐œ‹)=0 satisfying a Hรถlder condition for which the following norm is finite โ€–๐œ™โ€–โˆ˜๐ถ๐›ผ[0,๐œ‹]=โ€–๐œ™โ€–๐ถ[0,๐œ‹]+sup0<๐‘ฅ<๐‘ฅ+โ„Ž<๐œ‹||||๐œ™(๐‘ฅ+โ„Ž)โˆ’๐œ™(๐‘ฅ)โ„Ž๐›ผ,(2.1) where ๐ถ[0,๐œ‹] is the space of all continuous function ๐œ™(๐‘ฅ) defined on [0,๐œ‹] with the norm โ€–๐œ™โ€–๐ถ[0,๐œ‹]=max0โ‰ค๐‘ฅโ‰ค๐œ‹||||.๐œ™(๐‘ฅ)(2.2) With the help of a positive operator ๐ด, we introduce the fractional spaces ๐ธ๐›ผ,0<๐›ผ<1, consisting of all ๐‘ฃ in a Banach space ๐ธ for which the following norm is finite: โ€–๐‘ฃโ€–๐ธ๐›ผ=โ€–๐‘ฃโ€–๐ธ+sup๐œ†>0๐œ†1โˆ’๐›ผโ€–โ€–๐ดexp{โˆ’๐œ†๐ด}๐‘ฃ๐ธ.(2.3) Throughout the paper, positive constants will be indicated by ๐‘€๐‘–(๐›ผ,๐›ฝ,โ€ฆ). Here variables are used to focus on the fact that the constant depends only on ๐›ผ,๐›ฝ,โ€ฆ and the subindex ๐‘– is used to indicate a different constant.

Theorem 2.1. Let ๐œ‘โˆˆโˆ˜๐ถ2๐›ผ+2[0,๐œ‹],๐นโˆˆ๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ[0,๐œ‹]), and ๐œŒ๎…žโˆˆ๐ถ[0,๐‘‡]. Then for the solution of problem (1.1), the following coercive stability estimates โ€–โ€–๐‘ข๐‘กโ€–โ€–๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ[0,๐œ‹])+โ€–๐‘ขโ€–๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ+2[0,๐œ‹])๎€ท๐‘ฅโ‰ค๐‘€โˆ—๎€ธโ€–โ€–๐œŒ,๐‘ž๎…žโ€–โ€–๐ถ[0,๐‘‡]๎€ท+๐‘€๐‘Ž,๐›ฟ,๐œŽ,๐›ผ,๐‘ฅโˆ—๎€ธร—๎‚ต,๐‘ž,๐‘‡โ€–๐œ‘โ€–โˆ˜๐ถ2๐›ผ+2[0,๐œ‹]+โ€–๐นโ€–๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ[0,๐œ‹])+โ€–๐œŒโ€–๐ถ[0,๐‘‡]๎‚ถ,โ€–๐‘โ€–๐ถ[0,๐‘‡]๎€ท๐‘ฅโ‰ค๐‘€โˆ—๎€ธโ€–โ€–๐œŒ,๐‘ž๎…žโ€–โ€–๐ถ[0,๐‘‡]๎€ท+๐‘€๐‘Ž,๐›ฟ,๐œŽ,๐›ผ,๐‘ฅโˆ—๎€ธร—๎‚ธ,๐‘ž,๐‘‡โ€–๐œ‘โ€–โˆ˜๐ถ2๐›ผ+2[0,๐œ‹]+โ€–๐นโ€–๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ[0,๐œ‹])+โ€–๐œŒโ€–๐ถ[0,๐‘‡]๎‚น(2.4) hold.

Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [8]): ๐‘ข(๐‘ก,๐‘ฅ)=๐œ‚(๐‘ก)๐‘ž(๐‘ฅ)+๐‘ค(๐‘ก,๐‘ฅ),(2.5) where ๎€œ๐œ‚(๐‘ก)=๐‘ก0๐‘(๐‘ )๐‘‘๐‘ .(2.6) Using the overdetermined condition, we get ๐œŒ๎€ท๐œ‚(๐‘ก)=(๐‘ก)โˆ’๐‘ค๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—),๐œŒ(2.7)๐‘(๐‘ก)=๎…ž(๐‘ก)โˆ’๐‘ค๐‘ก๎€ท๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—).(2.8) Using identity (2.8) and the triangle inequality, it follows that ||||=||||๐œŒ๐‘(๐‘ก)๎…ž(๐‘ก)โˆ’๐‘ค๐‘ก๎€ท๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—)||||๎€ท๐‘ฅโ‰ค๐‘€โˆ—||๐œŒ,๐‘ž๎€ธ๎€ท๎…ž||+||๐‘ค(๐‘ก)๐‘ก๎€ท๐‘ก,๐‘ฅโˆ—๎€ธ||๎€ธ๎€ท๐‘ฅโ‰ค๐‘€โˆ—๎€ธ๎‚ต,๐‘žmax0โ‰ค๐‘กโ‰ค๐‘‡||๐œŒ๎…ž||(๐‘ก)+max0โ‰ค๐‘กโ‰ค๐‘‡max0โ‰ค๐‘ฅโ‰ค๐œ‹||๐‘ค๐‘ก||๎‚ถ๎€ท๐‘ฅ(๐‘ก,๐‘ฅ)โ‰ค๐‘€โˆ—๎€ธ๎‚ต,๐‘žmax0โ‰ค๐‘กโ‰ค๐‘‡||๐œŒ๎…ž||(๐‘ก)+max0โ‰ค๐‘กโ‰ค๐‘‡โ€–โ€–๐‘ค๐‘กโ€–โ€–(๐‘ก,โ‹…)โˆ˜๐ถ2๐›ผ[0,๐œ‹]๎‚ถ(2.9) for any ๐‘ก,๐‘กโˆˆ[0,๐‘‡]. Here, ๐‘ค(๐‘ก,๐‘ฅ) is the solution of the following problem: ๐œ•๐‘ค(๐‘ก,๐‘ฅ)๐œ•๐œ•๐‘กโˆ’๐‘Ž2๐‘ค(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ2๐œŒ๎€ทโˆ’๐‘Ž(๐‘ก)โˆ’๐‘ค๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—)๐‘‘2๐‘ž(๐‘ฅ)๐‘‘๐‘ฅ2โˆ’๐ท๐‘ก1/2โˆ’๐ท๐‘ค(๐‘ก,๐‘ฅ)๐‘ก1/2๐œŒ(๐‘ก)โˆ’๐ท๐‘ก1/2๐‘ค๎€ท๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—)๎€ท๐‘ž(๐‘ฅ)+๐œŽ๐œŒ(๐‘ก)โˆ’๐‘ค๐‘ก,๐‘ฅโˆ—๎€ธ๐‘ž(๐‘ฅโˆ—)๐‘ž(๐‘ฅ)+๐œŽ๐‘ค(๐‘ก,๐‘ฅ)=๐‘“(๐‘ก,๐‘ฅ),0<๐‘ฅ<๐œ‹,0<๐‘กโ‰ค๐‘‡,๐‘ค(๐‘ก,0)=๐‘ค(๐‘ก,๐œ‹)=0,0โ‰ค๐‘กโ‰ค๐‘‡,๐‘ค(0,๐‘ฅ)=๐œ‘(๐‘ฅ),0โ‰ค๐‘ฅโ‰ค๐œ‹.(2.10) For simplicity, we assign ๐น(๐‘ก,๐‘ฅ)=๐‘Ž๐œŒ(๐‘ก)๐‘ž(๐‘ฅโˆ—)๐‘‘2๐‘ž(๐‘ฅ)๐‘‘๐‘ฅ2โˆ’๐œŽ๐œŒ(๐‘ก)๐‘ž(๐‘ฅโˆ—)๐ท๐‘ž(๐‘ฅ)+๐‘ก1/2๐œŒ(๐‘ก)๐‘ž(๐‘ฅโˆ—)๐‘ž(๐‘ฅ)+๐‘“(๐‘ก,๐‘ฅ),๐บ(๐‘ก,๐‘ฅ)=๐‘„1๎€ท๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—๎€ธ๐‘ค๎€ท,๐‘ก๐‘ก,๐‘ฅโˆ—๎€ธ+๐‘„2๎€ท๐‘ž,๐‘ฅ,๐‘ฅโˆ—๎€ธ๐ท๐‘ก1/2๐‘ค๎€ท๐‘ก,๐‘ฅโˆ—๎€ธ+๐ท๐‘ก1/2๐‘ค(๐‘ก,๐‘ฅ),(2.11) where ๐‘„1๎€ท๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—๎€ธ=1,๐‘ก๐‘ž(๐‘ฅโˆ—)๎‚ต๐‘‘โˆ’๐‘Ž2๐‘ž(๐‘ฅ)๐‘‘๐‘ฅ2๎‚ถ,๐‘„+๐œŽ๐œŒ(๐‘ก)2๎€ท๐‘ž,๐‘ฅ,๐‘ฅโˆ—๎€ธ๐‘ž=โˆ’(๐‘ฅ)๐‘ž(๐‘ฅโˆ—).(2.12) Note that functions ๐น(๐‘ก,๐‘ฅ),๐‘„1(๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—,๐‘ก) and ๐‘„2(๐‘ž,๐‘ฅ,๐‘ฅโˆ—) only contain given functions. Then, we can rewrite problem (2.10) as ๐œ•๐‘ค(๐‘ก,๐‘ฅ)๐œ•๐œ•๐‘กโˆ’๐‘Ž2๐‘ค(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ2+๐œŽ๐‘ค(๐‘ก,๐‘ฅ)=๐น(๐‘ก,๐‘ฅ)+๐บ(๐‘ก,๐‘ฅ),0<๐‘ฅ<๐œ‹,0<๐‘กโ‰ค๐‘‡,๐‘ค(๐‘ก,0)=๐‘ค(๐‘ก,๐œ‹)=0,0โ‰ค๐‘กโ‰ค๐‘‡,๐‘ค(0,๐‘ฅ)=๐œ‘(๐‘ฅ),0โ‰ค๐‘ฅโ‰ค๐œ‹.(2.13) So, the end of proof of Theorem 2.1 is based on estimate (2.9) and the following theorem.

