Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

Abstract

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

1. Introduction

Nonlocal problems have been a major research area in modern physics, biology, chemistry, and engineering when it is impossible to determine the boundary values of the unknown function. Numerical methods and theory of solutions of the nonlocal boundary value problems for partial differential equations of variable type were carried out in for example, [1–10] and the references therein. Hyperbolic equations with nonlocal integral conditions are widely used for chemical heterogeneity, plasma physics, thermoelasticity, and so forth. The solutions of hyperbolic equations with nonlocal integral conditions were investigated in [11–15]. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been studied extensively (see, e.g., [16–28]).

In the present paper, the nonlocal boundary value problem for the multidimensional hyperbolic equation with nonlocal integral condition𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}(),𝑥=𝑓𝑡,𝑥𝑥=₁,…,𝑥_𝑚𝑢()=∈Ω,0<𝑡<1,0,𝑥¹₀𝛼𝜌𝑢𝜌,𝑥𝑑𝜌+𝜑(𝑥),𝑢_𝑡(0,𝑥)=𝜓(𝑥),𝑥∈𝑢()Ω,𝑡,𝑥=0,0<𝑡<1,𝑥∈𝑆 (1.1) is considered. Here Ω is the unit open cube in the 𝑚-dimensional Euclidean space ℝ^𝑚={𝑥=(𝑥₁,…,𝑥_𝑚)∶0<𝑥_𝑗<1,1≤𝑗≤𝑚} with boundary 𝑆, Ω=Ω∪𝑆, 𝑎_𝑟(𝑥)(𝑥∈Ω), 𝜑(𝑥), 𝜓(𝑥)(𝑥∈Ω), and 𝑓(𝑡,𝑥)(𝑡∈(0,1),𝑥∈Ω) are given smooth functions, and 𝑎_𝑟(𝑥)≥𝑎>0.

The first and second orders of approximation in t and the second order of approximation in space variables difference schemes for the approximate solution of nonlocal boundary value problem (1.1) are presented. Stability of these difference schemes and of the first- and second-order difference derivatives established. Error analysis is obtained by numerical solutions of one-dimensional hyperbolic equations with integral condition.

2. Difference Schemes and Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us define the grid sets Ω_ℎ=𝑥=𝑥_𝑟=ℎ₁𝑟₁,…,ℎ_𝑚𝑟_𝑚,𝑟=𝑟₁,…,𝑟_𝑚,0≤𝑟_𝑗≤𝑁_𝑗,ℎ_𝑗𝑁_𝑗,Ω=1,𝑗=1,…,𝑚_ℎ=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆. (2.1) We introduce the Hilbert space 𝐿_2ℎ=𝐿₂(Ω_ℎ) of the grid functions 𝜑^ℎ(𝑥)=𝜑ℎ₁𝑟₁,…,ℎ_𝑚𝑟_𝑚 (2.2) defined on Ω_ℎ, equipped with the norm 𝜑^ℎ_𝐿2(Ω_ℎ)=_𝑥∈Ωℎ𝜑^ℎ(𝑥)²ℎ₁,⋯,ℎ_𝑚^1/2. (2.3) To the differential operator 𝐴^𝑥 generated by problem (1.1), we assign the difference operator 𝐴^𝑥_ℎ by the formula𝐴^𝑥_ℎ𝑢^ℎ_𝑥=−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑢^ℎ_{𝑥𝑟𝑥𝑟𝑟𝑗} (2.4) acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the condition 𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆_ℎ. 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 problem𝑑²𝑣^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑣^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,𝑣^ℎ(0,𝑥)=¹₀𝛼𝜌𝑣^ℎ𝜌,𝑥𝑑𝜌+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑑𝑣^ℎ(0,𝑥)𝑑𝑡=𝜓^ℎ(𝑥),𝑥∈Ω_ℎ (2.5) for an infinite system of ordinary differential equations.

In the second step, we replace problem (2.5) by the difference scheme 𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘+1(𝑥)=𝑓^ℎ_𝑘+1(𝑥),𝑓^ℎ_𝑘+1(𝑥)=𝑓^ℎ𝑡_𝑘+1,𝑥,𝑡_𝑘+1=()Ω𝑘+1𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈_ℎ,𝑢^ℎ₀(𝑥)=^𝑁_𝑚=1𝛼𝑡_𝑚𝑢^ℎ_𝑚(𝑥)𝜏+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝐼+𝜏²𝐴^𝑥_ℎ𝑢^ℎ₁(𝑥)−𝑢^ℎ₀(𝑥)𝜏⁻¹=𝜓^ℎ(𝑥),𝑥∈Ω_ℎ (2.6) of the first order accuracy in t.

