Finite Difference Method for the Reverse Parabolic Problem

Abstract

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.

1. Introduction

In the study of boundary value problems for partial differential equations, the role played by the well-posedness (coercivity inequalities) is well known (see, e.g., [1–3]). Well-posedness of nonlocal boundary value problems for partial differential equations of parabolic type has been studied extensively by many researchers (see, e.g., [4–15] and the references therein).

In the paper [4], Ashyralyev studied the positivity of second-order differential and difference operators with nonlocal condition and the structure of interpolation spaces generated by these operators in a Banach space. Applying this result, he obtained the coercive inequalities for the solutions of the nonlocal boundary value problem for differential and difference equations.

In [5], Ashyralyev et al. considered the nonlocal boundary value problem 𝑣^′(𝑡)+𝐴𝑣(𝑡)=𝑓(𝑡),0<𝑡<1,𝑣(0)=𝑣(𝜆)+𝜇,0<𝜆≤1, (1.1) in a Banach space with strongly positive operator 𝐴. They established the well-posedness of problem (1.1) in Hölder spaces. Moreover, they obtained the exact Schauder's estimates in Hölder norms of solutions of the boundary values problem for 2𝑚-th order multidimensional parabolic equations.

Ashyralyev established in [6] the well-posedness of the nonlocal boundary-value problem (1.1) in Bochner spaces. He considered the first and second order of accuracy difference schemes for the approximate solutions of problem (1.1). He also established the coercive inequalities for the solutions of these difference schemes. Moreover, in applications, he obtained the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation.

Clément and Guerre-Delabriére studied in [8] maximal regularity (in the 𝐿_𝑝-sense) for abstract Cauchy problems of order one and boundary value problems of order two. As is well-known regularity of the first problems implies regularity of the second ones; they also proved that the converse to hold if the underlying Banach space has the UMD property. A stronger notion of regularity, which is introduced by Sobolevskii, plays an important role in the proofs.

In [9], Gulin et al. considered the linear heat equation: 𝑢_𝑡=𝑢_𝑥𝑥+𝑢_𝑦𝑦,0<𝑥<1,0<𝑦<1, (1.2) with Dirichlet condition 𝑢(𝑥,0,𝑡)=𝑢(𝑥,1,𝑡)=0, 0≤𝑥≤1 and nonlocal boundary conditions 𝑢(0,𝑦,𝑡)=0, 𝑢_𝑥(0,𝑦,𝑡)=𝑢_𝑥(1,𝑦,𝑡), 0≤𝑦≤1. They constructed an explicit difference scheme with second order of approximation with respect to the space variables and first order of approximation with respect to 𝑡. Moreover, using previous results of Ionkin and Morozova for the one-dimensional heat equation with nonlocal boundary conditions, they proved the stability of this scheme with respect to the 𝐷₁-norm ‖𝑦‖_𝐷1=(𝐷₁𝑦,𝑦)^1/2, which is induced by the symmetric and positive-definite matrix 𝐷₁.

Liu et al. studied in [10] a finite difference method for multidimensional nonlinear coupled system of parabolic and hyperbolic equations. By using a variational method, they obtained an a priori estimate. They also proved that the finite difference scheme is uniquely solvable and unconditionally stable. To support the theory, they gave numerical example of two-dimensional problem.

In [11, 12], Martin-Vaquero and Vigo-Aguiar provided algorithms improving the CPU time and accuracy of Crandall's formula. They studied the convergence of the algorithms and compared the efficiency of the methods with well-known numerical examples.

In [13], Sapagovas applied finite difference approximations to a nonhomogeneous heat equation in one space dimension, subject to nonlocal boundary conditions. He presented a stable difference approximation for the equation and a piecewise constant discretization of the integrals appearing in the boundary conditions. He discussed the stability of the complete problem with respect to two parameters included in the integral terms. He constructed a stability region in the plane of the parameters and gave practical examples with specific choices of the integral conditions. Sapagovas investigated in [14] the stability of implicit difference schemes for the equation of a thermoelastic rod, which is a parabolic equation subject to integral conditions for the boundaries.

In [15], Shakhmurov dealt with a nonlocal boundary value problem for a degenerate equation in a Banach space 𝐸 with unbounded operators in 𝐸. He proved the maximal 𝐿_𝑝 regularity and Fredholmness of the problem. He also applied the results to nonlocal boundary value problems for degenerate elliptic and quasielliptic differential equations and their finite or infinite systems on cylindrical domains.

It is well known that reverse problems arise in various applications, for example, boundary layer problems in fluid dynamics [16, 17], plasma physics, and astrophysics in the study of propagation of an electron beam through the solar corona [18]. For further applications of such problems, we refer the reader to [19–22] and the references therein.

In the paper [23], Ashyralyev et al. considered the multipoint nonlocal boundary value problem for reverse parabolic equations 𝑑𝑢(𝑡)(𝑡)(𝑡),(),𝑢(1)=𝑑𝑡−𝐴𝑢=𝑓0≤𝑡≤1_𝑝𝑘=1𝛼_𝑘𝑢𝜃_𝑘+𝜑,0≤𝜃₁<𝜃₂<⋯<𝜃_𝑝<1 (1.3) in a Hilbert space 𝐻 with self-adjoint positive definite operator 𝐴.