Theorem 2.2. For the solution of problem (2.10), the following coercive stability estimate โ€–โ€–๐‘ค๐‘กโ€–โ€–โˆ˜๐ถ2๐›ผ[0,๐œ‹]๎€ทโ‰ค๐‘€๐‘Ž,๐›ฟ,๐œŽ,๐›ผ,๐‘ฅโˆ—๎€ธร—๎‚ต,๐‘ž,๐‘‡โ€–๐œ‘โ€–โˆ˜๐ถ2๐›ผ+2[0,๐œ‹]+โ€–๐นโ€–๐ถ([0,๐‘‡],โˆ˜๐ถ2๐›ผ[0,๐œ‹])+โ€–๐œŒโ€–๐ถ[0,๐‘‡]๎‚ถ(2.14) holds.

Proof. In a Banach space ๐ธ=โˆ˜๐ถ[0,๐œ‹], with the help of the positive operator ๐ด defined by ๐œ•๐ด๐‘ข=โˆ’๐‘Ž(๐‘ฅ)2๐‘ข(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ2+๐œŽ๐‘ข,(2.15) with ๐ท๎€ฝ๐‘ข(๐ด)=(๐‘ฅ)โˆถ๐‘ข,๐‘ข๎…ž,๐‘ข๎…ž๎…ž[]๎€พโˆˆ๐ถ0,๐œ‹,๐‘ข(0)=๐‘ข(๐œ‹)=0,(2.16) where ๐œŽ is a positive constant, problem (2.10) can be written in the abstract form as an initial-value problem ๐‘ค๐‘ก๐‘ค+๐ด๐‘ค=๐น(๐‘ก)+๐บ(๐‘ก),0<๐‘กโ‰ค๐‘‡,(0)=๐œ‘.(2.17) By the Cauchy formula, the solution can be written as ๐‘ค(๐‘ก)=๐‘’โˆ’๐‘ก๐ด๎€œ๐œ‘โˆ’๐‘ก0๐‘’โˆ’(๐‘กโˆ’๐‘ )๐ด(๐น(๐‘ )+๐บ(๐‘ ))๐‘‘๐‘ .(2.18) Applying the formula ๐ท๐‘ก1/2๎€œ๐‘ข(๐‘ก)=๐‘ก0๐‘ข๎…ž(๐œ)๐‘‘๐œ‰โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2,(2.19) we get the following presentation of the solution of abstract problem (2.17): ๐ท1/2๎€œ๐‘ค(๐‘ก)=โˆ’๐‘ก0๐ด๐‘’โˆ’๐œ‰๐ด๐œ‘โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๎€œ๐‘‘๐œ‰โˆ’๐‘ก0๐น(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โˆ’๎€œ๐‘‘๐œ‰๐‘ก0๐บ(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๎€œ๐‘‘๐œ‰+๐‘ก0๎€œ๐œ‰0๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ด๐น(๐‘ )โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2+๎€œ๐‘‘๐‘ ๐‘‘๐œ‰๐‘ก0๎€œ๐œ‰0๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ด๐บ(๐‘ )โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐‘‘๐‘ ๐‘‘๐œ‰.(2.20) Changing the order of integration, we obtain that ๐ท1/2๎€œ๐‘ค(๐‘ก)=โˆ’๐‘ก0๐ด๐‘’โˆ’๐œ‰๐ด๐œ‘โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๎€œ๐‘‘๐œ‰โˆ’๐‘ก0๐น(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2+๎€œ๐‘‘๐œ‰๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๎€œ๐‘‘๐œ‰๐น(๐‘ )๐‘‘๐‘ โˆ’๐‘ก0๐บ(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2+๎€œ๐‘‘๐œ‰๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐‘‘๐œ‰๐บ(๐‘ )๐‘‘๐‘ =5๎“๐‘˜=1๐ฝ๐‘˜,(2.21) where ๐ฝ1๎€œ(๐‘ก)=โˆ’๐‘ก0๐ด๐‘’โˆ’๐œ‰๐ด๐œ‘โˆš๐œ‹(๐‘กโˆ’๐‘)1/2๐ฝ๐‘‘๐œ‰,2๎€œ(๐‘ก)=โˆ’๐‘ก0๐น(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐ฝ๐‘‘๐œ‰,3๎€œ(๐‘ก)=๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐ฝ๐‘‘๐œ‰๐น(๐‘ )๐‘‘๐‘ ,4๎€œ(๐‘ก)=โˆ’๐‘ก0๐บ(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐ฝ๐‘‘๐œ‰,5๎€œ(๐‘ก)=๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐‘‘๐œ‰๐บ(๐‘ )๐‘‘๐‘ .(2.22) Now, we estimate ๐ฝ๐‘˜(๐‘ก),๐‘˜=1,2,3,4,5 separately. It is known that [13] โ€–โ€–๐ด๐›ผ๐‘’โˆ’๐‘ก๐ดโ€–โ€–๐ธโ†’๐ธโ‰ค๐‘€,0โ‰ค๐›ผโ‰ค1.(2.23) Since operators ๐ด and exp(โˆ’๐‘ก๐ด) commute, โ€–โ€–๐ด๐‘’โˆ’๐‘ก๐ด๐œ‘โ€–โ€–๐ธ๐›ผโ‰คโ€–โ€–๐‘’โˆ’๐‘ก๐ดโ€–โ€–๐ธ๐›ผโ†’๐ธ๐›ผโ€–๐ด๐œ‘โ€–๐ธ๐›ผโ‰คโ€–โ€–๐‘’โˆ’๐‘ก๐ดโ€–โ€–๐ธโ†’๐ธโ€–๐ด๐œ‘โ€–๐ธ๐›ผ.(2.24) Applying the definition of norm of the spaces ๐ธ๐›ผ and (2.23) and (2.24), we get โ€–โ€–๐ฝ1โ€–โ€–(๐‘ก)๐ธ๐›ผ=โ€–โ€–โ€–โ€–๎€œ๐‘ก0๐ด๐‘’โˆ’๐œ‰๐ด๐œ‘โˆš๐œ‹(๐‘กโˆ’๐‘)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐ธ๐›ผโ‰ค๐‘€1โ€–๐ด๐œ‘โ€–๐ธ๐›ผ(2.25) for any ๐‘ก,๐‘กโˆˆ[0,๐‘‡]. Estimation of ๐ฝ2(๐‘ก) is as follows: โ€–โ€–๐ฝ2โ€–โ€–(๐‘ก)๐ธ๐›ผ=โ€–โ€–โ€–โ€–๎€œ๐‘ก0๐น(๐œ‰)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐ธ๐›ผโ‰คโ€–๐น(๐‘ก)โ€–๐ถ(๐ธ๐›ผ)๎€œ๐‘ก01โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐‘‘๐œ‰โ‰ค๐‘€2โ€–๐นโ€–๐ถ(๐ธ๐›ผ).