Theorem 2.1. Let 𝜏 and ℎ be sufficiently small numbers. Then, the solutions of the difference scheme (2.6) satisfy the following stability estimates:max_{0≤𝑘≤𝑁}𝑢^ℎ_𝑘𝐿2ℎ+max^𝑚_{0≤𝑘≤𝑁𝑟=1}𝑢^ℎ_{𝑘𝑥𝑟𝑟𝑗𝐿2ℎ}≤𝑀₁max_{1≤𝑘≤𝑁−1}𝑓^ℎ_𝑘𝐿2ℎ+𝜓^ℎ_𝐿2ℎ+^𝑚_𝑟=1𝜑^ℎ_{𝑥𝑟𝑟𝑗𝐿2ℎ},max_{1≤𝑘≤𝑁−1}𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_{𝑘−1𝐿2ℎ}+max^𝑚_{0≤𝑘≤𝑁𝑟=1}𝑢^ℎ_{𝑘𝑥𝑟𝑥𝑟𝑟𝑗𝐿2ℎ}≤𝑀₁𝑓^ℎ_1𝐿2ℎ+max_{2≤𝑘≤𝑁−1}𝜏⁻¹𝑓^ℎ_𝑘−𝑓^ℎ_{𝑘−1𝐿2ℎ}+^𝑚_𝑟=1𝜓^ℎ_{𝑥𝑟𝑟𝑗𝐿2ℎ}+^𝑚_𝑟=1𝜑^ℎ_{𝑥𝑟𝑥𝑟𝑟𝑗𝐿2ℎ}, (2.7) where 𝑀₁ does not depend on 𝜏, ℎ, 𝜑^ℎ(𝑥), 𝜓^ℎ(𝑥), and 𝑓^ℎ_𝑘(𝑥), 1≤𝑘<𝑁.

The proof of Theorem 2.1 is based on the symmetry property of difference operator 𝐴^𝑥_ℎ defined by the formula (2.4) and on the following theorem on coercivity inequality of the elliptic difference problem.

Theorem 2.2. For the solutions of the elliptic difference problem𝐴^𝑥_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(𝑥)=0,𝑥∈𝑆_ℎ, (2.8) the following coercivity inequality holds [29]: ^𝑚_𝑟=1𝑢^ℎ_{𝑥𝑟𝑥𝑟𝑟𝑗𝐿2ℎ}≤𝑀𝜔^ℎ_𝐿2ℎ. (2.9)

Moreover, the second order of accuracy difference schemes𝜏⁻²𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥+1)2𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)+14𝐴^𝑥_ℎ𝑢^ℎ_𝑘+1(𝑥)+𝑢^ℎ_𝑘−1(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑥∈Ω_ℎ,𝑓^ℎ_𝑘=𝑓^ℎ_𝑘𝑡_𝑘,𝑥,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,_ℎ0(𝑥)=^𝑁_𝑗=1𝜏2𝛼𝑡_𝑗𝑢^ℎ𝑡_𝑗,𝑥+𝛼𝑡_𝑗−1𝑢^ℎ𝑡_𝑗−1,𝑥+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝜏𝐼+²𝐴^𝑥_ℎ2𝜏⁻¹𝑢^ℎ₁(𝑥)−𝑢^ℎ₀(𝑥−𝜏)2𝑓^ℎ₀(𝑥)−𝐴^𝑥_ℎ𝑢^ℎ₀(𝑥)=𝜓^ℎ(𝑥),𝑓^ℎ₀=𝑓^ℎ(0,𝑥),𝑓^ℎ_𝑁=𝑓^ℎ(1,𝑥),𝑥∈Ω_ℎ, (2.10) and 𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−1+𝐴^𝑥_ℎ𝑢^ℎ_𝑘+𝜏²4𝐴^𝑥_ℎ²𝑢^ℎ_𝑘+1=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘=𝑓^ℎ_𝑘𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ₀(𝑥)=^𝑁_𝑗=1𝜏2𝛼𝑡_𝑗𝑢^ℎ𝑡_𝑗,𝑥+𝛼𝑡_𝑗−1𝑢^ℎ𝑡_𝑗−1,𝑥+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝜏𝐼+²𝐴^𝑥_ℎ4𝜏𝐼+²𝐴^𝑥_ℎ4𝜏⁻¹𝑢^ℎ₁(𝑥)−𝑢^ℎ₀(𝑥−𝜏)2𝑓^ℎ₀(𝑥)−𝐴^𝑥_ℎ𝑢^ℎ₀(𝑥)=𝜓^ℎ(𝑥),𝑓^ℎ₀=𝑓^ℎ(0,𝑥),𝑓^ℎ_𝑁=𝑓^ℎ(1,𝑥),𝑥∈Ω_ℎ (2.11) for approximately solving the boundary value problem (1.1) is presented.

