A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

Abstract

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert space H with the self-adjoint positive definite operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

1. Introduction

Hyperbolic partial differential equations play an important role in many branches of science and engineering and can be used to describe a wide variety of phenomena such as acoustics, electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics (see [1–5] and the references given therein).

While applying mathematical modelling to several phenomena of physics, biology, and ecology, there often arise problems with nonclassical boundary conditions, which the values of unknown function on the boundary are connected with inside of the given domain. Such type of boundary conditions are called nonlocal boundary conditions. Over the last decades, boundary value problems with nonlocal boundary conditions have become a rapidly growing area of research (see, e.g., [6–16] and the references given therein).

In the present work, we consider the nonlocal boundary value problem 𝑑²𝑢(𝑡)𝑑𝑡²𝑢(0)=+_𝑛𝐴𝑢(𝑡)=𝑓(𝑡)(0≤𝑡≤1),_𝑗=1𝛼_𝑗𝑢𝜆_𝑗+𝜑,𝑢_𝑡(0)=^𝑛_𝑗=10<𝜆₁𝛽_𝑗𝑢_𝑡𝜆_𝑗+𝜓,<𝜆₂<⋯<𝜆_𝑛≤1,(1.1) where 𝐴 is a self-adjoint positive definite operator in a Hilbert space 𝐻.

A function 𝑢(𝑡) is called a solution of the problem (1.1), if the following conditions are satisfied:(i)𝑢(𝑡) is twice continuously differentiable on the segment [0,1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii) The element 𝑢(𝑡) belongs to 𝐷(𝐴), independent of 𝑡, and dense in 𝐻 for all 𝑡∈[0,1] and the function 𝐴𝑢(𝑡) is continuous on the segment [0,1].(iii)𝑢(𝑡) satisfies the equation and nonlocal boundary conditions (1.1).

In the paper of [8], the following theorem on the stability estimates for the solution of the nonlocal boundary value problem (1.1) was proved.

Theorem 1.1. Suppose that 𝜑∈𝐷(𝐴), 𝜓∈𝐷(𝐴^1/2), and 𝑓(𝑡) is a continuously differentiable function on [0,1] and the assumption ^𝑛_𝑘=1||𝛼_𝑘+𝛽_𝑘||+^𝑛_𝑚₌₁||𝛼_𝑚||^𝑛_𝑘_{=1𝑘≠𝑚}||𝛽_𝑘||<|||||1+^𝑛_𝑘=1𝛼_𝑘𝛽_𝑘|||||(1.2) holds. Then, there is a unique solution of problem (1.1) and the stability inequalities max_0≤𝑡≤1‖𝑢(𝑡)‖_𝐻≤𝑀‖𝜑‖_𝐻+‖‖𝐴^−1/2𝜓‖‖_𝐻+max_0≤𝑡≤1‖‖𝐴^−1/2𝑓(𝑡)‖‖_𝐻max_0≤𝑡≤1,‖‖𝐴^1/2𝑢(𝑡)‖‖_𝐻≤𝑀‖‖𝐴^1/2𝜑‖‖_𝐻+‖𝜓‖_𝐻+max_0≤𝑡≤1‖𝑓(𝑡)‖_𝐻max_0≤𝑡≤1,‖‖‖𝑑²𝑢(𝑡)𝑑𝑡²‖‖‖_𝐻+max_0≤𝑡≤1‖𝐴𝑢(𝑡)‖_𝐻≤𝑀‖𝐴𝜑‖_𝐻+‖‖𝐴^1/2𝜓‖‖_𝐻+‖𝑓(0)‖_𝐻+¹⁰‖‖𝑓^(𝑡)‖‖_𝐻𝑑𝑡(1.3) hold, where 𝑀 does not depend on 𝜑,  𝜓, and 𝑓(𝑡),  𝑡∈[0,1].

Moreover, the first order of accuracy difference scheme 𝑡𝜏⁻²_𝑘𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+𝐴𝑢_𝑘+1=𝑓_𝑘,𝑓_𝑘=𝑓𝑡_𝑘,𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₀=^𝑛_𝑟=1𝜏𝛼_𝑟𝑢_{[𝜆𝑟/𝜏]}+𝜑,₋₁𝑢₁−𝑢₀=^𝑛_𝑟=1𝛽_𝑟𝑢_{[𝜆𝑟/𝜏]+1}−𝑢_{[𝜆𝑟/𝜏]}𝜏1+𝜓,(1.4) for the approximate solution of problem (1.1) was presented. The stability estimates for the solution of this difference scheme, under the assumption ^𝑛_𝑘=1||𝛼_𝑘||+^𝑛_𝑘=1||𝛽_𝑘||+^𝑛_𝑘=1||𝛼_𝑘||^𝑛_𝑘=1||𝛽_𝑘||<1,(1.5) were established.

In the development of numerical techniques for solving PDEs, the stability has been an important research topic (see [6–31]). A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid steps 𝜏 and ℎ with respect to the time and space variables, are connected. In abstract terms, this particularly means that 𝜏‖𝐴_ℎ‖→0 when 𝜏→0.

We are interested in studying the high order of accuracy difference schemes for hyperbolic PDEs, in which stability is established without any assumption with respect to the grid steps 𝜏 and ℎ. Particularly, a convenient model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator.

In the present paper, the second order of accuracy unconditionally stable difference schemes for approximately solving boundary value problem (1.1) is presented. The stability estimates for the solutions of these difference schemes and their first and second order difference derivatives are established. This operator approach permits one to obtain the stability estimates for the solutions of difference schemes of nonlocal boundary value problems, for one-dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables.

Some results of this paper without proof were presented in [7].

Note that nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–16, 20–24, 32–38] and the references therein).