(2.26) Let us estimate ๐ฝ3(๐‘ก): โ€–โ€–๐ฝ3โ€–โ€–(๐‘ก)๐ธ๐›ผ=โ€–โ€–โ€–โ€–๎€œ๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐น(๐‘ )๐‘‘๐‘ ๐ธ๐›ผโ‰คโ€–โ€–โ€–โ€–๎€œ๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐‘‘๐‘ ๐ธ๐›ผโ†’๐ธ๐›ผโ€–๐นโ€–๐ถ(๐ธ๐›ผ).(2.27) It is proven that (see [28]) โ€–โ€–โ€–โ€–๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐ธโ†’๐ธโ‰ค๐‘€โˆš.๐‘กโˆ’๐‘ (2.28) Using the definition of norm of the spaces ๐ธ๐›ผ, we can obtain that โ€–โ€–โ€–โ€–๎€œ๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐‘‘๐‘ ๐ธ๐›ผโ†’๐ธ๐›ผ=๎€œ๐‘ก0โ€–โ€–โ€–โ€–๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐‘กโˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐ธโ†’๐ธ๐‘‘๐‘ +sup๐œ†>0๎€œ๐‘ก0โ€–โ€–โ€–โ€–๎€œ๐‘ก๐‘ ๐œ†1โˆ’๐›ผ๐ด๐‘’โˆ’๐œ†๐ด๐ด๐‘’โˆ’(๐‘กโˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐ธโ†’๐ธ๐‘‘๐‘ .(2.29) Using estimates (2.23) and (2.28), we get โ€–โ€–๐ฝ3โ€–โ€–(๐‘ก)๐ธ๐›ผโ‰คโ€–โ€–โ€–โ€–๎€œ๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐‘‘๐œ‰๐‘‘๐‘ ๐ธ๐›ผโ†’๐ธ๐›ผโ€–๐นโ€–๐ถ(๐ธ๐›ผ)โ‰ค๐‘€3โ€–๐นโ€–๐ถ(๐ธ๐›ผ).(2.30) Expanding ๐บ(๐‘ ), estimation of ๐ฝ4(๐‘ก) is as follows: โ€–โ€–๐ฝ4โ€–โ€–(๐‘ก)๐ธ๐›ผโ‰ค๎€œ๐‘ก0โ€–โ€–โ€–โ€–๐‘„1๎€ท๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—๎€ธ๐‘ค๎€ท,๐‘ก๐œ‰,๐‘ฅโˆ—๎€ธโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐ธ๐›ผ+๎€œ๐‘‘๐œ‰๐‘ก0โ€–โ€–โ€–โ€–๐‘„2๎€ท๐‘ž,๐‘ฅ,๐‘ฅโˆ—๎€ธ๐ท๐‘ก1/2๐‘ค๎€ท๐œ‰,๐‘ฅโˆ—๎€ธโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐ธ๐›ผ๎€œ๐‘‘๐œ‰+๐‘ก0โ€–โ€–โ€–โ€–๐ท๐‘ก1/2๐‘ค(๐œ‰,๐‘ฅ)โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–โ€–โ€–๐ธ๐›ผ๐‘‘๐œ‰.(2.31) It is known that (see [34]) โ€–๐‘คโ€–๐ธ๐›ผโ€–โ€–๐ทโ‰ค๐‘€๐‘ก1/2๐‘คโ€–โ€–๐ธ๐›ผ.(2.32) Since ๐‘„1(๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—,๐‘ก) and ๐‘„2(๐‘ž,๐‘ฅ,๐‘ฅโˆ—) are known functions, it is easy to see that โ€–โ€–๐ฝ4(โ€–โ€–๐‘ก)๐ธ๐›ผโ‰ค๐‘€4๎€ท๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—๎€ธ๎€œ,๐‘‡๐‘ก01โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2โ€–โ€–๐ท๐‘ก1/2โ€–โ€–๐‘ค(๐œ‰)๐ธ๐›ผ๐‘‘๐œ‰.(2.33) Estimation of ๐ฝ5(๐‘ก) can be given similar to the estimation of ๐ฝ4(๐‘ก). By (2.23) and (2.32), โ€–โ€–๐ฝ5(โ€–โ€–๐‘ก)๐ธ๐›ผโ‰ค๎€œ๐‘ก0๎€œ๐‘ก๐‘ ๐ด๐‘’โˆ’(๐œ‰โˆ’๐‘ )๐ดโˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2๐‘‘๐œ‰๐บ(๐‘ )๐‘‘๐‘ โ‰ค๐‘€5๎€ท๐‘ž,๐œŒ,๐‘ฅ,๐‘ฅโˆ—๎€ธ๎€œ,๐‘‡๐‘ก0โ€–โ€–๐ท๐‘ก1/2โ€–โ€–๐‘ค(๐‘ )๐ธ๐›ผ๐‘‘๐‘ .(2.34) Finally combining estimates (2.25), (2.26), (2.30), (2.33), and (2.34), we get โ€–โ€–๐ท๐‘ก1/2๐‘คโ€–โ€–๐ธ๐›ผโ‰ค๐‘€1โ€–๐ด๐œ‘โ€–๐ธ๐›ผ+๎€ท๐‘€2+๐‘€3๎€ธโ€–๐นโ€–๐ถ(๐ธ๐›ผ)+๎€œ๐‘ก0๎ƒฉ๐‘€4โˆš๐œ‹(๐‘กโˆ’๐œ‰)1/2+๐‘€5๎ƒชโ€–โ€–๐ท๐‘ก1/2๐‘คโ€–โ€–(๐‘ )๐ธ๐›ผ๐‘‘๐‘ .(2.35) Using the Gronwallโ€™s inequality, we can write โ€–โ€–๐ท๐‘ก1/2๐‘คโ€–โ€–๐ธ๐›ผโ‰ค๐‘’๐‘€6๎€ท๐‘€1โ€–๐ด๐œ‘โ€–๐ธ๐›ผ+๐‘€7โ€–๐นโ€–๐ถ(๐ธ๐›ผ)๎€ธ.(2.36) From the last estimate, we can obtain the estimate for ๐‘ค๐‘ก(๐‘ก) by using problem (2.17) and well-posedness of the Cauchy problem in ๐ถ(๐ธ๐›ผ) (see [35]). So the following theorem finishes the proof of Theorem 2.2.

Theorem 2.3 (see, [36]). For 0<๐›ผ<1/2, the norms of the spaces ๐ธ๐›ผ(๐ถ[0,๐œ‹],๐ด) and ๐ถ2๐›ผ[0,๐œ‹] are equivalent.