𝑢(𝑡) is called a solution of problem (1.3) if the following conditions hold: (1)𝑢(𝑡) is continuously differentiable on the segment [0,1]. The derivatives at the end points of the segment are understood as the appropriate unilateral derivatives.(2)The element 𝑢(𝑡) belongs to 𝐷(𝐴) for all 𝑡∈[0,1] and the function 𝐴𝑢(𝑡) is continuous on the segment [0,1].(3)𝑢(𝑡) satisfies the equation and the nonlocal boundary conditions (1.3).

A solution of problem (1.3) defined in this manner will be from now referred to as a solution of problem (1.3) in the space 𝐶(𝐻)=𝐶([0,1],𝐻) of all continuous functions 𝜑(𝑡) defined on [0,1] with values in 𝐻 equipped with the norm 𝜑_𝐶(𝐻)=max_0≤𝑡≤1𝜑(𝑡)_𝐻. (1.4)

Problem (1.3) is well posed in 𝐶(𝐻), if for the solutions of (1.3), we have the following coercivity inequality:𝑢^′_𝐶(𝐻)+‖𝐴𝑢(𝑡)‖_𝐶(𝐻)≤𝑀𝑓_𝐶(𝐻)+𝐴𝜑_𝐻. (1.5) Here, 1≤𝑀 is independent of 𝑓(𝑡)∈𝐶(𝐻),𝜑∈𝐷(𝐴).

Throughout the paper, 𝑀 indicates positive constants which can be different from time to time and we are not interested to make precise. We write 𝑀(𝛼,𝛽,…) to stress the fact that the constant depends only on 𝛼,𝛽,…

Under the assumption: ^𝑝_𝑘=1𝛼_𝑘≤1. (1.6)

Ashyralyev et al. established in [23] the well-posedness of these problems in the space of smooth functions. In applications, they obtained coercivity estimates for the solution of parabolic differential equations.

Moreover, in [24], Ashyralyev et al. considered the first order of accuracy Rothe difference scheme: 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1−𝐴𝑢_𝑘−1=𝜑_𝑘,𝜑_𝑘=𝑓𝑡_𝑘,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=1,_𝑁=^𝑝_𝑚=1𝛼_𝑚𝑢_ℓ𝑚ℓ+𝜑,_𝑚=𝜃_𝑚𝜏,1≤𝑚≤𝑝, (1.7) for approximately solving problem (1.3). They established some stability estimates and almost coercivity of the solution for the difference scheme.

In the present paper, multipoint nonlocal boundary value problem for the multidimensional parabolic equation with Dirichlet condition, 𝑢_𝑡(𝑡,𝑥)+^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}(),𝑥=𝑓𝑡,𝑥𝑥=₁,…,𝑥_𝑛𝑢()=∈Ω,0<𝑡<1,1,𝑥^𝑝_𝑖=1𝛼_𝑖𝑢𝜃_𝑖,𝑥+𝜑(𝑥),𝑥∈Ω,0≤𝜃₁<𝜃₂<⋯<𝜃_𝑝𝑢()<1,𝑡,𝑥=0,𝑥∈𝑆,0≤𝑡≤1 (1.8) under the condition (1.6) is considered. Here, 𝑎_𝑟(𝑥),(𝑥∈Ω),𝜑(𝑥)(𝑥∈Ω), and 𝑓(𝑡,𝑥)(𝑡∈(0,1),𝑥∈Ω) are given smooth functions and 𝑎_𝑟(𝑥)≥𝑎>0, and Ω=(0,ℓ)×⋯×(0,ℓ) is the open cube in the 𝑛-dimensional Euclidean space with boundary 𝑆,Ω=Ω∪𝑆. In the Hilbert space 𝐻=𝐿₂(Ω), we introduce the self-adjoint positive definite operator 𝐴 defined by 𝐴𝑢(𝑥)=−^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_𝑥𝑟(𝑥)_𝑥𝑟, (1.9) with domain 𝐷(𝐴)=𝑢,𝑢_𝑥𝑟,𝑢_{𝑥𝑟𝑥𝑟}∈𝐿₂Ω,1≤𝑟≤𝑛∶𝑢(𝑥)|_𝑥∈𝑆=0. (1.10) Then, problem (1.8) can be written in the abstract form as the nonlocal boundary value problem for reverse parabolic equation (1.3).

The first and second orders of accuracy in 𝑡 and the second order of accuracy in space variables for the approximate solution of problem (1.8) are presented. Applying the method of papers [23, 24], the stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are obtained. The modified Gauss elimination method for solving these difference schemes in the case of one-dimensional parabolic partial differential equations is used.

