The Numerical Solution of the Bitsadze-Samarskii Nonlocal Boundary Value Problems with the Dirichlet-Neumann Condition

Abstract

We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem with the Dirichlet-Neumann condition for the multidimensional elliptic equation. The first and second orders of accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

1. Introduction

Methods of solution of the Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [1–22] and the references given therein).

Let  Ω be the unit open cube in 𝑅^𝑚(𝑥=(𝑥₁,…,𝑥_𝑚)∶0<𝑥_𝑘<1,1≤𝑘≤𝑚) with boundary 𝑆,Ω=Ω∪𝑆. In [0,1]×Ω, the Bitsadze-Samarskii-type nonlocal boundary value problem for the multidimensional elliptic equation −𝑢_𝑡𝑡−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}+𝜂𝑢=𝑓(𝑡,𝑥),0<𝑡<1,𝑥=𝑥₁,…,𝑥_𝑚𝑢()(𝑥)()=∈Ω,0,𝑥=𝜑,𝑢1,𝑥^𝐽_𝑗=1𝛼_𝑗𝑢𝜆_𝑗,𝑥+𝜓(𝑥),𝑥∈Ω,_𝐽𝑗=1𝛼_𝑗≤1,0<𝜆₁<⋯<𝜆_𝐽𝑢()∣<1,𝑡,𝑥_𝑥∈𝑆1()=0,𝜕𝑢𝑡,𝑥𝜕^→𝑛_𝑥∈𝑆²=0,𝑆¹∪𝑆²=𝑆 (1.1) is considered. Here 𝑎_𝑟(𝑥),(𝑥∈Ω),𝜓(𝑥),𝜑(𝑥)(𝑥∈Ω), and 𝑓(𝑡,𝑥)(𝑡∈(0,1),𝑥∈Ω) are given smooth functions, 𝑎_𝑟(𝑥)≥𝑎>0, 𝜂 is a positive number, and ^→𝐧 is the normal vector to Ω. We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem (1). The first and second orders of accuracy difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of two-dimensional elliptic partial differential equations.