3. Numerical Results

We have not been able to obtain a sharp estimate for the constants figuring in the stability inequalities. So we will provide the following results of numerical experiments of the following problem: ๐œ•๐‘ข(๐‘ก,๐‘ฅ)=๐œ•๐œ•๐‘ก2๐‘ข(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ2โˆ’๐‘ข(๐‘ก,๐‘ฅ)+๐ท๐‘ก1/2๎ƒฉ1๐‘ข(๐‘ก,๐‘ฅ)+๐‘(๐‘ก)sin๐‘ฅ+๐‘“(๐‘ก,๐‘ฅ),๐‘“(๐‘ก,๐‘ฅ)=โˆ’3๐‘กโˆ’โˆš๐œ‹๐‘กโˆ’1/2+2โˆš๐œ‹๐‘ก1/2๎ƒช],[],[],๐‘ข๎‚€๐œ‹sin๐‘ฅ,๐‘ฅโˆˆ(0,๐œ‹),๐‘กโˆˆ(0,1๐‘ข(0,๐‘ฅ)=sin๐‘ฅ,๐‘ฅโˆˆ0,๐œ‹๐‘ข(๐‘ก,0)=๐‘ข(๐‘ก,๐œ‹)=0,๐‘กโˆˆ0,1๐‘ก,2๎‚=1โˆ’๐‘ก.(3.1) The exact solution of the given problem is ๐‘ข(๐‘ก,๐‘ฅ)=(1โˆ’๐‘ก)sin๐‘ฅ and for the control parameter ๐‘(๐‘ก) is 1+๐‘ก.

3.1. The First Order of Accuracy Difference Scheme

For the approximate solution of the problem (3.1), the Rothe difference scheme ๐‘ข๐‘˜๐‘›โˆ’๐‘ข๐‘›๐‘˜โˆ’1๐œ=๐‘ข๐‘˜๐‘›+1โˆ’2๐‘ข๐‘˜๐‘›+๐‘ข๐‘˜๐‘›โˆ’1โ„Ž2โˆ’๐‘ข๐‘˜๐‘›+๐ท๐œ1/2๐‘ข๐‘˜๐‘›+๐‘๐‘˜๐‘ž๐‘›๎€ท๐‘ก+๐‘“๐‘˜,๐‘ฅ๐‘›๎€ธ,๐‘“๎€ท๐‘ก๐‘˜,๐‘ฅ๐‘›๎€ธ=๎ƒฉโˆ’3๐‘ก๐‘˜โˆ’1โˆš๐œ‹๐‘ก๐‘˜โˆ’1/2+2โˆš๐œ‹๐‘ก๐‘˜1/2๎ƒชsin๐‘ฅ๐‘›,๐‘๐‘˜๎€ท๐‘ก=๐‘๐‘˜๎€ธ,๐‘ž๐‘›=sin๐‘ฅ๐‘›,๐‘ฅ๐‘›=๐‘›โ„Ž,๐‘ก๐‘˜๐‘ข=๐‘˜๐œ,1โ‰ค๐‘˜โ‰ค๐‘,1โ‰ค๐‘›โ‰ค๐‘€โˆ’1,๐‘€โ„Ž=๐œ‹,๐‘๐œ=1,0๐‘›=sin๐‘ฅ๐‘›๐‘ข,0โ‰ค๐‘›โ‰ค๐‘€,๐‘˜0=๐‘ข๐‘˜๐‘€๐‘ข=0,0โ‰ค๐‘˜โ‰ค๐‘,๐‘˜๐‘ ๎€ท๐‘ก=๐œŒ๐‘˜๎€ธ๎€ท๐‘ก,๐œŒ๐‘˜๎€ธ=1โˆ’๐‘ก๐‘˜๎‚ž๐œ‹,0โ‰ค๐‘˜โ‰ค๐‘,๐‘ =๎‚Ÿ,2โ„Ž(3.2) where โŒŠ๐‘ฅโŒ‹ denotes greatest integer less than ๐‘ฅ is constructed. Throughout the paper, let us denote ๐œŒ๎€ท๐‘ก๐‘˜๎€ธ=1โˆ’๐‘ก๐‘˜,๐‘ž๐‘›=sin๐‘ฅ๐‘›,๐‘ก๐‘˜=๎€ฝ๐‘ก๐‘˜๎€พ,๐‘ฅ=๐‘˜๐œ,0โ‰ค๐‘˜โ‰ค๐‘,๐‘๐œ=1๐‘›=๎€ฝ๐‘ฅ๐‘›๎€พ,๐‘“๎€ท๐‘ก=๐‘›โ„Ž,0โ‰ค๐‘›โ‰ค๐‘€โˆ’1,๐‘€โ„Ž=๐œ‹๐‘˜,๐‘ฅ๐‘›๎€ธ=๎ƒฉโˆ’3๐‘ก๐‘˜โˆ’1โˆš๐œ‹๐‘ก๐‘˜โˆ’1/2+2โˆš๐œ‹๐‘ก๐‘˜1/2๎ƒชsin๐‘ฅ๐‘›,๐น๎€ท๐‘ก๐‘˜,๐‘ฅ๐‘›๎€ธ=๐œŒ๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธ๎ƒฉ๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธโ„Ž2๎€ท๐‘ฅโˆ’sin๐‘›๎€ธ๎ƒชโˆ’1โˆš๐œ‹๐‘˜๎“๐‘š=1ฮ“(๐‘˜โˆ’๐‘š+(1/2))๐œ(๐‘˜โˆ’๐‘š)!1/2๎€ท๐‘ฅsin๐‘ ๎€ธ๎€ท๐‘ฅsin๐‘›๎€ธ๎€ท๐‘ก+๐‘“๐‘˜,๐‘ฅ๐‘›๎€ธ.(3.3) We search the solution of (3.2) in the following form: ๐‘ข๐‘˜๐‘›=๐œ‚๐‘˜๐‘ž๐‘›+๐‘ค๐‘˜๐‘›,(3.4) where ๐œ‚๐‘˜=๐‘˜๎“๐‘–=1๐‘๐‘–๐œ,1โ‰ค๐‘˜โ‰ค๐‘,๐œ‚0=0.(3.5) Moreover for the interior grid point ๐‘ข๐‘˜๐‘ , we have that ๐‘ข๐‘˜๐‘ =๐œ‚๐‘˜๐‘ž๐‘ +๐‘ค๐‘˜๐‘ ๎€ท๐‘ก=๐œŒ๐‘˜๎€ธ.(3.6) From (3.4), (3.5), and the condition ๐‘ข๐‘˜๐‘ =๐œŒ(๐‘ก๐‘˜), it follows that ๐œ‚๐‘˜=๐œŒ๎€ท๐‘ก๐‘˜๎€ธโˆ’๐‘ค๐‘˜๐‘ ๐‘ž๐‘ ,๐‘(3.7)๐‘˜=๐œ‚๐‘˜โˆ’๐œ‚๐‘˜โˆ’1๐œ๐‘ข,1โ‰ค๐‘˜โ‰ค๐‘,(3.8)๐‘˜๐‘›=๐œŒ๎€ท๐‘ก๐‘˜๎€ธโˆ’๐‘ค๐‘˜๐‘ ๐‘ž๐‘ ๐‘ž๐‘›+๐‘ค๐‘˜๐‘›,0โ‰ค๐‘˜โ‰ค๐‘,0โ‰ค๐‘›โ‰ค๐‘€,(3.9) where ๐‘ค๐‘˜๐‘›,0โ‰ค๐‘˜โ‰ค๐‘,0โ‰ค๐‘›โ‰ค๐‘€ is the solution of the difference scheme ๐‘ค๐‘˜๐‘›โˆ’๐‘ค๐‘›๐‘˜โˆ’1๐œ=๐‘ค๐‘˜๐‘›+1โˆ’2๐‘ค๐‘˜๐‘›+๐‘ค๐‘˜๐‘›โˆ’1โ„Ž2โˆ’๐‘ค๐‘˜๐‘›โˆ’๐‘ค๐‘˜๐‘ ๎€ท๐‘ฅsin๐‘ ๎€ธ๎ƒฉ๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธโ„Ž2๎€ท๐‘ฅโˆ’sin๐‘›๎€ธ๎ƒชโˆ’1โˆš๐œ‹๐‘˜๎“๐‘š=1ฮ“(๐‘˜โˆ’๐‘š+(1/2))๎ƒฉ๐‘ค(๐‘˜โˆ’๐‘š)!๐‘š๐‘ โˆ’๐‘ค๐‘ ๐‘šโˆ’1๎€ท๐‘ฅsin๐‘ ๎€ธ๐œ1/2๎€ท๐‘ฅsin๐‘›๎€ธโˆ’๐‘ค๐‘š๐‘›โˆ’๐‘ค๐‘›๐‘šโˆ’1๐œ1/2๎ƒช๎€ท๐‘ก+๐น๐‘˜,๐‘ฅ๐‘›๎€ธ๐‘ค,1โ‰ค๐‘˜โ‰ค๐‘,1โ‰ค๐‘›โ‰ค๐‘€โˆ’1,๐‘˜0=๐‘ค๐‘˜๐‘€๐‘ค=0,0โ‰ค๐‘˜โ‰ค๐‘,0๐‘›๎€ท๐‘ฅ=sin๐‘›๎€ธ,0โ‰ค๐‘›โ‰ค๐‘€.(3.10) First, applying the first order of accuracy difference scheme (3.10), we obtain (๐‘€+1)ร—(๐‘€+1) system of linear equations and we write them in the matrix form ๐ด๐‘ค๐‘˜+๐‘˜โˆ’1๎“๐‘—=0๐ต๐‘—๐‘ค๐‘—=๐ท๐œ‘๐‘˜,1โ‰ค๐‘˜โ‰ค๐‘,๐‘ค0=๎€ฝ๎€ท๐‘ฅsin๐‘›๎€ธ๎€พ๐‘€๐‘›=0,(3.11) where โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ด=1000.0.000๐‘ฅ๐‘ฆ๐‘ฅ0.