2. Difference Schemes: Stability Estimates

We will discretize problem (1.8) in two steps. In the first step, we define the grid spaces Ω_ℎ=𝑥=𝑥_𝑚=ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛;𝑚=𝑚₁,…,𝑚_𝑛,𝑚_𝑟=0,…,𝑁_𝑟,ℎ_𝑟𝑁_𝑟,Ω=1,𝑟=1,…,𝑛_ℎ=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆. (2.1)

We denote that |ℎ|=ℎ²₁+⋯+ℎ²₁. Let 𝐿_2ℎ=𝐿₂(Ω_ℎ) denote the Banach space of grid functions: 𝜑^ℎ(𝑥)=𝜑ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛, (2.2) defined on Ω_ℎ, equipped with the norm 𝜑^ℎ_𝐿2ℎ=_𝑥∈Ω_ℎ𝜑^ℎ(𝑥)²ℎ₁⋯ℎ_𝑛^1/2. (2.3)

To the differential operator 𝐴 generated by problem (1.8), we assign the second-order approximation difference operator 𝐴^𝑥_ℎ=𝐶^𝑥_ℎ+𝐵^𝑥_ℎ acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the condition 𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆_ℎ. Assume that 𝐶^𝑥_ℎ is self-adjoint, positive-definite operator in 𝐿_2ℎ and (𝐶^𝑥_ℎ)⁻¹𝐵^𝑥_ℎ is bounded operator in 𝐿_2ℎ.

By using 𝐴^𝑥_ℎ, we arrive at the multipoint nonlocal boundary value problem: 𝑑𝑢^ℎ(𝑡,𝑥)𝑑𝑡−𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,𝑢^ℎ(1,𝑥)=^𝑝_𝑚=1𝛼_𝑚𝑢^ℎ𝜃_𝑚,𝑥+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ, (2.4) for a finite system of ordinary differential equations with a fixed |ℎ|. Note that |ℎ|=ℎ²₁+⋯+ℎ²_𝑛→0. Therefore, we will try to obtain stability, coercivity stability, and almost coercivity estimates with constants independent of |ℎ|.

In the second step, problem (2.4) is replaced by the first order of accuracy difference scheme 𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)𝜏−𝐴^𝑥_ℎ𝑢^ℎ_𝑘−1(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁,𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=^𝑝_𝑚=1𝛼_𝑚𝑢^ℎ_ℓ𝑚(𝑥)+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,ℓ_𝑚=𝜃_𝑚𝜏,𝑚=1,…,𝑝, (2.5) and the second order of accuracy difference scheme 𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)𝜏−𝐴^𝑥_ℎ𝐵^𝑥_ℎ𝑢^ℎ_𝑘−1(𝑥)=𝑓^ℎ_𝑘(𝑥),𝐵^𝑥_ℎ=𝐼+𝜏𝐴^𝑥_ℎ2,𝑓^ℎ_𝑘(𝑥)=𝐵^𝑥_ℎ𝑓^ℎ𝑡_{𝑘−(𝜏/2)},𝑡,𝑥_𝑘=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=^𝑝_𝑚=1𝛼_𝑚𝐼+𝑑_𝑚𝐴^𝑥_ℎ𝑢^ℎ_ℓ𝑚(𝑥)+𝑑_𝑚𝐵^𝑥_ℎ𝜓_ℓ𝑚+1+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑑_𝑚=𝜃_𝑚−𝜃_𝑚𝜏𝜏,ℓ_𝑚=𝜃_𝑚𝜏,𝑚=1,…,𝑝, (2.6) where [⋅] denotes the greatest integer function.

To formulate our results, let 𝐿_2ℎ=𝐿₂(Ω_ℎ) and 𝑊²_2ℎ=𝑊²₂(Ω_ℎ) be spaces of the grid functions 𝜑^ℎ(𝑥)={𝜑(ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛)} defined on Ω_ℎ, equipped with the norms 𝜑^ℎ_𝐿2ℎ=_𝑥∈Ω_ℎ𝜑^ℎ(𝑥)²ℎ₁⋯ℎ_𝑛^1/2,𝜑^ℎ_𝑊22ℎ=𝜑^ℎ_𝐿2ℎ+_𝑥∈Ω_ℎ^𝑛𝑟=1𝜑^ℎ_𝑥𝑟²ℎ₁⋯ℎ_𝑛^1/2+_𝑥∈Ω_ℎ^𝑛𝑟=1𝜑^ℎ(𝑥)_{𝑥𝑟𝑥𝑟,𝑚𝑟}²ℎ₁⋯ℎ_𝑛^1/2. (2.7)

Furthermore, let [0,1]_𝜏={𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=1} be the uniform grid space with step size 𝜏>0, where 𝑁 is a fixed positive integer. We denote ℱ_𝜏(𝐻)=ℱ([0,1]_𝜏,𝐻) for the linear space of grid functions 𝜑^𝜏={𝜑_𝑘}^𝑁₁ with values in the Hilbert space 𝐻.

For 𝛼∈[0,1], let 𝒞^𝛼(𝐻)=𝒞^𝛼([0,1]_𝜏,𝐻) and 𝒞^𝛼₁(𝐻)=𝒞^𝛼₁([0,1]_𝜏,𝐻) be, respectively, the Hölder space and the weighted Hölder space with the norms 𝜑^𝜏_{𝒞𝛼(𝐻)}=𝜑^𝜏_{𝒞𝜏(𝐻)}+max_{1≤𝑘<𝑘+𝑟≤𝑁}𝜑_𝑘+𝑟−𝜑_𝑘𝐻(𝑟𝜏)^𝛼,𝜑^𝜏_{𝒞𝛼1(𝐻)}=𝜑^𝜏_{𝒞𝜏(𝐻)}+max_{1≤𝑘<𝑘+𝑟≤𝑁}((𝑁−𝑘)𝜏)^𝛼𝜑_𝑘+𝑟−𝜑_𝑘𝐻(𝑟𝜏)^𝛼. (2.8) Here, 𝒞_𝜏(𝐻)=𝒞([0,1]_𝜏,𝐻) is the Banach space of bounded grid functions with norm: 𝜑^𝜏_{𝒞𝜏(𝐻)}=max_{1≤𝑘≤𝑁}𝜑_𝑘𝐻. (2.9)