2. Difference Schemes: The Stability and Coercive Stability Estimates

The discretization of problem (1) is carried out in two steps. In the first step, let us define the grid sets Ω_ℎ=𝑥=𝑥_𝑚=ℎ₁𝑚₁,…,ℎ_𝑚𝑚_𝑚,𝑚=𝑚₁,…,𝑚_𝑚,0≤𝑚_𝑟≤𝑁_𝑟,ℎ_𝑟𝑁_𝑟,Ω=1,𝑟=1,…,𝑚_ℎ=Ω_ℎ∩Ω,𝑆^𝑟_ℎ=Ω_ℎ∩𝑆^𝑟,𝑟=1,2. (2.1) We introduce the Hilbert space 𝐿_2ℎ=𝐿₂(Ω_ℎ) and 𝑊²_2ℎ=𝑊²₂(Ω_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑(ℎ₁𝑚₁,…,ℎ_𝑚𝑚_𝑚)} defined on Ω_ℎ, equipped with the norms 𝜑^ℎ_𝑊22ℎ=𝜑^ℎ_𝐿2ℎ+_𝑥∈Ωℎ^𝑚_𝑟=1𝜑^ℎ_𝑥𝑟²ℎ₁⋯ℎ_𝑚^1/2+_𝑥∈Ωℎ^𝑚_𝑟=1𝜑^ℎ_{𝑥𝑟𝑥𝑟,𝑚𝑟}²ℎ₁⋯ℎ_𝑚^1/2,𝜑^ℎ_𝐿2(Ω_ℎ)=_𝑥∈Ωℎ𝜑^ℎ(𝑥)²ℎ₁⋯ℎ_𝑚^1/2. (2.2) To the differential operator 𝐴 generated by problem (1), we assign the difference operator 𝐴^𝑥_ℎ by the formula 𝐴^𝑥_ℎ𝑢^ℎ=−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑢^ℎ_{𝑥𝑟𝑥𝑟,𝑗𝑟}+𝜂𝑢^ℎ(𝑥), (2.3) acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the conditions 𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆¹_ℎ and 𝐷_ℎ𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆²_ℎ. Here, 𝐷_ℎ𝑢^ℎ(𝑥) is an approximation to 𝜕𝑢/𝜕^→𝐧. It is known that 𝐴^𝑥_ℎ is a self-adjoint positive definite operator in 𝐿₂(Ω_ℎ). With the help of 𝐴^𝑥_ℎ, we arrive at the nonlocal boundary value problem −𝑑²𝑢^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,𝑢^ℎ(0,𝑥)=𝜑^ℎ(𝑥),𝑢^ℎ(1,𝑥)=^𝐽_𝑗=1𝛼_𝑗𝑢^ℎ𝜆_𝑗,𝑥+𝜓^ℎ(𝑥),𝑥∈Ω_ℎ,_𝐽𝑗=1𝛼_𝑗≤1,0<𝜆₁<⋯<𝜆_𝐽<1, (2.4) for an infinite system of ordinary differential equations. In the second step, we replace problem (2.4) by the first and second orders of accuracy difference schemes −𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ₀(𝑥)=𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=^𝐽_𝑗=1𝛼_𝑗𝑢^ℎ_{[𝜆𝑗/𝜏]}(𝑥)+𝜓^ℎ(𝑥),𝑥∈Ω_ℎ,−𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑡,𝑥_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ₀(𝑥)=𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=^𝐽_𝑗=1𝛼_𝑗𝑢^ℎ_{[𝜆𝑗/𝜏]}(𝑥)+𝑢^ℎ_{[𝜆𝑗/𝜏]+1}(𝑥)−𝑢^ℎ_{[𝜆𝑗/𝜏]}(𝑥)𝜆_𝑗𝜏−𝜆_𝑗𝜏+𝜓^ℎ(𝑥),𝑥∈Ω_ℎ. (2.5)(2.6) To formulate our result on well-posedness, we will give definition of 𝐶^𝛼₀₁([0,1]_𝜏,𝐻) and 𝐶([0,1]_𝜏,𝐻). Let 𝐹([0,1]_𝜏,𝐻) be the linear space of mesh functions 𝜑^𝜏={𝜑_𝑘}₁^𝑁−1 with values in the Hilbert space 𝐻. We denote 𝐶([0,1]_𝜏,𝐻) normed space with the norm 𝜑^𝜏_{𝐶([0,1]𝜏,𝐻)}=max_{1≤𝑘≤𝑁−1}𝜑_𝑘𝐻, (2.7) and 𝐶^𝛼₀₁([0,1]_𝜏,𝐻) normed space with the norm 𝜑^𝜏_{𝐶𝛼01([0,1]𝜏,𝐻)}=𝜑^𝜏_{𝐶([0,1]𝜏,𝐻)}+sup_{1≤𝑘≤𝑘+𝑟≤𝑁−1}((𝑁−𝑘)𝜏)^𝛼((𝑘+𝑟)𝜏)^𝛼(𝑟𝜏)^𝛼𝜑_𝑘+𝑟−𝜑_𝑘𝐻. (2.8)

Theorem 2.1. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, the solutions of difference schemes (2.5) and (2.6) satisfy the following stability and almost coercive stability estimates 𝑢^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)}≤𝑀₁𝜑^ℎ_𝐿2ℎ+𝜓^ℎ_𝐿2ℎ+𝑓^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)},𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1𝜏²₁^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝑊22ℎ)}≤𝑀₂𝜑^ℎ_𝑊22ℎ+𝜓^ℎ_𝑊22ℎ1+ln|ℎ|𝑓𝜏+^{ℎ𝑘1𝑁−1𝐶([0,1]}_𝜏^,𝐿_2ℎ⁾. (2.9) Here, 𝑀₁ and 𝑀₂ do not depend on𝜏,ℎ,𝜓^ℎ(𝑥),𝜑^ℎ(𝑥), and 𝑓^ℎ_𝑘(𝑥),1≤𝑘≤𝑁−1.

Theorem 2.2. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, the solution of difference schemes (2.5) and (2.6) satisfies the following coercive stability estimate: 𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1𝜏²₁^𝑁−1_{𝐶𝛼01([0,1]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝐶𝛼01([0,1]𝜏,𝑊22ℎ)}≤𝑀₃𝜑^ℎ_𝑊22ℎ+𝜓^ℎ_𝑊22ℎ+1𝛼()𝑓1−𝛼^{ℎ𝑘1𝑁−1𝐶𝛼}₀₁^([0,1]_𝜏^,𝐿_2ℎ⁾. (2.10)𝑀₃ is independent of 𝜏,ℎ,𝜓^ℎ(𝑥),𝜑^ℎ(𝑥), and 𝑓^ℎ_𝑘(𝑥),1≤𝑘≤𝑁−1.