We have the following theorem.

Theorem 2.3. Let 𝜏 and ℎ be sufficiently small numbers. Then, for the solution of the difference schemes (2.10) and (2.11) the stability inequalitiesmax_{0≤𝑘≤𝑁}𝑢^ℎ_𝑘𝐿2ℎ+max^𝑚_{0≤𝑘≤𝑁𝑟=1}𝑢^ℎ_{𝑘𝑥𝑟,𝑗𝑟𝐿2ℎ}≤𝑀₂max_{0≤𝑘≤𝑁}𝑓^ℎ_𝑘𝐿2ℎ+𝜓^ℎ_𝐿2ℎ+^𝑚_𝑟=1𝜑^ℎ_{𝑥𝑟,𝑗𝑟𝐿2ℎ},max_{1≤𝑘≤𝑁−1}𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_{𝑘−1𝐿2ℎ}≤𝑀₂𝑓^ℎ_0𝐿2ℎ+max_{1≤𝑘≤𝑁−1}𝜏⁻¹𝑓^ℎ_𝑘−𝑓^ℎ_{𝑘−1𝐿2ℎ}+^𝑚_𝑟=1𝜓^ℎ_{𝑥𝑟,𝑗𝑟𝐿2ℎ}+^𝑚_𝑟=1𝜓^ℎ_{𝑥𝑟,𝑥𝑟,𝑗𝑟𝐿2ℎ} (2.12) hold, where 𝑀₂ is independent of 𝜏, ℎ, 𝜑^ℎ(𝑥), 𝜓^ℎ(𝑥), and 𝑓^ℎ_𝑘(𝑥), 0≤𝑘≤𝑁.

The proof of Theorem 2.3 is based on the symmetry property of difference operator 𝐴^𝑥_ℎ defined by formula (2.4) and on Theorem 2.2 on coercivity inequality of elliptic difference problem (2.8).

In Theorems 2.1 and 2.3, the constants 𝑀₁ and 𝑀₂ cannot be obtained sharply. Therefore, in the following section, we will study the accuracy of these difference schemes for solving the one-dimensional hyperbolic equations with the integral condition. Moreover, the method is supported by numerical experiments.

3. Numerical Analysis

3.1. The First Order of Accuracy in Time Difference Scheme

In this section, the nonlocal boundary value problem𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−()𝜕1+𝑥²𝑢(𝑡,𝑥)𝜕𝑥²−𝑢_𝑥()()(),𝑓()=)]𝑒𝑡,𝑥+𝑢𝑡,𝑥=𝑓𝑡,𝑥𝑡,𝑥[(𝑥+2sin𝑥−cos𝑥^𝑡−1−𝑡+𝑒^𝑡𝑢()=sin𝑥,0<𝑡<1,0<𝑥<𝜋,0,𝑥¹₀𝑒^−𝑠𝑢(𝑠,𝑥)𝑑𝑠+𝜓(𝑥),𝜓(𝑥)=1−3𝑒⁻¹𝑢sin𝑥,_𝑡()𝑢()()0,𝑥=0,0≤𝑥≤𝜋,𝑡,0=𝑢𝑡,𝜋=0,0≤𝑡≤1 (3.1) for one dimensional hyperbolic equation is considered.