Theorem 2.1. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, for the solutions of difference schemes (2.5) and (2.6), the following stability estimate holds: 𝑢^ℎ_𝑘^𝑁_{0𝒞𝜏(𝐿2ℎ)}≤𝑀𝛿,𝜃_𝑝𝜑^ℎ_𝐿2ℎ+𝑓^ℎ_𝑘^𝑁_{1𝒞𝜏(𝐿2ℎ)}, (2.10) where 𝑀(𝛿,𝜃_𝑝) is independent of 𝜏,ℎ,𝜑^ℎ(𝑥), and 𝑓^ℎ_𝑘(𝑥) and 𝑘=1,…,𝑁.

Proof. The proof of Theorem 2.1 is based on the formulas for the solution of difference scheme (2.5) 𝑢^ℎ_𝑘=𝑅^𝑁−𝑘𝑢^ℎ_𝑁−^𝑁_𝑗=𝑘+1𝑅^𝑗−𝑘𝑓^ℎ_𝑗𝑢𝜏,0≤𝑘≤𝑁−1,_ℎ𝑁=𝑇_𝜏−^𝑝𝑁_{𝑘=1𝑗=ℓ𝑘+1}𝛼_𝑘𝑅^{𝑗−ℓ𝑘}𝑓^ℎ_𝑗𝜏+𝜑^ℎ, (2.11)(2.12) and for the solution of difference scheme (2.6) 𝑢^ℎ_𝑘=𝐷^𝑁−𝑘𝑢^ℎ_𝑁−^𝑁_𝑗=𝑘+1𝐷^𝑗−𝑘𝑓^ℎ_𝑗𝑢𝜏,0≤𝑘≤𝑁−1,_ℎ𝑁=𝑇^′_𝜏−^𝑝𝑁_{𝑚=1𝑗=ℓ𝑚+1}𝛼_𝑚𝐼+𝑑_𝑚𝐴^𝑥_ℎ𝐷^{𝑗−ℓ𝑚}𝑓^ℎ_𝑗𝜏+^𝑝_𝑚=1𝛼_𝑚𝑑_𝑚𝐵^𝑥_ℎ⁻¹𝑓^ℎ_ℓ𝑚+1+𝜑^ℎ. (2.13)(2.14) Here, 𝑅=𝐼+𝜏𝐴^𝑥_ℎ⁻¹,𝐷=𝐼+𝜏𝐴^𝑥_ℎ+𝜏𝐴^𝑥_ℎ²2⁻¹,𝑇_𝜏=𝐼−^𝑝_𝑘=1𝛼_𝑘𝑅^{𝑁−[𝜃𝑘/𝜏]−1},𝑇′_𝜏=𝐼−^𝑝_𝑘=1𝛼_𝑘𝐼+𝑑_𝑘𝐴^𝑥_ℎ𝐷^{𝑁−[𝜃𝑘/𝜏]−1}. (2.15)
By the spectral representation of self-adjoint positive definite operator and the triangle inequality, we have 𝑇_{𝜏𝐻→𝐻}≤sup_𝛿≤𝜇11−∑^𝑝_𝑘=1𝛼_𝑘1+𝜏𝜇^{−𝑁+[𝜃𝑝/𝜏]}≤𝑀𝛿,𝜃_𝑝. (2.16)
Similarly, we have 𝑇^′_{𝜏𝐻→𝐻}≤𝑀𝛿,𝜃_𝑝. (2.17) Estimates (2.16) and (2.17) conclude the proof of Theorem 2.1.

Theorem 2.2. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, for the solutions of difference problem (2.5) and (2.6), the following almost coercivity inequality 𝜏⁻¹𝑢^ℎ_𝑘−𝑢^𝑘_𝑘−1^𝑁_{1𝒞𝜏(𝐿2ℎ)}≤𝑀𝛿,𝜃_𝑝𝜑^ℎ_𝑊22ℎ+𝑓^ℎ_𝑘^𝑁_{1𝒞𝜏(𝐿2ℎ)}1ln𝜏+|ℎ| (2.18) is valid, where 𝑀(𝛿,𝜃_𝑝) does not depend on 𝜏,ℎ,𝜑^ℎ(𝑥),𝑓^ℎ_𝑘(𝑥),𝑘=1,…,𝑁.

Proof. Using formulas (2.11)–(2.14), estimates (2.16) and (2.17), the triangle inequality, assumption (1.6), we obtain 𝜏⁻¹𝑢^ℎ_𝑘−𝑢^𝑘_𝑘−1^𝑁_{1𝒞𝜏(𝐿2ℎ)}≤𝑀𝛿,𝜃_𝑝𝜑^ℎ_𝑊22ℎ1+minln𝜏,1+ln𝐴^ℎ_{𝑥𝐿2ℎ→𝐿2ℎ}𝑓^ℎ_𝑘^𝑁_{1𝒞𝜏(𝐿2ℎ)}. (2.19) Since ln𝐴^ℎ_{𝑥𝐿2ℎ→𝐿2ℎ}1≤𝑀ln|ℎ|, (2.20) we have that 1minln𝜏,1+ln𝐴^ℎ_{𝑥𝐿2ℎ→𝐿2ℎ}≤𝑀₁1ln|ℎ|.𝜏+ (2.21) From that, inequality (2.19), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in 𝐿_2ℎ it follows inequality (2.18). Theorem 2.2 is proved.