๐‘ง1.0000๐‘ฅ๐‘ฆ๐‘ฅ.๐‘ง2.000..........0000.๐‘ง๐‘€โˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.๐‘ฅ๐‘ฆ๐‘ฅ0000.0.001(๐‘€+1)ร—(๐‘€+1),๐ต0=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ000.0.000๐‘Ž0.๐‘“1.0000๐‘Ž.๐‘“2.00........000.๐‘“๐‘ +๐‘Ž.00........000.๐‘“๐‘€โˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.0๐‘Ž000.0.00(๐‘€+1)ร—(๐‘€+1),๐ต๐‘—=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ000.0.000๐‘0.๐‘‘1.0000๐‘.๐‘‘2.00........000.๐‘‘๐‘ +๐‘.00........000.๐‘‘๐‘€โˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.0๐‘000.0.00(๐‘€+1)ร—(๐‘€+1),(3.12) for any ๐‘—=1,2,โ€ฆ,๐‘˜โˆ’2, and ๐ต๐‘˜โˆ’1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ000.0.000๐‘ฃ0.๐‘1.0000๐‘ฃ.๐‘2.00........000.๐‘๐‘ +๐‘ฃ.00........000.๐‘๐‘€โˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.0๐‘ฃ000.0.00(๐‘€+1)ร—(๐‘€+1).(3.13) Here, for any ๐‘›=1,2,โ€ฆ,๐‘€โˆ’1, 1๐‘ฅ=โˆ’โ„Ž21,๐‘ฆ=๐œ+2โ„Ž21+1+โˆš๐œ,๐‘ง๐‘›=๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธโ„Ž2โˆ’๎€ท๐‘ฅsin๐‘›๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธโˆ’๎€ท๐‘ฅsin๐‘›๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธโˆš๐œฮ“,in(๐‘ +1)thcolumn,๐‘Ž=(๐‘˜โˆ’1/2)โˆš๐œ๐œ‹(๐‘˜โˆ’1)!,๐‘“๐‘›๎€ท๐‘ฅ=โˆ’sin๐‘›๎€ธฮ“(๐‘˜โˆ’1/2)โˆš๎€ท๐‘ฅ๐œ๐œ‹sin๐‘ ๎€ธ,1(๐‘˜โˆ’1)!๐‘=โˆš๎‚ต๐œ๐œ‹ฮ“(๐‘˜โˆ’๐‘—โˆ’1/2)โˆ’(๐‘˜โˆ’๐‘—โˆ’1)!ฮ“(๐‘˜โˆ’๐‘—+1/2)๎‚ถ,๐‘‘(๐‘˜โˆ’๐‘—)!๐‘›=๎€ท๐‘ฅsin๐‘›๎€ธโˆš๎€ท๐‘ฅ๐œ๐œ‹sin๐‘ ๎€ธ๎‚ตฮ“(๐‘˜โˆ’๐‘—โˆ’1/2)โˆ’(๐‘˜โˆ’๐‘—โˆ’1)!ฮ“(๐‘˜โˆ’๐‘—+1/2)๎‚ถ,1(๐‘˜โˆ’๐‘—)!๐‘ฃ=โˆ’โˆš๐œโˆ’1๐œ,๐‘๐‘›=๎€ท๐‘ฅsin๐‘›๎€ธโˆš๎€ท๐‘ฅ๐œsin๐‘ ๎€ธ,๐‘ค๐‘Ÿ=โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ค๐‘Ÿ0โ‹ฎ๐‘ค๐‘Ÿ๐‘€โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ(๐‘€+1)ร—1๐œ‘forany๐‘Ÿ=0,1,โ€ฆ,๐‘˜,๐‘˜=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0๐œ™๐‘˜1โ‹ฎ๐œ™๐‘˜๐‘€โˆ’10โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ(๐‘€+1)ร—1,๐œ™๐‘˜๐‘›=๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธ๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธโ„Ž2โˆ’๐œŒ๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ฅsin๐‘ ๎€ธ๎€ท๐‘ฅsin๐‘›๎€ธ๎ƒชโˆ’1โˆš๐œ‹๐‘˜๎“๐‘š=1ฮ“(๐‘˜โˆ’๐‘š+1/2)๐œ(๐‘˜โˆ’๐‘š)!1/2๎€ท๐‘ฅsin๐‘ ๎€ธ๎€ท๐‘ฅsin๐‘›๎€ธ๎€ท๐‘ก+๐‘“๐‘˜,๐‘ฅ๐‘›๎€ธ,(3.14) and ๐ท is (๐‘€+1)ร—(๐‘€+1) identity matrix. Using (3.11), we can obtain that ๐‘ค๐‘˜=๐ดโˆ’1๎ƒฉ๐ท๐œ‘๐‘˜โˆ’๐‘˜โˆ’1๎“๐‘—=0๐ต๐‘—๐‘ค๐‘—๎ƒช,๐‘˜=1,2,โ€ฆ,๐‘,๐‘ค0=๎€ฝsin๐‘ฅ๐‘›๎€พ๐‘€๐‘›=0.(3.15) To solve the resulting difference equations, we apply the method given in (3.15) step by step for ๐‘˜=1,2,โ€ฆ,๐‘. For the evaluation of ๐‘ค๐‘Ÿ,๐‘Ÿ=2,3,โ€ฆ,๐‘,๐‘ค๐‘Ÿโˆ’1 is needed. It is obtained in the previous step. Then, the solution pairs (๐‘ข,๐‘) are obtained by using the last formulas (3.9) and (3.8).

3.2. The Second Order of Accuracy Difference Scheme

For the approximate solution of the problem (3.1), the Crank-Nicholson difference scheme ๐‘ข๐‘˜๐‘›โˆ’๐‘ข๐‘›๐‘˜โˆ’1๐œ=๐‘ข๐‘˜๐‘›+1โˆ’2๐‘ข๐‘˜๐‘›+๐‘ข๐‘˜๐‘›โˆ’12โ„Ž2+๐‘ข๐‘˜โˆ’1๐‘›+1โˆ’2๐‘ข๐‘›๐‘˜โˆ’1+๐‘ข๐‘˜โˆ’1๐‘›โˆ’12โ„Ž2โˆ’๐‘ข๐‘˜๐‘›+๐‘ข๐‘›๐‘˜โˆ’12+๐‘๐‘˜+๐‘๐‘˜โˆ’12๐‘ž๐‘›+๐ท๐œ1/2๐‘ข๎‚€๐‘ก๐‘˜โˆ’๐œ2,๐‘ฅ๐‘›๎‚๎‚€๐‘ก+๐‘“๐‘˜โˆ’๐œ2,๐‘ฅ๐‘›๎‚,๐‘๐‘˜๎€ท๐‘ก=๐‘๐‘˜๎€ธ๐‘ข,1โ‰ค๐‘˜โ‰ค๐‘,1โ‰ค๐‘›โ‰ค๐‘€โˆ’10๐‘›๎€ท๐‘ฅ=sin๐‘›๎€ธ๐‘ข,0โ‰ค๐‘›โ‰ค๐‘€,๐‘˜0=๐‘ข๐‘˜๐‘€๐‘ข=0,0โ‰ค๐‘˜โ‰ค๐‘,๐‘˜๐‘ +๐‘ข๐‘˜๐‘ +1โˆ’๐‘ข๐‘˜๐‘ โ„Ž๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘กโˆ’๐‘ โ„Ž=๐œŒ๐‘˜๎€ธ๎‚ž๐œ‹,0โ‰ค๐‘˜โ‰ค๐‘,๐‘ =๎‚Ÿ2โ„Ž(3.16) is constructed.