The exact solution of problem (3.1) is𝑢(𝑡,𝑥)=𝑒^𝑡−1−𝑡sin𝑥. (3.2)

Applying the formulas𝑢𝑥_𝑛+1−𝑢𝑥_𝑛−12ℎ−𝑢^′𝑥_𝑛=𝑂ℎ²,𝑢(𝜏)−𝑢(0)𝜏−𝑢^′(0)(𝜏),𝑢𝑥=𝑂_𝑛+1−2𝑢𝑥_𝑛+𝑢𝑥_𝑛−1ℎ²−𝑢^′′𝑥_𝑛=𝑂ℎ² (3.3) and using the first order of accuracy in 𝑡 implicit difference scheme (2.6), we obtain the difference scheme first order of accuracy in 𝑡 and second order of accuracy in 𝑥𝑢_𝑛^𝑘+1−2𝑢^𝑘_𝑛+𝑢_𝑛^𝑘−1𝜏²−1+𝑥_𝑛𝑢^𝑘+1_𝑛+1−2𝑢_𝑛^𝑘+1+𝑢^𝑘+1_𝑛−1ℎ²−𝑢^𝑘+1_𝑛+1−𝑢^𝑘+1_𝑛−12ℎ+𝑢_𝑛^𝑘+1=𝑓𝑡_𝑘+1,𝑥_𝑛,𝑓𝑡_𝑘+1,𝑥_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘+1−1−𝑡_𝑘+1+𝑒^𝑡𝑘+1sin𝑥_𝑛,𝑁𝜏=1,𝑥_𝑛𝑢=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=𝜋,_0𝑛−^𝑁_𝑘=1𝑒^−𝑘𝜏𝜏𝑢^𝑘_𝑛=𝜓𝑥_𝑛,𝑡_𝑘𝜓𝑥=𝑘𝜏,1≤𝑘≤𝑁−1,_𝑛=1−3𝑒⁻¹sin𝑥_𝑛𝑢,0≤𝑛≤𝑀,_𝑘0=𝑢^𝑘_𝑀=0,𝑢¹_𝑛−𝑢⁰_𝑛=0,0≤𝑘≤𝑁,1≤𝑛≤𝑀−1 (3.4) for approximate solutions of nonlocal boundary value problem (3.1). It can be written in the matrix form𝐴_𝑛𝑢_𝑛+1+𝐵_𝑛𝑢_𝑛+𝐶_𝑛𝑢_𝑛−1=𝐷𝜑_𝑛𝑢,1≤𝑛≤𝑀−1,₀=0,𝑢_𝑀=0. (3.5) Here𝐴_𝑛=0000⋅⋅⋅0000𝑎_𝑛0⋅⋅⋅00000𝑎_𝑛⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑎_𝑛00000⋅⋅⋅0𝑎_𝑛0000⋅⋅⋅00_{(𝑁+1)×(𝑁+1)},𝐵_𝑛=1−𝜏𝑒^−𝜏−𝜏𝑒^−2𝜏−𝜏𝑒^−3𝜏⋅⋅⋅−𝜏𝑒^{−(𝑁−1)𝜏}−𝜏𝑒^−𝑁𝜏𝑏𝑐𝑑_𝑛0⋅⋅⋅000𝑏𝑐𝑑_𝑛⋅⋅⋅0000𝑏𝑐⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑑_𝑛00000⋅⋅⋅𝑐𝑑_𝑛−1100⋅⋅⋅00_{(𝑁+1)×(𝑁+1)},𝐶_𝑛=0000⋅⋅⋅0000𝑒_𝑛0⋅⋅⋅00000𝑒_𝑛⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑒_𝑛00000⋅⋅⋅0𝑒_𝑛0000⋅⋅⋅00_{(𝑁+1)×(𝑁+1)},𝑎_𝑛=−1+𝑥_𝑛ℎ²−112ℎ,𝑏=𝜏²,𝑐=−2𝜏²,𝑑_𝑛=1𝜏²+21+𝑥_𝑛ℎ²+1,𝑒_𝑛=−1+𝑥_𝑛ℎ²+1,𝜑2ℎ_𝑛=𝜑⁰_𝑛𝜑¹_𝑛𝜑²_𝑛⋮𝜑^𝑁_{𝑛(𝑁+1)×1},𝜑_𝑛^𝑘+1=𝑓𝑡_𝑘+1,𝑥_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘−1−𝑡_𝑘+𝑒^𝑡𝑘sin𝑥_𝑛,𝑥_𝑛=𝑛ℎ,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1, (3.6) and 𝐷=𝐼_𝑁+1 is the identity matrix.𝑈_𝑠=𝑢⁰_𝑠𝑢¹_𝑠𝑢²_𝑠⋮𝑢^𝑁_{𝑠(𝑁+1)×1},𝑠=𝑛−1,𝑛,𝑛+1. (3.7) This type system was used by Samarskii and Nikolaev [30] for difference equations. For the solution of the matrix equation (3.5), we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form: 𝑢_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1,𝑛=𝑀−1,…,2,1, (3.8) where 𝑢_𝑀=0,  𝛼_𝑗(𝑗=1,…,𝑀−1) are (𝑁+1)×(𝑁+1) square matrices, 𝛽_𝑗(𝑗=1,…,𝑀−1) are (𝑁+1)×1column matrices,  𝛼₁,𝛽₁ are zero matrices, and𝛼_𝑛+1=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛⁻¹𝐴_𝑛,𝛽_𝑛+1=𝐵_𝑛+𝐶_𝑛𝛼_𝑛⁻¹𝐷_𝑛𝜑_𝑛−𝐶_𝑛𝛽_𝑛,𝑛=1,2,3,…,𝑀−1. (3.9)