Theorem 2.3 (see [25, 26]). For the solution of the elliptic difference problem: 𝐴^𝑥_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(𝑥)=0,𝑥∈𝑆_ℎ, (2.22) the following coercivity inequality holds: ^𝑛_𝑟=1𝑢^ℎ_{𝑘𝑥𝑟𝑥𝑟,𝑗𝑟𝐿2ℎ}≤𝑀𝜔^ℎ_𝐿2ℎ, (2.23) where 𝑀 does not depend on ℎ and 𝜔^ℎ.

Theorem 2.4. Let 𝜏 and |ℎ| be sufficiently small positive numbers. Then, the solutions of difference problem (2.5) and (2.6) satisfy the following coercivity stability estimate: 𝜏⁻¹𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1^𝑁_{1𝒞𝛼1(𝐿2ℎ)}≤𝑀𝛿,𝜃_𝑝,𝛼𝜑^ℎ_𝑊22ℎ+𝑓^ℎ_𝑘^𝑁_{1𝒞𝛼1(𝐿2ℎ)}, (2.24) where 𝑀(𝛿,𝜃_𝑝,𝛼) is independent of 𝜏,ℎ,𝑓^ℎ_𝑘(𝑥), and 𝜑^ℎ(𝑥),𝑘=1,…,𝑁.

Theorem 2.5. Let 𝐴^𝑥_ℎ𝜑^ℎ(𝑥)=𝑓^ℎ_𝑁(𝑥)−∑^𝑝_𝑘=1𝛼_𝑚𝑓^ℎ_ℓ𝑚(𝑥). Then, for solutions of problem (2.5) and (2.6), the following coercive stability estimate holds: 𝜏⁻¹𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1^𝑁_{1𝒞𝛼(𝐿2ℎ)}+𝑢^ℎ_𝑘^𝑁_{1𝒞𝛼(𝑊22ℎ)}≤𝑀𝛿,𝜃_𝑝,𝛼𝑓^ℎ_𝑘^𝑁_{1𝒞𝛼(𝐿2ℎ)}, (2.25) where 𝑀(𝛿,𝜃_𝑝,𝛼) does not depend on 𝜏,ℎ,𝑓^ℎ_𝑘(𝑥), and 𝜑^ℎ(𝑥),𝑘=1,…,𝑁.

The proofs of Theorems 2.4–2.5 are based on the formulas: 𝐴^𝑥_ℎ𝑢^ℎ_𝑘−1=𝑅^{𝑁−𝑘+1}𝐴^𝑥_ℎ𝑢^ℎ_𝑁−^𝑁_𝑗=𝑘+1𝜏𝐴^𝑥_ℎ𝑅^{𝑗−𝑘+1}𝑓^ℎ_𝑗−𝑓^ℎ_𝑘+𝑅^{𝑁−𝑘+1}−𝐼𝑓^ℎ_𝑘,𝐴^𝑥_ℎ𝐵^𝑥_ℎ𝑢^ℎ_𝑘−1=𝐷^{𝑁−𝑘+1}𝐴^𝑥_ℎ𝐵^𝑥_ℎ𝑢^ℎ_𝑁−^𝑁_𝑗=𝑘+1𝜏𝐴^𝑥_ℎ𝐼+𝜏𝐴^𝑥_ℎ2𝐷^{𝑗−𝑘+1}𝑓^ℎ_𝑗−𝑓^ℎ_𝑘+𝐷^{𝑁−𝑘+1}−𝐼𝑓^ℎ_𝑘, (2.26) the self-adjoint positive definiteness of the operator 𝐴^𝑥_ℎ in 𝐿_2ℎ, estimates (2.16) and (2.17), the triangle inequality, and assumption (1.6).

3. Numerical Results

For the numerical result, we consider the nonlocal boundary value problem: 𝜕𝑢(𝑡,𝑥)+((𝑥𝜕𝜕𝑡2+cos))²𝑢(𝑡,𝑥)𝜕𝑥²(𝑥)()−sin𝜕𝑢𝑡,𝑥()𝑓()=𝜕𝑥=𝑓𝑡,𝑥,0<𝑥<𝜋,0<𝑡<1,𝑡,𝑥2𝑡−6𝑡²+4𝑡³sin(𝑥)−𝑡²(1−𝑡)²(2+cos(𝑥))sin(𝑥)−𝑡²(1−𝑡)²(𝑥)(𝑥),𝑢()()𝑢()()cossin0,𝑥=𝑢1,𝑥,0≤𝑥≤𝜋,𝑡,0=𝑢𝑡,𝜋=0,0≤𝑡≤1, (3.1) for the reverse parabolic equation. It is easy to see that 𝑢(𝑡,𝑥)=𝑡²(1−𝑡)²sin(𝑥) is the exact solution of (3.1).