Proofs of Theorems 2.1 and 2.2 are based on the symmetry properties of operator 𝐴^𝑥_ℎ defined by formula (2.3) and on the following formulas: 𝑢^ℎ_𝑘(𝑥)=𝐼−𝑅^2𝑁−1×𝑅^𝑘−𝑅^2𝑁−𝑘𝜑^ℎ(𝑥)+𝑅^𝑁−𝑘−𝑅^𝑁+𝑘𝑢^ℎ_𝑁(𝑥)−𝑅^𝑁−𝑘−𝑅^𝑁+𝑘×(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+i𝑓^ℎ_𝑖(𝑥)𝜏+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑖=1𝑅^{|𝑘−𝑖|}−𝑅^𝑘+𝑖𝑓^ℎ_𝑖(𝑥)𝑢𝜏,_ℎ𝑁(𝑥)=𝐷^−1𝐽_𝑘=1𝛼_𝑘𝐼−𝑅^2𝑁−1×𝑅^{[𝜆𝑘/𝜏]}−𝑅^{2𝑁−[𝜆𝑘/𝜏]}𝜑^ℎ(𝑥)−𝑅^{𝑁−[𝜆𝑘/𝜏]}−𝑅^{𝑁+[𝜆𝑘/𝜏]}(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵⁻¹×^𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵⁻¹×^{[𝜆𝑘/𝜏]}_𝑖=1𝑅^{[𝜆𝑘/𝜏]−𝑖}𝑓^ℎ_𝑖(𝑥)𝜏+^𝑁−1_{𝜆𝑖=𝑘/𝜏+1}𝑅^{𝑖−[𝜆𝑘/𝜏]}𝑓^ℎ_𝑖(𝑥)𝜏−^𝑁−1_𝑖=1𝑅^{[𝜆𝑘/𝜏]+𝑖}𝑓^ℎ_𝑖(𝑥)𝜏+𝜓^ℎ(𝑥), (2.11) for difference scheme (2.5), and 𝑢^ℎ_𝑁(𝑥)=𝐷^−1𝐽_𝑘=1𝛼_𝑘𝐼−𝑅^2𝑁−1×𝑅^{[𝜆𝑘/𝜏]}−𝑅^{2𝑁−[𝜆𝑘/𝜏]}𝜑^ℎ(𝑥)−𝑅^{𝑁−[𝜆𝑘/𝜏]}−𝑅^{𝑁+[𝜆𝑘/𝜏]}(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹×𝐵^−1𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵⁻¹×^{[𝜆𝑘/𝜏]}_𝑖=1𝑅^{[𝜆𝑘/𝜏]−𝑖}𝑓^ℎ_𝑖(𝑥)𝜏+^𝑁−1_{𝜆𝑖=𝑘/𝜏+1}𝑅^{𝑖−[𝜆𝑘/𝜏]}𝑓^ℎ_𝑖(𝑥)𝜏−^𝑁−1_𝑖=1𝑅^{[𝜆𝑘/𝜏]+𝑖}𝑓^ℎ_𝑖(𝑥)𝜏+𝜆_𝑘𝜏−𝜆_𝑘𝜏𝐼−𝑅^2𝑁−1×𝜏𝐵𝑅^{𝑘[𝜆/𝜏]+1}−𝑅^{2𝑁−[𝜆𝑘/𝜏]}𝜑^ℎ(𝑥)−𝑅^{𝑁−[𝜆𝑘/𝜏]−1}−𝑅^{𝑁+[𝜆𝑘/𝜏]}(2𝐼+𝜏𝐵)⁻¹×^𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏²+(2𝐼+𝜏𝐵)^{−1[𝜆𝑘/𝜏]}_𝑖=1𝑅^{[𝜆𝑘/𝜏]−𝑖}𝑓^ℎ_𝑖(𝑥)𝜏²+^𝑁−1_{𝜆𝑖=𝑘/𝜏+1}𝑅^{𝑖−[𝜆𝑘/𝜏]−1}𝑓^ℎ_𝑖(𝑥)𝜏²−^𝑁−1_𝑖=1𝑅^{[𝜆𝑘/𝜏]+𝑖}𝑓^ℎ_𝑖(𝑥)𝜏²+𝜓^ℎ(𝑥), (2.12) for difference scheme (2.6). Here, 𝑅=(𝐼+𝜏𝐵)⁻¹,𝐵=𝜏𝐴2+𝜏²𝐴²4+𝐴,𝐴=𝐴^𝑥_ℎ,𝐷=𝐼−𝑅^2𝑁−^𝐽_𝑘=1𝛼_𝑘𝑅^{𝑘𝑁−[𝜆/𝜏]}−𝑅^{𝑘𝑁+[𝜆/𝜏]}for(2.5),𝐷=𝐼−𝑅^2𝑁−^𝐽_𝑘=1𝛼_𝑘𝑅^{𝑘𝑁−[𝜆/𝜏]}−𝑅^{𝑘𝑁+[𝜆/𝜏]}−1𝜏𝜆_𝑘−𝜆_𝑘𝜏𝜏×𝐵𝑅^{𝑁−[𝜆𝑘/𝜏]}−𝑅^{𝑁+[𝜆𝑘/𝜏]+1}for(2.6), (2.13) and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in 𝐿_2ℎ.