Here, ฮ“๎‚€1๐‘˜โˆ’๐‘Ÿ+2๎‚=๎€œโˆž0๐‘ก๐‘˜โˆ’๐‘Ÿ+1/2๐‘’โˆ’๐‘ก๐‘‘๐‘ก.(3.17) Moreover, applying the second order of approximation formula for ๐ท๐‘ก1/2๐‘ข๎‚€๐‘ก๐‘˜โˆ’๐œ2๎‚=1๎€œฮ“(1/2)๐‘ก๐‘˜0โˆ’๐œ/2๎‚€๐‘ก๐‘˜โˆ’๐œ2๎‚โˆ’๐‘ โˆ’1/2๐‘ข๎…ž(๐‘ )๐‘‘๐‘ ,(3.18) it is obtained (see [27]) ๐ท๐œ1/2โŽงโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽฉโˆš๐‘ข=โˆ’๐‘‘23๐‘ข0+๐‘‘โˆš23๐‘ข1+๐‘‘๐œ3โˆš2๎€ท๐‘ฅsin๐‘›๎€ธโˆ’โˆš,๐‘˜=1,2๐‘‘65๐‘ข0+๐‘‘โˆš65๐‘ข1+๐‘‘โˆš65๐‘ข2โˆ’โˆš๐‘‘๐œ6๎€ท๐‘ฅ10sin๐‘›๎€ธ๐‘‘,๐‘˜=2,๐‘˜โˆ’1โˆ‘๐‘š=2๎€ฝ๎€บ(๐‘˜โˆ’๐‘š)๐‘1+๐‘2๎€ป๐‘ข๐‘šโˆ’2+๎€บ(2๐‘šโˆ’2๐‘˜โˆ’1)๐‘1โˆ’2๐‘2๎€ป๐‘ข๐‘šโˆ’1+๎€บ(๐‘˜โˆ’๐‘š+1)๐‘1+๐‘2๎€ป๐‘ข๐‘š๎€พ+๐‘‘6โˆš2๎€บโˆ’๐‘ข๐‘˜โˆ’2โˆ’4๐‘ข๐‘˜โˆ’1+5๐‘ข๐‘˜๎€ป,3โ‰ค๐‘˜โ‰ค๐‘.(3.19) Here and throughout the paper, ๐ท๐‘ก1/2๐‘ข=๐ท๐œ1/2๐‘ข๎‚€๐‘ก๐‘˜โˆ’๐œ2,๐‘ฅ๐‘›๎‚,2๐‘‘=โˆš๐œ‹๐œ,๐‘1=๎‚™1๐‘˜โˆ’๐‘š+2โˆ’๎‚™1๐‘˜โˆ’๐‘šโˆ’2,๐‘21=โˆ’3๎‚ต๎‚€1๐‘˜โˆ’๐‘š+2๎‚3/2โˆ’๎‚€1๐‘˜โˆ’๐‘šโˆ’2๎‚3/2๎‚ถ.(3.20) We search the solution of (3.16) in the following form: ๐‘ข๐‘˜๐‘›=๐œ‚๐‘˜๐‘ž๐‘›+๐‘ค๐‘˜๐‘›,(3.21) where ๐œ‚๐‘˜=๐‘˜๎“๐‘–=1๐‘๐‘–+๐‘๐‘–โˆ’12๐œ,1โ‰ค๐‘˜โ‰ค๐‘,๐œ‚0=0.(3.22) We have that ๐‘ข๐‘˜๐‘ +๐‘ข๐‘˜๐‘ +1โˆ’๐‘ข๐‘˜๐‘ โ„Ž๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ โ„Ž=๐œ‚๐‘˜๐‘ฅ๎‚ต๎‚ต1โˆ’โˆ—โˆ’๐‘ โ„Žโ„Ž๎‚ถ๐‘ž๐‘ +๐‘ฅโˆ—โˆ’๐‘ โ„Žโ„Ž๐‘ž๐‘ +1๎‚ถ+๎‚ต๐‘ฅ1โˆ’โˆ—โˆ’๐‘ โ„Žโ„Ž๎‚ถ๐‘ค๐‘˜๐‘ +๐‘ฅโˆ—โˆ’๐‘ โ„Žโ„Ž๐‘ค๐‘˜๐‘ +1๎€ท๐‘ก=๐œŒ๐‘˜๎€ธ.(3.23) Let us denote ๐‘ฅ๐‘ฆ=โˆ—โˆ’๐‘ โ„Žโ„Ž=๐‘ฅโˆ—โ„Žโˆ’๎ƒ“๐‘ฅโˆ—โ„Ž๎ƒ”โ„Ž,(3.24) where 0โ‰ค๐‘ฆ<1. Then, one can write ๐œ‚๐‘˜=๐œŒ๎€ท๐‘ก๐‘˜๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘˜๐‘ โˆ’๐‘ฆ๐‘ค๐‘˜๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1.(3.25) So the values of (๐‘(๐‘ก๐‘˜)+๐‘(๐‘ก๐‘˜โˆ’1))/2,1โ‰ค๐‘˜โ‰ค๐‘ can be obtained by the following formula: ๐‘๐‘˜+๐‘๐‘˜โˆ’12=๎€ท๐œŒ๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘กโˆ’๐œŒ๐‘˜โˆ’1๐‘ค๎€ธ๎€ธ/๐œโˆ’(1โˆ’๐‘ฆ)๎€ท๎€ท๐‘˜๐‘ โˆ’๐‘ค๐‘ ๐‘˜โˆ’1๎€ธ๎€ธ๐‘ค/๐œโˆ’๐‘ฆ๎€ท๎€ท๐‘˜๐‘ +1โˆ’๐‘ค๐‘˜โˆ’1๐‘ +1๎€ธ๎€ธ/๐œ(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1.(3.26) Let ๐‘ค๐‘Ÿ denote ๐‘ค๐‘Ÿ=โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ค๐‘Ÿ0โ‹ฎ๐‘ค๐‘Ÿ๐‘€โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ(๐‘€+1)ร—1for๐‘Ÿ=0,1,โ€ฆ,๐‘.(3.27) For ๐‘˜=1, one can show that ๐‘ค1 is the solution of the difference scheme ๐‘ค1๐‘›โˆ’๐‘ค0๐‘›๐œ=๐‘ค1๐‘›+1โˆ’2๐‘ค1๐‘›+๐‘ค1๐‘›โˆ’12โ„Ž2+๐‘ค0๐‘›+1โˆ’2๐‘ค0๐‘›+๐‘ค0๐‘›โˆ’12โ„Ž2โˆ’๐‘ค1๐‘›+๐‘ค0๐‘›2+๎‚ต๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’12โ„Ž2โˆ’๐‘ž๐‘›2๎‚ถร—๎ƒฉ๐œŒ๎€ท๐‘ก1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค1๐‘ โˆ’๐‘ฆ๐‘ค1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1+๐œŒ๎€ท๐‘ก0๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค0๐‘ โˆ’๐‘ฆ๐‘ค0๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๐‘‘โˆš23๎ƒฉ๐œŒ๎€ท๐‘ก1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค1๐‘ โˆ’๐‘ฆ๐‘ค1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค1๐‘›๎ƒชโˆ’๐‘‘โˆš23๎ƒฉ๐œŒ๎€ท๐‘ก0๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค0๐‘ โˆ’๐‘ฆ๐‘ค0๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค0๐‘›๎ƒช+๐‘‘๐œ3โˆš2๎‚€๐‘ก๐‘ž(๐‘›)+๐‘“1โˆ’๐œ2,๐‘ฅ๐‘›๎‚๐‘ค,1โ‰ค๐‘›โ‰ค๐‘€โˆ’1,10=๐‘ค1๐‘€๐‘ค=0,0๐‘›๎€ท๐‘ฅ=sin๐‘›๎€ธ,0โ‰ค๐‘›โ‰ค๐‘€.(3.28) We have the system of linear equations and we write them in the matrix form ๐ด1๐‘ค1+๐ต1๐‘ค0=๐ท๐œ‘1,(3.29) where ๐ด1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ1000โ‹…0โ‹…โ‹…000๐‘Ž๐‘ฆ1๐‘Ž0โ‹…๐‘™1๐‘1โ‹…0000๐‘Ž๐‘ฆ1๐‘Žโ‹…๐‘™2๐‘2โ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…000๐‘Žโ‹…๐‘™๐‘ +๐‘Ž๐‘๐‘ โ‹…0000000โ‹…๐‘™๐‘ +1+๐‘ฆ1๐‘๐‘ +1+๐‘Žโ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘™๐‘€โˆ’1๐‘๐‘€โˆ’1โ‹…๐‘Ž๐‘ฆ1๐‘ŽโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ0000โ‹…00โ‹…001(๐‘€+1)ร—(๐‘€+1),๐ต1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0000โ‹…0โ‹…โ‹…000๐‘Ž๐‘ฃ1๐‘Ž0โ‹…๐‘‘1๐‘’1โ‹…0000๐‘Ž๐‘ฃ1๐‘Žโ‹…๐‘‘2๐‘’2โ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…000๐‘Žโ‹…๐‘‘๐‘ +๐‘Ž๐‘’๐‘ โ‹…0000000โ‹…๐‘‘๐‘ +1+๐‘ฃ1๐‘’๐‘ +1+๐‘Žโ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘‘๐‘€โˆ’1๐‘’๐‘€โˆ’1โ‹…๐‘Ž๐‘ฃ1๐‘ŽโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ0000โ‹…00โ‹…000(๐‘€+1)ร—(๐‘€+1).