For the approximate solution of nonlocal boundary value problem (3.1), consider the set [0,1]_𝜏×[0,𝜋]_ℎ of a family of grid points depending on the small parameters 𝜏 and ℎ[0,1]_𝜏×[0,𝜋]_ℎ=𝑡,𝑥_𝑛∶𝑡_𝑘=𝑘𝜏,𝑘=1,…,𝑁−1,𝑁𝜏=1,𝑥_𝑛=𝑛ℎ,𝑛=1,…,𝑀−1,𝑀ℎ=𝜋. (3.2)

Applying (2.5), we get the first order of accuracy in 𝑡 and the second order of accuracy in 𝑥𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏+2+cos𝑥_𝑛𝑢^𝑘−1_𝑛+1−2𝑢_𝑛^𝑘−1+𝑢^𝑘−1_𝑛−1ℎ²−sin𝑥_𝑛𝑢^𝑘−1_𝑛+1−𝑢^𝑘−1_𝑛−1𝑡2ℎ=𝑓_𝑘,𝑥_𝑛𝑢,𝑘=1,…,𝑁,𝑛=1,…,𝑀−1,_𝑘0=𝑢^𝑘_𝑀𝑢=0,𝑘=0,…,𝑁,_0𝑛=𝑢^𝑁_𝑛,𝑛=0,…,𝑀, (3.3) for the approximate solutions of the nonlocal boundary value problem (3.1).

Note that for difference scheme (3.3), we have that −2+cos𝑥_𝑛𝑢^𝑘−1_𝑛+1−2𝑢_𝑛^𝑘−1+𝑢^𝑘−1_𝑛−1ℎ²−sin𝑥_𝑛𝑢^𝑘−1_𝑛+1−𝑢^𝑘−1_𝑛−12ℎ₁^𝑀−1=𝐶^𝑥_ℎ𝑢^ℎ_𝑘−1(𝑥)+𝐵^𝑥_ℎ𝑢^ℎ_𝑘−1(𝑥), (3.4) where 𝐶^𝑥_ℎ𝑢^ℎ(𝑥)=−2+cos𝑥_𝑛+1𝑢_𝑛+1−𝑢_𝑛/ℎ−2+cos𝑥_𝑛𝑢_𝑛−𝑢_𝑛−1/ℎℎ₁^𝑀−1,𝐵^𝑥_ℎ𝑢^ℎ(𝑥)=cos(𝑥_𝑛+1)−cos(𝑥_𝑛)ℎ𝑢_𝑛−𝑢_𝑛−1ℎ−sin𝑥_𝑛𝑢_𝑛+1−𝑢_𝑛2ℎ₁^𝑀−1. (3.5)

It is easy to see that 𝐶^𝑥_ℎ=(𝐶^𝑥_ℎ)^∗ and 𝐶^𝑥_ℎ≥𝛿𝐼_ℎ, and 𝐶^𝑥_ℎ⁻¹𝐵^𝑥_{ℎ𝐿2ℎ→𝐿2ℎ}≤𝑀, (3.6) where 𝐼_ℎ is the identity operator.

So, Theorems 2.1, 2.2, 2.4, and 2.5 are compatible for the solution of (3.3).

We can write (3.3) as in the matrix form 𝐴_𝑛𝑢_𝑛+1+𝐵_𝑛𝑢_𝑛+𝐶_𝑛𝑢_𝑛−1=𝐼𝜑_𝑛𝑢,𝑛=1,…,𝑀−1,₀=^→0,𝑢_𝑀=^→0. (3.7) Here, 𝜑_𝑛 is an (𝑁+1)×1 column matrix, 𝐴_𝑛,𝐵_𝑛,𝐶_𝑛 are (𝑁+1)×(𝑁+1) square matrices, 𝐴_𝑛=𝑎_𝑛𝑅,𝐶_𝑛=𝑐_𝑛𝑅, ,𝐵𝑅=0000⋯00001000⋯00000100⋯00000010⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯00000000⋯10000000⋯01000000⋯0010_𝑛=𝑏1000⋯000−1_𝑛𝑑00⋯00000𝑏_𝑛𝑑0⋯000000𝑏_𝑛𝑑⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯𝑑0000000⋯𝑏_𝑛𝑑000000⋯0𝑏_𝑛𝑑00000⋯00𝑏_𝑛𝑑,𝑎_𝑛=2+cos𝑥_𝑛ℎ²−sin𝑥_𝑛2ℎ,𝑏_𝑛1=−𝜏−22+cos𝑥_𝑛ℎ²,𝑐_𝑛=2+cos𝑥_𝑛ℎ²+sin𝑥_𝑛12ℎ,𝑑=𝜏.𝜑_𝑛=𝜑⁰_𝑛⋮𝜑^𝑁_𝑛,𝜑⁰_𝑛𝜑=0,𝑛=1,…,𝑀−1,_𝑘𝑛=𝑓𝑡_𝑘,𝑥_𝑛,𝑛=1,…,𝑁,𝑛=1,…,𝑀−1, (3.8) here and in the future 𝐼 is the (𝑁+1)×(𝑁+1) identity matrix, 𝑢_𝑠=𝑢⁰_𝑠⋮𝑢^𝑁_{𝑠(𝑁+1)×1},𝑠=𝑛−1,𝑛,𝑛+1. (3.9)