3.2. The Second Order of Accuracy in Time Difference Scheme

Applying (3.3) and using the second order of accuracy in 𝑡 implicit difference scheme (2.10), we obtain the second order of accuracy difference scheme in 𝑡 and in 𝑥𝑢_𝑛^𝑘+1−2𝑢^𝑘_𝑛+𝑢^𝑘−1_𝑛−1𝜏²−1+𝑥_𝑛𝑢^𝑘−1_𝑛+1−2𝑢_𝑛^𝑘−1+𝑢^𝑘−1_𝑛−14ℎ²−𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−12ℎ²−𝑢^𝑘+1_𝑛+1−2𝑢_𝑛^𝑘+1+𝑢^𝑘+1_𝑛−14ℎ²−𝑢^𝑘−1_𝑛+1−𝑢^𝑘−1_𝑛−1+𝑢8ℎ^𝑘𝑛+1−𝑢^𝑘_𝑛−1+𝑢4ℎ^𝑘+1𝑛+1−𝑢^𝑘+1_𝑛−1+18ℎ2𝑢^𝑘_𝑛+𝑢_𝑛^𝑘+1+𝑢_𝑛^𝑘−14=𝜑^𝑘_𝑛,𝜑^𝑘_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘−1−𝑡_𝑘+𝑒^𝑡𝑘sin𝑥_𝑛,𝑀ℎ=𝜋,𝑥_𝑛=𝑛ℎ,1≤𝑛≤𝑀−1,𝑁𝜏=1,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,_0𝑛=^𝑁_𝑘=1𝜏2𝑒^−𝑘𝜏𝑢^𝑘_𝑛+𝑒^{−(𝑘−1)𝜏}𝑢_𝑛^𝑘−1+𝜓𝑥_𝑛𝜓𝑥,0≤𝑛≤𝑀,_𝑛=1−3𝑒⁻¹sin𝑥_𝑛𝑢,0≤𝑛≤𝑀,_1𝑛−𝑢⁰_𝑛=𝜏²2𝑢⁰_𝑛+1−𝑢⁰_𝑛−1+2ℎ1+𝑥_𝑛𝑢⁰_𝑛+1−2𝑢⁰_𝑛+𝑢⁰_𝑛−1ℎ²−𝑢⁰_𝑛+sin𝑥_𝑛𝑢,1≤𝑛≤𝑀−1,_𝑘0=𝑢^𝑘_𝑀=0,0≤𝑘≤𝑁 (3.10) for approximate solutions of the nonlocal boundary value problem (3.1). We have again (𝑁+1)×(𝑀+1) system of linear equations. We can write the system as a matrix equation (3.5).