Theorem 2.3 (see [22]). For the solution of the elliptic difference problem 𝐴^𝑥_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(𝑥)|_𝑥∈𝑆¹_ℎ=0,𝐷_ℎ𝑢^ℎ(𝑥)|_𝑥∈𝑆²_ℎ=0,𝑆¹_ℎ∪𝑆²_ℎ=𝑆_ℎ, (2.14) the following coercivity inequality holds: ^𝑚_𝑟=1𝑢^ℎ_{𝑥𝑟𝑥𝑟,𝑗𝑟𝐿2ℎ}≤𝑀₄𝜔^ℎ_𝐿2ℎ, (2.15) where 𝑀₄ does not depend on ℎ and 𝜔^ℎ(𝑥).

Note that we have not been able to obtain sharp estimate for the constants figuring in the stability estimates. Hence, in the following section, we study difference schemes (2.5) and (2.6) by numerical experiments.

3. Numerical Results

For the numerical result, we consider the nonlocal boundary value problem −𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−𝜕²𝑢(𝑡,𝑥)𝜕𝑥²()1+𝑢=2exp−𝑡𝑥−2𝑥²+𝑡2,𝑢()−10<𝑡<1,0<𝑥<1,0,𝑥=𝑥²𝑢()1−2𝑥,1,𝑥=𝑢2+𝑥,𝑥²2−𝑥exp(−1)−3𝑥²4−3𝑥2−1exp2𝑢(),0≤𝑥≤1,𝑡,0=𝑢_𝑥(𝑡,1)=0,0≤𝑡≤1, (3.1) for the elliptic equation. The exact solution of (3.1) is 𝑢(𝑡,𝑥)=𝑡𝑥−𝑡𝑥²2+𝑥²−2𝑥exp(−𝑡). (3.2) For the approximate solution of the nonlocal boundary Bitsadze-Samarskii problem (3.1), we consider the set [0,1]_𝜏×[0,1]_ℎ of a family of grid points depending on the small parameters 𝜏 and ℎ[0,1]_𝜏×[0,1]_ℎ=𝑡_𝑘,𝑥_𝑛∶𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁_𝜏𝑥=1_𝑛=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=1}. (3.3) Firstly, applying difference scheme (2.5), we present the first order of accuracy difference scheme for the approximate solution of problem (3.1) is −𝑢_𝑛^𝑘+1−2𝑢^𝑘_𝑛+𝑢_𝑛^𝑘−1𝜏²−𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝑢^𝑘_𝑛=𝑓𝑡_𝑘,𝑥_𝑛,𝑢1≤𝑘≤𝑁−1,1≤𝑛≤𝑀−1,_0𝑛=𝜑𝑥_𝑛𝑢,0≤𝑛≤𝑀,_𝑁𝑛=𝑢_𝑛^[𝑁/2]+𝑥²_𝑛2−𝑥_𝑛()−exp−13𝑥²_𝑛4−3𝑥_𝑛2−1exp2𝑢,0≤𝑛≤𝑀,_𝑘0=𝑢^𝑘_𝑀−𝑢^𝑘_𝑀−1ℎ𝑓𝑡=0,0≤𝑘≤𝑁,_𝑘,𝑥_𝑛=2exp−𝑡_𝑘𝑥_𝑛−𝑥²_𝑛2+𝑡_𝑘2,𝜑𝑥−1_𝑛=𝑥²_𝑛−2𝑥_𝑛. (3.4) Then, we have an (𝑁+1)×(𝑀+1) system of linear equations and we will write them in the matrix form 𝐴𝑈_𝑛+1+𝐵𝑈_𝑛+𝐶𝑈_𝑛−1=𝐷𝜑_𝑛𝑈,1≤𝑛≤𝑀−1,₀=0,𝑈_𝑀−𝑈_𝑀−1=0, (3.5) where 