(3.30) Here, for any ๐‘›=1,2,โ€ฆ,๐‘€โˆ’1, ๎‚ตโˆ’1๐‘Ž=2โ„Ž2๎‚ถ,๐‘ฆ1=๎ƒฉ1๐œ+1โ„Ž2+12โˆšโˆ’๐‘‘23๎ƒช,๐‘ฃ1=๎ƒฉโˆ’1๐œ+1โ„Ž2+12โˆš+๐‘‘23๎ƒช,๐‘™๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ(1โˆ’๐‘ฆ)2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›(1โˆ’๐‘ฆ)2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโˆš+๐‘‘23๐‘ž๐‘›,๐‘๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ๐‘ฆ2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›๐‘ฆ2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโˆš+๐‘‘23๐‘ž๐‘›,๐‘‘๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ(1โˆ’๐‘ฆ)2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›(1โˆ’๐‘ฆ)2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโˆšโˆ’๐‘‘23๐‘ž๐‘›,๐‘’๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ๐‘ฆ2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›๐‘ฆ2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโˆšโˆ’๐‘‘23๐‘ž๐‘›,๐œ‘1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0๐œ™11โ‹ฎ๐œ™1๐‘€โˆ’10โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ(๐‘€+1)ร—1,๐œ™1๐‘›=๎ƒฉ๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธ2โ„Ž2โˆ’๎€ท๐‘ฅsin๐‘›๎€ธ2๎ƒช๐œŒ๎€ท๐‘ก1๎€ธ๎€ท๐‘ก+๐œŒ0๎€ธ(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1+๐‘‘โˆš2๐‘ž๐‘›3๐œŒ๎€ท๐‘ก1๎€ธ๎€ท๐‘กโˆ’๐œŒ0๎€ธ(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1+๐‘‘๐œ3โˆš2๐‘ž๐‘›๎‚€๐‘ก+๐‘“1โˆ’๐œ2,๐‘ฅ๐‘›๎‚(3.31) and ๐ท is (๐‘€+1)ร—(๐‘€+1) identity matrix. Using (3.29), we can obtain that ๐‘ค1=๐ด1โˆ’1๎€ท๐ท๐œ‘1โˆ’๐ต1๐‘ค0๎€ธ,๐‘ค0=๎€ฝsin๐‘ฅ๐‘›๎€พ๐‘€๐‘›=0.(3.32) For ๐‘˜=2,โ€‰โ€‰๐‘ค2 is the solution of the difference scheme ๐‘ค2๐‘›โˆ’๐‘ค1๐‘›๐œ=๐‘ค2๐‘›+1โˆ’2๐‘ค2๐‘›+๐‘ค2๐‘›โˆ’12โ„Ž2โˆ’๐‘ค2๐‘›+๐‘ค1๐‘›2+๐‘ค1๐‘›+1โˆ’2๐‘ค1๐‘›+๐‘ค1๐‘›โˆ’12โ„Ž2+๎‚ต๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’12โ„Ž2โˆ’๐‘ž๐‘›2๎‚ถ๎ƒฉ๐œŒ๎€ท๐‘ก2๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค2๐‘ โˆ’๐‘ฆ๐‘ค2๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1+๐œŒ๎€ท๐‘ก1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค1๐‘ โˆ’๐‘ฆ๐‘ค1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๐‘‘โˆš65๎ƒฉ๐œŒ๎€ท๐‘ก2๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค2๐‘ โˆ’๐‘ฆ๐‘ค2๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค2๐‘›๎ƒช+๐‘‘โˆš65๎ƒฉ๐œŒ๎€ท๐‘ก1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค1๐‘ โˆ’๐‘ฆ๐‘ค1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค1๐‘›๎ƒชโˆ’โˆš2๐‘‘65๎ƒฉ๐œŒ๎€ท๐‘ก0๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค0๐‘ โˆ’๐‘ฆ๐‘ค0๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค0๐‘›๎ƒชโˆ’โˆš๐‘‘๐œ6๎‚€๐‘ก10๐‘ž(๐‘›)+๐‘“2โˆ’๐œ2,๐‘ฅ๐‘›๎‚๐‘ค,1โ‰ค๐‘›โ‰ค๐‘€โˆ’1,20=๐‘ค2๐‘€๐‘ค=0,0๐‘›๎€ท๐‘ฅ=sin๐‘›๎€ธ,0โ‰ค๐‘›โ‰ค๐‘€.(3.33) The system of linear equations given above can be written in the matrix form ๐ด2๐‘ค2+๐ต2๐‘ค1+๐ถ2๐‘ค0=๐ท๐œ‘2,(3.34) where ๐ด2=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ1000โ‹…0โ‹…โ‹…000๐‘Ž๐‘ฆ2๐‘Ž0โ‹…๐‘”1โ„Ž1โ‹…0000๐‘Ž๐‘ฆ2๐‘Žโ‹…๐‘”2โ„Ž2โ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘”๐‘ +๐‘Žโ„Ž๐‘ โ‹…0000000โ‹…๐‘”๐‘ +1+๐‘ฆ2โ„Ž๐‘ +1+๐‘Žโ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘”๐‘€โˆ’1โ„Ž๐‘€โˆ’1โ‹…๐‘Ž๐‘ฆ2๐‘ŽโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ0000โ‹…00โ‹…001(๐‘€+1)ร—(๐‘€+1),๐ต2=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0000โ‹…0โ‹…โ‹…000๐‘Ž๐‘ฃ2๐‘Ž0โ‹…๐‘”1โ„Ž1โ‹…0000๐‘Ž๐‘ฃ2๐‘Žโ‹…๐‘”2โ„Ž2โ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘”๐‘ +๐‘Žโ„Ž๐‘ โ‹…0000000โ‹…๐‘”๐‘ +1+๐‘ฃ2โ„Ž๐‘ +1+๐‘Žโ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘”๐‘€โˆ’1โ„Ž๐‘€โˆ’1โ‹…๐‘Ž๐‘ฃ2๐‘ŽโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ0000โ‹…00โ‹…000(๐‘€+1)ร—(๐‘€+1),๐ถ2=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0000โ‹…0โ‹…โ‹…0000๐‘ง00โ‹…๐‘–1๐‘—1โ‹…00000๐‘ง0โ‹…๐‘–2๐‘—2โ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘–๐‘ +๐‘ง๐‘—๐‘ โ‹…0000000โ‹…๐‘–๐‘ +1๐‘—๐‘ +1+๐‘งโ‹…000โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…โ‹…0000โ‹…๐‘–๐‘€โˆ’1๐‘—๐‘€โˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆโ‹…0๐‘ง00000โ‹…00โ‹…000(๐‘€+1)ร—(๐‘€+1).