Here𝐴_𝑛=𝑎0000⋅⋅⋅000_𝑛2𝑎_𝑛𝑎_𝑛0⋅⋅⋅0000𝑎_𝑛2𝑎_𝑛𝑎_𝑛⋅⋅⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑎_𝑛2𝑎_𝑛𝑎_𝑛𝑤_𝑛000⋅⋅⋅000_{(𝑁+1)×(𝑁+1)},𝐵_𝑛=𝜏1−2−𝜏𝑒^−𝜏−𝜏𝑒^−2𝜏⋅⋅⋅−𝜏𝑒^{−(𝑁−1)𝜏}−𝜏2𝑒^−𝑁𝜏𝑏_𝑛𝑑_𝑛𝑏_𝑛⋅⋅⋅000𝑏_𝑛𝑑_𝑛⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅𝑏_𝑛0000⋅⋅⋅𝑑_𝑛𝑏_𝑛𝑦_𝑛10⋅⋅⋅00_{(𝑁+1)×(𝑁+1)},𝐶_𝑛=𝑐0000⋅⋅⋅000_𝑛2𝑐_𝑛𝑐_𝑛0⋅⋅⋅0000𝑐_𝑛2𝑐_𝑛𝑐_𝑛⋅⋅⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑐_𝑛2𝑐_𝑛𝑐_𝑛𝑧_𝑛000⋅⋅⋅000_{(𝑁+1)×(𝑁+1)},𝑎_𝑛=−1+𝑥_𝑛4ℎ²−18ℎ,𝑏_𝑛=1𝜏²+1+𝑥_𝑛2ℎ²+14,𝑐_𝑛=−1+𝑥_𝑛4ℎ²+18ℎ,𝑑_𝑛=−2𝜏²+1+𝑥_𝑛ℎ²+12,𝑦_𝑛=−1+1+𝑥_𝑛𝜏²ℎ²+𝜏²2,𝑤_𝑛𝜏=−²−4ℎ1+𝑥_𝑛𝜏²2ℎ²,𝑧_𝑛=𝜏²−4ℎ1+𝑥_𝑛𝜏²2ℎ²,𝜑_𝑛=𝜑⁰_𝑛𝜑¹_𝑛⋮𝜑^𝑁_{𝑛(𝑁+1)×1},𝜑^𝑘_𝑛=𝑓𝑡_𝑘,𝑥_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘−1−𝑡_𝑘+𝑒^𝑡𝑘sin𝑥_𝑛,1≤𝑘≤𝑁−1,𝐷=𝐼_𝑁+1,𝑈_𝑠=𝑢⁰_𝑠𝑢¹_𝑠𝑢²_𝑠⋮𝑢^𝑁_{𝑠(𝑁+1)×1},𝑠=𝑛−1,𝑛,𝑛+1. (3.11) For the solution of the matrix equation (3.5), we used the same algorithm as in the first order of accuracy difference scheme.

3.3. The Second Order of Accuracy in Time Difference Scheme Generated by 𝐴²

Applying (3.3) and formulas𝑢𝑥_𝑛+2−4𝑢𝑥_𝑛+1+6𝑢𝑥_𝑛−4𝑢𝑥_𝑛−1+𝑢𝑥_𝑛−2ℎ⁴−𝑢^(𝑖𝑣)𝑥_𝑛=𝑂ℎ²,2𝑢(0)−5𝑢(ℎ)+4𝑢(2ℎ)−𝑢(3ℎ)ℎ²−𝑢^′′(0)=𝑂ℎ²,2𝑢(𝜋)−5𝑢(𝜋−ℎ)+4𝑢(𝜋−2ℎ)−𝑢(𝜋−3ℎ)ℎ²−𝑢^′′(𝜋)=𝑂ℎ² (3.12) and using difference scheme (2.11), we obtain the second order of accuracy difference scheme𝑢_𝑛^𝑘+1−2𝑢^𝑘_𝑛+𝑢_𝑛^𝑘−1𝜏²−1+𝑥_𝑛𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²−𝑢^𝑘_𝑛+1−𝑢^𝑘_𝑛−12ℎ+𝑢^𝑘_𝑛+𝜏²41+𝑥_𝑛²𝑢^𝑘+1_𝑛+2−4𝑢^𝑘+1_𝑛+1+6𝑢_𝑛^𝑘+1−4𝑢^𝑘+1_𝑛−1+𝑢^𝑘+1_𝑛−2ℎ⁴+41+𝑥_𝑛𝑢^𝑘+1_𝑛+2−2𝑢^𝑘+1_𝑛+1+2𝑢^𝑘+1_𝑛−1−𝑢^𝑘+1_𝑛−22ℎ³−2𝑥_𝑛𝑢^𝑘+1_𝑛+1−2𝑢_𝑛^𝑘+1+𝑢^𝑘+1_𝑛−1ℎ²−𝑢^𝑘+1_𝑛+1−𝑢^𝑘+1_𝑛−1ℎ+𝑢_𝑛^𝑘+1=𝜑^𝑘_𝑛,𝜑^𝑘_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘−1−𝑡_𝑘+𝑒^𝑡𝑘sin𝑥_𝑛,𝑀ℎ=𝜋,𝑥_𝑛=𝑛ℎ,2≤𝑛≤𝑀−2,𝑁𝜏=1,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,_0𝑛=^𝑁_𝑘=1𝜏2𝑒^−𝑘𝜏𝑢^𝑘_𝑛+𝑒^{−(𝑘−1)𝜏}𝑢_𝑛^𝑘−1=𝜓𝑥_𝑛𝜓𝑥,0≤𝑛≤𝑀,_𝑛=1−3𝑒⁻¹sin𝑥_𝑛𝑢,0≤𝑛≤𝑀,_1𝑛−𝑢⁰_𝑛=𝜏²2𝑢⁰_𝑛+1−𝑢⁰_𝑛−1+2ℎ1+𝑥_𝑛𝑢⁰_𝑛+1−2𝑢⁰_𝑛+𝑢⁰_𝑛−1ℎ²−𝑢⁰_𝑛+sin𝑥_𝑛𝑢,2≤𝑛≤𝑀−2,_𝑘0=𝑢^𝑘_𝑀𝑢=0,0≤𝑘≤𝑁,_𝑘1=45𝑢^𝑘₂−15𝑢^𝑘₃𝑢,0≤𝑘≤𝑁,_𝑘𝑀−1=45𝑢^𝑘_𝑀−2−15𝑢^𝑘_𝑀−3,0≤𝑘≤𝑁 (3.13) for approximate solutions of problem (3.1). One can write the (𝑁+1)×(𝑀+1) system of linear equations (3.13) as the matrix equation𝐴_𝑛𝑢_𝑛+2+𝐵_𝑛𝑢_𝑛+1+𝐶_𝑛𝑢_𝑛+𝐷_𝑛𝑢_𝑛−1+𝐸_𝑛𝑢_𝑛−2=𝑅𝜑_𝑛,𝑢₀=0,𝑢_𝑀=𝑢0,2≤𝑛≤𝑀−2,₁=45𝑢₂−15𝑢₃,𝑢_𝑀−1=45𝑢_𝑀−2−15𝑢_𝑀−3. (3.14)