𝐴=000⋅0⋅0000𝑎0⋅0⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅0⋅0𝑎0000⋅0⋅000_{(𝑁+1)×(𝑁+1)},𝐵=100⋅0⋅000𝑐𝑏𝑐⋅0⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅0⋅𝑐𝑏𝑐000⋅−1⋅001_{(𝑁+1)×(𝑁+1)}, (3.6) and 𝐶=𝐴,𝐷 is an (𝑁+1)×(𝑁+1) identity matrix and 𝑈_𝑠=𝑢⁰_𝑠𝑢¹_𝑠⋅𝑢_𝑠^𝑁−1𝑢^𝑁_{𝑠(𝑁+1)×1}, (3.7) where 𝑠=𝑛−1,𝑛,𝑛+1, 𝜑_𝑛=𝜑⁰_𝑛𝜑¹_𝑛⋅𝜑_𝑛^𝑁−1𝜑^𝑁_{𝑛(𝑁+1)×1}. (3.8) Here, 1𝑎=−ℎ²2,𝑏=𝜏²+2ℎ²1+1,𝑐=−𝜏²,𝜑^𝑘_𝑛=𝑥²_𝑛−2𝑥_𝑛𝑓𝑡,𝑘=0,_𝑘,𝑥_𝑛𝑥,1≤𝑘≤𝑁−1,_2𝑛2−𝑥_𝑛exp(−1)−3𝑥²_𝑛4−3𝑥_𝑛2−1exp2,𝑘=𝑁. (3.9) So, we have a second-order difference equation with respect to 𝑛 matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to 𝑛 matrix coefficients. Hence, we seek a solution of the matrix equation in the following form: 𝑈_𝑗=𝛼_𝑗+1𝑈_𝑗+1+𝛽_𝑗+1𝑈,𝑗=𝑀−1,…,1,_𝑀=𝐼−𝛼_𝑀⁻¹𝛽_𝑀,𝛼_𝑗+1=−𝐵+𝐶𝛼_𝑗⁻¹𝛽𝐴,_𝑗+1=𝐵+𝐶𝛼_𝑗⁻¹𝐷_𝜑𝑗−𝐶𝛽_𝑗,𝑗=1,…,𝑀−1, (3.10) where 𝛼_𝑗(𝑗=1,…,𝑀) are (𝑁+1)×(𝑁+1) square matrix and 𝛽_𝑗(𝑗=1,…,𝑀) are (𝑁+1)×1 column matrix and 𝛼₁ is the (𝑁+1)×(𝑁+1) zero matrix and 𝛽_𝑗 is the (𝑁+1)×1 zero matrix. Secondly, applying difference scheme (2.6), we present the following second order of accuracy difference scheme for the approximate solutions of problem (3.1): −𝑢_𝑛^𝑘+1−2𝑢^𝑘_𝑛+𝑢_𝑛^𝑘−1𝜏²−𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝑢^𝑘_𝑛=𝑓𝑡_𝑘,𝑥_𝑛,𝑢1≤𝑘≤𝑁−1,1≤𝑛≤𝑀−1,_0𝑛=𝜑𝑥_𝑛𝑢,0≤𝑛≤𝑀,_𝑁𝑛=𝑢_𝑛^[𝑁/2]+𝑢_𝑛^[𝑁/2]+1−𝑢_𝑛^[𝑁/2]𝑁2−𝑁2+𝑥²_𝑛2−𝑥_𝑛()−exp−13𝑥²_𝑛4−3𝑥_𝑛2−1exp2𝑢,0≤𝑛≤𝑀,_𝑘0=0,𝑢^𝑘_𝑀−2−4𝑢^𝑘_𝑀−1+3𝑢^𝑘_𝑀𝑓𝑡=0,0≤𝑘≤𝑁,_𝑘,𝑥_𝑛=2exp−𝑡_𝑘𝑥_𝑛−𝑥²_𝑛2+𝑡_𝑘2,𝜑𝑥−1_𝑛=𝑥²_𝑛−2𝑥_𝑛. (3.11) So, we have again an (𝑁+1)×(𝑀+1) system of linear equations and we will write in the matrix form 𝐴𝑈_𝑛+1+𝐵𝑈_𝑛+𝐶𝑈_𝑛−1=𝑅𝜑_𝑛𝑈,1≤𝑛≤𝑀−1,₀=0,𝑈_𝑀−2−4𝑈_𝑀−1+3𝑈_𝑀=0, (3.12) where 𝐴=000⋅00⋅0000𝑎0⋅00⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅00⋅0𝑎0000⋅00⋅000_{(𝑁+1)×(𝑁+1)},𝐵=100⋅00⋅000𝑐𝑏𝑐⋅00⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅00⋅𝑐𝑏𝑐000⋅𝑑𝑒⋅001_{(𝑁+1)×(𝑁+1)},𝑈𝐶=𝐴,𝑅=𝐷,_𝑠=𝑢⁰_𝑠𝑢¹_𝑠⋅𝑢_𝑠^𝑁−1𝑢^𝑁_{𝑠(𝑁+1)×1}, (3.13) where 𝑠=𝑛−1,𝑛,𝑛+1 and 𝜑_𝑛=^{𝜑0𝑛𝜑1𝑛⋅}_{𝜑𝑛𝑁−1𝜑𝑁𝑛(𝑁+1)×1}.