(3.35) Here, for any ๐‘›=1,2,โ€ฆ,๐‘€โˆ’1, 1๐‘Ž=โˆ’2โ„Ž2,๐‘ฆ2=1๐œ+1โ„Ž2+12โˆ’๐‘‘โˆš65,๐‘ฃ21=โˆ’๐œ+1โ„Ž2+12โˆ’๐‘‘โˆš65,๐‘”๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ(1โˆ’๐‘ฆ)2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›(1โˆ’๐‘ฆ)2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘‘โˆš6๐‘ž๐‘›(1โˆ’๐‘ฆ)5๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโ„Ž,in(๐‘ +1)thcolumn,๐‘›=๎€ท๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’1๎€ธ๐‘ฆ2โ„Ž2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘ž๐‘›๐‘ฆ2๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ+๐‘‘โˆš6๐‘ž๐‘›๐‘ฆ5๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ๐‘–,in(๐‘ +2)thcolumn,๐‘›โˆš=โˆ’2๐‘‘6๐‘ž๐‘›(1โˆ’๐‘ฆ)5๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธ,๐‘—๐‘›2โˆš=โˆ’6๐‘ž๐‘›๐‘ฆ5๎€ท(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎€ธโˆš,๐‘ง=2๐‘‘65,๐œ‘2=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ0๐œ™21โ‹ฎ๐œ™2๐‘€โˆ’10โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ(๐‘€+1)ร—1,๐œ™2๐‘›=๎ƒฉ๎€ท๐‘ฅsin๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’2sin๐‘›๎€ธ๎€ท๐‘ฅ+sin๐‘›โˆ’1๎€ธ2โ„Ž2โˆ’๎€ท๐‘ฅsin๐‘›๎€ธ2+๐‘‘โˆš6๐‘ž๐‘›5๎ƒชร—๐œŒ๎€ท๐‘ก2๎€ธ๎€ท๐‘ก+๐œŒ1๎€ธ(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1โˆ’โˆš2๐‘‘6๐‘ž๐‘›5๐œŒ๎€ท๐‘ก0๎€ธ(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1โˆ’โˆš๐‘‘๐œ6๐‘ž10๐‘›๎‚€๐‘ก+๐‘“2โˆ’๐œ2,๐‘ฅ๐‘›๎‚.(3.36) Using (3.34), we can obtain that ๐‘ค2=๐ด2โˆ’1๎€ท๐ท๐œ‘2โˆ’๐ต2๐‘ค1โˆ’๐ถ2๐‘ค0๎€ธ,๐‘ค0=๎€ฝsin๐‘ฅ๐‘›๎€พ๐‘€๐‘›=0.(3.37) For 3โ‰ค๐‘˜โ‰ค๐‘, we can obtain the following difference scheme: ๐‘ค๐‘˜๐‘›โˆ’๐‘ค๐‘›๐‘˜โˆ’1๐œ=๐‘ค๐‘˜๐‘›+1โˆ’2๐‘ค๐‘˜๐‘›+๐‘ค๐‘˜๐‘›โˆ’12โ„Ž2+๐‘ค๐‘˜โˆ’1๐‘›+1โˆ’2๐‘ค๐‘›๐‘˜โˆ’1+๐‘ค๐‘˜โˆ’1๐‘›โˆ’12โ„Ž2โˆ’๐‘ค๐‘˜๐‘›+๐‘ค๐‘›๐‘˜โˆ’12+๎‚ต๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’12โ„Ž2โˆ’๐‘ž๐‘›2๎‚ถร—๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘˜๐‘ โˆ’๐‘ฆ๐‘ค๐‘˜๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๎‚ต๐‘ž๐‘›+1โˆ’2๐‘ž๐‘›+๐‘ž๐‘›โˆ’12โ„Ž2โˆ’๐‘ž๐‘›2๎‚ถร—๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜โˆ’1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘ ๐‘˜โˆ’1โˆ’๐‘ฆ๐‘ค๐‘˜โˆ’1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๐‘‘๐‘˜โˆ’1๎“๐‘š=2๎€ฝ๎€ท(๐‘˜โˆ’๐‘š)๐‘1+๐‘2๎€ธร—๎ƒฉ๐œŒ๎€ท๐‘ก๐‘šโˆ’2๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘ ๐‘šโˆ’2โˆ’๐‘ฆ๐‘ค๐‘šโˆ’2๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๎€ท๐‘ž(๐‘›)(2๐‘šโˆ’2๐‘˜โˆ’1)๐‘1โˆ’2๐‘2๎€ธร—๎ƒฉ๐œŒ๎€ท๐‘ก๐‘šโˆ’1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘ ๐‘šโˆ’1โˆ’๐‘ฆ๐‘ค๐‘šโˆ’1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๎€ท๐‘ž(๐‘›)(2๐‘šโˆ’2๐‘˜โˆ’1)๐‘1โˆ’2๐‘2๎€ธ๐‘ค๐‘›๐‘šโˆ’1ร—๎ƒฉ๐œŒ๎€ท๐‘ก๐‘š๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘š๐‘ โˆ’๐‘ฆ๐‘ค๐‘š๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๎ƒช+๎€ท๐‘ž(๐‘›)(๐‘˜โˆ’๐‘š)๐‘1+๐‘2๎€ธ๐‘ค๐‘›๐‘šโˆ’2+๎€ท(๐‘˜โˆ’๐‘šโˆ’1)๐‘1+๐‘2๎€ธ๐‘ค๐‘š๐‘›+๎€ท(2๐‘šโˆ’2๐‘˜โˆ’1)๐‘1โˆ’2๐‘2๎€ธ๐‘ค๐‘›๐‘šโˆ’1๎€พโˆ’๐‘‘6โˆš2๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜โˆ’2๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘ ๐‘˜โˆ’2โˆ’๐‘ฆ๐‘ค๐‘˜โˆ’2๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค๐‘›๐‘˜โˆ’2๎ƒชโˆ’4๐‘‘6โˆš2๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜โˆ’1๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘ ๐‘˜โˆ’1โˆ’๐‘ฆ๐‘ค๐‘˜โˆ’1๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค๐‘›๐‘˜โˆ’1๎ƒช+5๐‘‘6โˆš2๎ƒฉ๐œŒ๎€ท๐‘ก๐‘˜๎€ธโˆ’(1โˆ’๐‘ฆ)๐‘ค๐‘˜๐‘ โˆ’๐‘ฆ๐‘ค๐‘˜๐‘ +1(1โˆ’๐‘ฆ)๐‘ž๐‘ +๐‘ฆ๐‘ž๐‘ +1๐‘ž(๐‘›)+๐‘ค๐‘˜๐‘›๎ƒช๎‚€๐‘ก+๐‘“๐‘˜โˆ’๐œ2,๐‘ฅ๐‘›๎‚,๐‘ค1โ‰ค๐‘›โ‰ค๐‘€โˆ’1,๐‘˜0=๐‘ค๐‘˜๐‘€๐‘ค=0,3โ‰ค๐‘˜โ‰ค๐‘,0๐‘›๎€ท๐‘ฅ=sin๐‘›๎€ธ,0