Here, 𝐴_𝑛,𝐵_𝑛,𝐶_𝑛,  𝐷_𝑛,  𝐸_𝑛, and 𝑅 are (𝑁+1)×(𝑁+1) square matrices∶𝐴_𝑛=0000⋅⋅⋅000𝑎_𝑛0⋅⋅⋅0000𝑎_𝑛⋅⋅⋅0⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑎_𝑛0000⋅⋅⋅0,𝐵_𝑛=000⋅⋅⋅000𝑏_𝑛𝑐_𝑛⋅⋅⋅0000𝑏_𝑛⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅𝑐_𝑛0000⋅⋅⋅𝑏_𝑛𝑐_𝑛𝑦_𝑛,𝐶00⋅⋅⋅00_𝑛=𝜏1−2−𝜏𝑒^−𝜏−𝜏𝑒^−2𝜏⋅⋅⋅−𝜏𝑒^{−(𝑁−2)𝜏}−𝜏𝑒^{−(𝑁−1)𝜏}−𝜏2𝑒^−𝑁𝜏𝑑𝑒_𝑛𝑓_𝑛⋅⋅⋅0000𝑑𝑒_𝑛⋅⋅⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅𝑒_𝑛𝑓_𝑛0000⋅⋅⋅𝑑𝑒_𝑛𝑓_𝑛𝑤_𝑛,𝐷10⋅⋅⋅000_𝑛=000⋅⋅⋅0000𝑔_𝑛𝑙_𝑛⋅⋅⋅00000𝑔_𝑛⋅⋅⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅𝑔_𝑛𝑙_𝑛0000⋅⋅⋅0𝑔_𝑛𝑙_𝑛𝑧_𝑛,𝐸00⋅⋅⋅000_𝑛=0000⋅⋅⋅0000𝑚_𝑛0⋅⋅⋅00000𝑚_𝑛⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅0000⋅⋅⋅𝑚_𝑛00000⋅⋅⋅0𝑚_𝑛.0000⋅⋅⋅00 (3.15) We denote 𝑎_𝑛=𝜏²41+𝑥_𝑛²ℎ⁴+21+𝑥_𝑛ℎ³,𝑏_𝑛=−1+𝑥_𝑛ℎ²−1,12ℎ𝑑=𝜏²,𝑒_𝑛2=−𝜏²+21+𝑥_𝑛ℎ²+1,𝑔_𝑛=1−2ℎ1+𝑥_𝑛ℎ²,𝑐_𝑛=𝜏²4−41+𝑥_𝑛²ℎ⁴−41+𝑥_𝑛ℎ³−2𝑥_𝑛ℎ²−1ℎ,𝑙_𝑛=𝜏²4−41+𝑥_𝑛²ℎ⁴−41+𝑥_𝑛ℎ³−2𝑥_𝑛ℎ²+1ℎ,𝑓_𝑛=1𝜏²+𝜏²461+𝑥_𝑛²ℎ⁴+4𝑥_𝑛ℎ²,𝑚+1_𝑛=𝜏²41+𝑥_𝑛²ℎ⁴−21+𝑥_𝑛ℎ³,𝑧_𝑛=𝜏²−4ℎ1+𝑥_𝑛𝜏²2ℎ²,𝑤_𝑛=−1+1+𝑥_𝑛𝜏²ℎ²+𝜏²2,𝑦_𝑛𝜏=−²−4ℎ1+𝑥_𝑛𝜏²2ℎ²,𝜑_𝑛=𝜑⁰_𝑛𝜑¹_𝑛⋮𝜑^𝑁_{𝑛(𝑁+1)×1},𝜑^𝑘_𝑛=𝑓𝑡_𝑘,𝑥_𝑛=𝑥_𝑛+2sin𝑥_𝑛−cos𝑥_𝑛𝑒^𝑡𝑘−1−𝑡_𝑘+𝑒^𝑡𝑘sin𝑥_𝑛,1≤𝑘≤𝑁−1,𝑅=𝐼_𝑁+1istheidentitymatrix,𝑈_𝑠=𝑈⁰_𝑠𝑈¹_𝑠⋮𝑈^𝑁_{𝑠(𝑁+1)×1}, (3.16) where 𝑠=𝑛±2, 𝑛±1, 𝑛.