Samarskii and Nikolaev studied this type of system in [27] for difference equations. We seek the solution of (3.7) by the formula 𝑢_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1,𝑛=𝑀−1,…,1, (3.10) where 𝑢_𝑀=^→0,𝛼_𝑛(𝑛=1,…,𝑀−1) are (𝑁+1)×(𝑁+1) square matrices and 𝛽_𝑛(𝑛=1,…,𝑀−1) are (𝑁+1)×1 column matrices. For the solution of difference equation (3.7) we need to use the following formulas for 𝛼_𝑛,𝛽_𝑛: 𝛼_𝑛=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛−1⁻¹𝐴_𝑛,𝛽_𝑛=𝐵_𝑛+𝐶_𝑛𝛼_𝑛−1⁻¹𝐼𝜑_𝑛−𝐶_𝑛𝛽_𝑛−1,𝑛=2,…,𝑀−1, (3.11) where 𝛼₁ is the (𝑁+1)×(𝑁+1) zero matrix and 𝛽₁ is the (𝑁+1)×1 zero column vector.

Second, we consider again the nonlocal boundary value problem (3.1). Applying (2.6) and formulas: 𝑢𝑥_𝑛+1−𝑢𝑥_𝑛−12ℎ−𝑢^′𝑥_𝑛=𝑂ℎ²,𝑢𝑥_𝑛+1−2𝑢𝑥_𝑛+𝑢𝑥_𝑛−1ℎ²−𝑢^′′𝑥_𝑛=𝑂ℎ²,𝑢𝑥_𝑛+2−4𝑢𝑥_𝑛+1+6𝑢𝑥_𝑛−4𝑢𝑥_𝑛−1+𝑢𝑥_𝑛−2ℎ⁴−𝑢⁽⁴⁾𝑥_𝑛=𝑂ℎ²,2𝑢(0)−5𝑢(ℎ)+4𝑢(2ℎ)−𝑢(3ℎ)ℎ²−𝑢^′′(0)=𝑂ℎ²,2𝑢(1)−5𝑢(1−ℎ)+4𝑢(1−2ℎ)−𝑢(1−3ℎ)ℎ²−𝑢^′′(1)=𝑂ℎ², (3.12) we get the second order of accuracy in 𝑡 and 𝑥𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏−sin𝑥_𝑛+𝜏sin𝑥_𝑛+𝜏sin𝑥_𝑛cos𝑥_𝑛𝑢^𝑘−1_𝑛+1−𝑢^𝑘−1_𝑛−1+𝑥2ℎ2+cos_𝑛+𝜏26cos𝑥_𝑛+5cos²𝑥_𝑛𝑢−2^{𝑘−1𝑛+1}−2𝑢_𝑛^𝑘−1+𝑢^𝑘−1_𝑛−1ℎ²+𝜏4sin𝑥_𝑛+2sin𝑥_𝑛cos𝑥_𝑛𝑢^𝑘−1_𝑛+2−2𝑢^𝑘−1_𝑛+1+2𝑢^𝑘−1_𝑛−1−𝑢^𝑘−1_𝑛−22ℎ³−𝜏22+cos𝑥_𝑛²𝑢^𝑘−1_𝑛+2−4𝑢^𝑘−1_𝑛+1+6𝑢_𝑛^𝑘−1−4𝑢^𝑘−1_𝑛−1+𝑢^𝑘−1_𝑛−2ℎ⁴=𝜑^𝑘_𝑛,𝜑^𝑘_𝑛=𝑓𝑡_𝑘−𝜏2,𝑥_𝑛−𝜏21ℎ²2+cos𝑥_𝑛×𝑓𝑡_𝑘−𝜏2,𝑥_𝑛+1−2𝑓𝑡_𝑘−𝜏2,𝑥_𝑛+𝑓𝑡_𝑘−𝜏2,𝑥_𝑛−1−1𝑥2ℎsin_𝑛𝑓𝑡_𝑘−𝜏2,𝑥_𝑛+1−𝑓𝑡_𝑘−𝜏2,𝑥_𝑛−1,𝑢𝑘=1,…,𝑁,𝑛=2,…,𝑀−2,_𝑘0=𝑢^𝑘_𝑀=0,𝑢^𝑘₁=45𝑢^𝑘₂−15𝑢^𝑘₃𝑢,𝑘=0,…,𝑁,_𝑘𝑀−1=45𝑢^𝑘_𝑀−2−15𝑢^𝑘_𝑀−3𝑢,𝑘=0,…,𝑁,_0𝑛−𝑢^𝑁_𝑛=0,𝑛=0,…,𝑀, (3.13) for the approximate solutions of the nonlocal boundary value problem (3.1).