Here, 1𝑎=−ℎ²2,𝑏=ℎ²+2𝜏²1+1,𝑐=−𝜏²𝑁,𝑑=2−𝑁2𝜑,𝑒=−1−𝑑,_𝑘𝑛=𝑥²_𝑛−2𝑥_𝑛𝑓𝑡,𝑘=0,_𝑘,𝑥_𝑛𝑥,1≤𝑘≤𝑁−1,_2𝑛2−𝑥_𝑛exp(−1)−3𝑥²_𝑛4−3𝑥_𝑛2−1exp2,𝑘=𝑁. (3.14) Thus, we have a second-order difference equation with respect to 𝑛 matrix coefficients. To solve this difference equation, we have applied the same procedure of modified Gauss elimination method (3.10) for difference equation with respect to 𝑛 matrix coefficients with 𝑈_𝑀=3𝐼+𝛼_𝑀𝛼_𝑀−1−4𝛼_𝑀⁻¹−𝛽_𝑀𝛼_𝑀−1−𝛽_𝑀−1+4𝛽_𝑀. (3.15) Now, we will give the results of the numerical analysis. The errors computed by 𝐸^𝑁_𝑀=max_{1≤𝑘≤𝑁−1}^𝑀−1_𝑛=1𝑢(𝑡_𝑘,𝑥_𝑛)−𝑢^𝑘_𝑛²ℎ^1/2 (3.16) of the numerical solutions for different values of 𝑀 and 𝑁, where 𝑢(𝑡_𝑘,𝑥_𝑛) represents the exact solution and 𝑢^𝑘_𝑛 represents the numerical solution at (𝑡_𝑘,𝑥_𝑛). Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes for 𝑁=𝑀=20, 40, and 60, respectively.

Table 1 

Error analysis.

4. Conclusion

In this work, the first and second orders of accuracy difference schemes for the approximate solution of the Bitsadze-Samarskii nonlocal boundary value problem for elliptic equations are presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations are proved. The theoretical statements for the solution of these difference schemes are supported by the results of numerical examples. The second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme. As a future work, high orders of accuracy difference schemes for the approximate solutions of this problem could be established. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations could be proved.