For the solution of matrix equation (3.14), we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following formula: 𝑈_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1𝑢_𝑛+2+𝛾_𝑛+1,𝑛=𝑀−2,…,2,1, (3.17) where 𝛼_𝑗,𝛽_𝑗(𝑗=1,…,𝑀−1) are (𝑁+1)×(𝑁+1) square matrices, 𝛾_𝑗(𝑗=1,…,𝑀−1) are (𝑁+1)×1column matrices, and 𝛼₁, 𝛽₁, 𝛾₁, 𝛾₂ are zero matrices. 𝛼₂, 𝛽₂ are𝛼₂=45𝐼_𝑁+1,𝛽₂1=−5𝐼_𝑁+1,𝐹_𝑛=𝐶_𝑛+𝐷_𝑛𝛼_𝑛+𝐸_𝑛𝛼_𝑛−1𝛼_𝑛+𝐸_𝑛𝛽_𝑛−1𝛼_𝑛+1=𝐹_𝑛⁻¹−𝐵_𝑛−𝐷_𝑛𝛽_𝑛−𝐸_𝑛𝛼_𝑛−1𝛽_𝑛,𝛽_𝑛+1=−𝐹_𝑛⁻¹𝐴_𝑛,𝛾_𝑛+1=𝐹_𝑛⁻¹𝑅𝜑_𝑛−𝐷_𝑛𝛾_𝑛+𝐸_𝑛𝛼_𝑛−1𝛾_𝑛+𝐸_𝑛𝛾_𝑛−1. (3.18) For solution of the last difference equation, we need to find 𝑢_𝑀, 𝑢_𝑀−1𝑢_𝑀=𝑢0,_𝑀−1=𝛽_𝑀−2+5𝐼−4𝐼−𝛼_𝑀−2𝛼_𝑀−1⁻¹4𝐼−𝛼_𝑀−2𝛾_𝑀−1−𝛾_𝑀−2. (3.19)

3.4. Error Analysis

The errors are computed by 𝐸^𝑁_𝑀=max_{1≤𝑘≤𝑁−1,1≤𝑛≤𝑀−1}𝑢𝑡_𝑘,𝑥_𝑛−𝑢^𝑘_𝑛 (3.20) of the numerical solutions, where 𝑢(𝑡_𝑘,𝑥_𝑛) represents the exact solution and 𝑢^𝑘_𝑛 represents the numerical solution at (𝑡_𝑘,𝑥_𝑛) and the results are given in Table 1.

Table 1 

Thus, the results show that the second order of accuracy difference schemes (3.10) and (3.13) are more accurate comparing with the first order of accuracy difference scheme (3.4).

Acknowledgment

The authors would like to thank Professor P. E. Sobolevskii (Jerusalem, Israel), for the helpful suggestions to the improvement of this paper.