We can rewrite this system in the following matrix form: 𝐴_𝑛𝑢_𝑛+2+𝐵_𝑛𝑢_𝑛+1+𝐶_𝑛𝑢_𝑛+𝐷_𝑛𝑢_𝑛−1+𝐸_𝑛𝑢_𝑛−2=𝐼𝜑_𝑛𝑢,𝑛=2,…,𝑀−2,₀=^→0,𝑢_𝑀=^→0,𝑢₁=45𝑢₂−15𝑢₃,𝑢_𝑀−1=45𝑢_𝑀−2−15𝑢_𝑀−3, (3.14) where 𝜑_𝑛 is an (𝑁+1)×1 column matrix, 𝐴_𝑛,  𝐵_𝑛,𝐶_𝑛,𝐷_𝑛,𝐸_𝑛 are (𝑁+1)×(𝑁+1) square matrices 𝜑_𝑛=𝜑⁰_𝑛⋮𝜑^N_𝑛, (3.15)𝐴_𝑛=𝑣_𝑛𝑅,𝐵_𝑛=𝑦_𝑛𝑅,𝐷_𝑛=𝑧_𝑛𝑅,𝐸_𝑛=𝑤_𝑛𝑅, 𝐶_𝑛=𝑟100⋯0−1_𝑛𝑝0⋯000𝑟_𝑛𝑝⋯00⋮⋮⋮⋯⋮⋮000⋯𝑝0000⋯𝑟_𝑛𝑝, (3.16) where 𝑝=1/𝜏, 𝑣_𝑛=𝜏2ℎ³4sin𝑥_𝑛+2sin𝑥_𝑛cos𝑥_𝑛−𝜏2ℎ⁴2+cos𝑥_𝑛²,𝑦_𝑛1=−𝑥2ℎsin_𝑛+𝜏sin𝑥_𝑛+𝜏sin𝑥_𝑛cos𝑥_𝑛+1ℎ²2+cos𝑥_𝑛+𝜏26cos𝑥_𝑛+5cos²𝑥_𝑛−𝜏−2ℎ³4sin𝑥_𝑛+2sin𝑥_𝑛cos𝑥_𝑛+2𝜏ℎ⁴2+cos𝑥_𝑛²,𝑟_𝑛1=−𝜏−2ℎ²2+cos𝑥_𝑛+𝜏26cos𝑥_𝑛+5cos²𝑥_𝑛−−23𝜏ℎ⁴2+cos𝑥_𝑛²,𝑧_𝑛=1𝑥2ℎsin_𝑛+𝜏sin𝑥_𝑛+𝜏sin𝑥_𝑛cos𝑥_𝑛+1ℎ²2+cos𝑥_𝑛+𝜏26cos𝑥_𝑛+5cos²𝑥_𝑛+𝜏−2ℎ³4sin𝑥_𝑛+2sin𝑥_𝑛cos𝑥_𝑛+2𝜏ℎ⁴2+cos𝑥_𝑛²,𝑤_𝑛𝜏=−2ℎ⁴2+cos𝑥_𝑛²−𝜏2ℎ³4sin𝑥_𝑛+2sin𝑥_𝑛cos𝑥_𝑛. (3.17)

For the solution of the last matrix equation, we use the modified variant of Gauss elimination method. We seek a solution of the matrix equation of the matrix equation in the following form: 𝑢_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1𝑢_𝑛+2+𝛾_𝑛+1𝑢,𝑛=𝑀−2,…,0,_𝑀=^→0,𝐷_𝑀=𝛽_𝑀−2+5𝐼−4𝐼−𝛼_𝑀−2𝛼_𝑀−1,𝑢_𝑀−1=𝐷_𝑀⁻¹4𝐼−𝛼_𝑀−2𝛾_𝑀−1−𝛾_𝑀−2, (3.18) where 𝛾₁=𝛾₂=^→0,𝛼₁=𝛽₁ are (𝑁+1)×(𝑁+1) zero matrices, 𝛼₂=−4𝛽₂=(4/5)𝐼, and 𝛽_𝑛+1=−𝐹_𝑛⁻¹𝐴_𝑛,𝛼_𝑛+1=−𝐹_𝑛⁻¹𝐵_𝑛+𝐷_𝑛𝛽_𝑛+𝐸_𝑛𝛼_𝑛−1𝛽_𝑛,𝛾_𝑛+1=−𝐹_𝑛⁻¹𝐼𝜑_𝑛−𝐷_𝑛𝛾_𝑛−𝐸_𝑛𝛼_𝑛−1𝛾_𝑛−𝐸_𝑛𝛾_𝑛−1,𝐹_𝑛=𝐶_𝑛+𝐷_𝑛𝛼_𝑛+𝐸_𝑛𝛽_𝑛−1+𝐸_𝑛𝛼_𝑛−1𝛼_𝑛,𝑛=2,…,𝑀−2. (3.19)

Now, let us give the results of the numerical analysis. In order to get the solution, we used MATLAB programs. The numerical solutions are recorded for different values of 𝑁=𝑀 and 𝑢^𝑘_𝑛 represents the numerical solutions of these difference schemes at (𝑡_𝑘,𝑥_𝑛). For their comparison, the errors are computed by 𝐸^𝑁_𝑀=max_{−𝑁≤𝑘≤𝑁,1≤𝑛≤𝑀−1}𝑢𝑡_𝑘,𝑥_𝑛−𝑢^𝑘_𝑛. (3.20) Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes. Table 1 is constructed for 𝑁=𝑀=20, 40, and 60, respectively. Hence, the second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme.

Table 1 

Error analysis.

Acknowledgment

The authors would like to thank Prof. Dr. Allaberen Ashyralyev (Fatih University, Turkey) on his very helpful comments and suggestions in improving the quality of this work.