2. The Second Order of Accuracy Difference Scheme Generated by 𝐴²

Throughout this paper for simplicity 𝜆₁>2𝜏 and 𝜆_𝑛<1 will be considered. Let us associate boundary value problem (1.1) with the second order of accuracy difference scheme 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+𝐴𝑢_𝑘+𝜏²4𝐴²𝑓_𝑘𝑢_𝑘+1=𝑓_𝑘,𝑢=𝑓𝑡_𝑘,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₀=^𝑛_𝑚₌₁𝛼_𝑚𝑢^{[𝜆𝑚/𝜏]}+𝜏⁻¹𝑢_{[𝜆𝑚/𝜏]}−𝑢_{[𝜆𝑚/𝜏]−1}×𝜆_𝑚−𝜆^𝑚𝐼+𝜏₂𝜏𝜏+𝜑,𝐴2𝜏⁻¹𝑢₁−𝑢₀−𝜏2𝑓⁰=_𝑛−𝐴𝑢₀_𝑘=1𝛽_𝑘𝜏⁻¹𝑢_{[𝜆𝑘/𝜏]}−𝑢_{[𝜆𝑘/𝜏]−1}+𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏×𝑓^{[𝜆𝑘/𝜏]}𝑓−𝐴𝑢_{[𝜆𝑘/𝜏]}+𝜓,₀=𝑓(0).(2.1) A study of discretization, over time only, of the nonlocal boundary value problem also permits one to include general difference schemes in applications, if the differential operator in space variables 𝐴 is replaced by the difference operator 𝐴_ℎ that act in the Hilbert space and are uniformly self-adjoint positive definite in ℎ for 0<ℎ≤ℎ₀.

In general, we have not been able to obtain the stability estimates for the solution of difference scheme (2.1) under assumption (1.5). Note that the stability of solution of difference scheme (2.1) will be obtained under the strong assumption ^𝑛_𝑘=1||𝛼_𝑘||1+||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||+^𝑛_𝑘=1||𝛽_𝑘||1+||||𝜆_𝑘−𝜆^𝑘+_𝑛𝜏𝜏||||_𝑘=1||𝛼_𝑘||^𝑛_𝑘=1||𝛽_𝑘||1+||||𝜆_𝑘−𝜆^𝑘+_𝑛𝜏𝜏||||_𝑘=1||𝛼_𝑘||||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||^𝑛_𝑘=1||𝛽_𝑘||||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||<1.(2.2) Now, let us give some lemmas that will be needed in the sequel.

Lemma 2.1. The following estimates hold: ‖‖‖𝑅𝑅‖_𝐻↦𝐻≤1,‖‖𝑅‖‖_𝐻↦𝐻≤1,₋₁‖‖𝜏𝐴𝑅_1/2‖‖_𝐻↦𝐻≤1,‖‖𝑅⁻¹𝑅‖‖_𝐻↦𝐻≤1,𝑅‖‖_𝐻↦𝐻≤1,‖‖𝜏𝐴^1/2𝑅‖‖_𝐻↦𝐻≤1.(2.3) Here, 𝐻 is the Hilbert space, 𝑅=(𝐼+𝑖𝜏𝐴^1/2−(𝜏²/2)𝐴)⁻¹, and 𝑅=(𝐼−𝑖𝜏𝐴^1/2−(𝜏²/2)𝐴)⁻¹.

Lemma 2.2. Suppose that assumption (2.2) holds. Denote 𝐵_𝜏=^𝑛_𝑚₌₁𝛼_𝑚𝐼+𝜏²𝐴2⁻¹⎡⎢⎢⎣𝑅^{[𝜆𝑚/𝜏]−1}𝐼−𝑖𝜏𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝐼+𝑖𝜏𝐴^1/2+_𝑛2⎤⎥⎥⎦_𝑘=1𝛽_𝑘𝐼+𝜏²𝐴2⁻¹𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹+𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−122^𝑛_𝑛𝑚=1_𝑘=1𝛼_𝑚𝛽_𝑘𝐼+𝜏²𝐴2⁻¹𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}+𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}+_𝑛_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝑖𝐴^1/2𝐼+𝜏²𝐴2⁻¹×⎡⎢⎢⎣𝑅^{[𝜆𝑚/𝜏]−1}𝐼−(𝑖𝜏/2)𝐴^1/2𝐼+𝑖𝜏𝐴^1/2−𝑅^{[𝜆𝑚/𝜏]−1}𝐼+(𝑖𝜏/2)𝐴^1/2𝐼−𝑖𝜏𝐴^1/2+_𝑛2⎤⎥⎥⎦_𝑘=1𝛽_𝑘𝜆^𝑘−𝜆^𝑘𝜏𝜏𝑖𝐴^1/2𝐼+𝜏²𝐴2⁻¹𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹+124^𝑛_𝑛𝑚=1_𝑘=1𝛼_𝑚𝛽_𝑘𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻²×𝑖𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖𝜏2𝐴^1/2−𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏−12𝐴^1/22^𝑛_𝑛𝑚=1_𝑘=1𝛼_𝑚𝛽_𝑘𝜆^𝑘−𝜆^𝑘𝜏𝜏𝑖𝐴^1/2𝐼+𝜏²𝐴2⁻¹−×1𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}−𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}2^𝑛_𝑛𝑚=1_𝑘=1𝛼_𝑚𝛽_𝑘𝜆^𝑚−𝜆^𝑚𝜏𝜏𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴𝐼+𝜏²𝐴2⁻¹×𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏2𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖2𝜏𝐴^1/2.(2.4) Then, the operator 𝐼−𝐵_𝜏 has an inverse 𝑇_𝜏=𝐼−𝐵_𝜏⁻¹,(2.5) and the following estimate holds: ‖‖𝑇_𝜏‖‖_𝐻↦𝐻≤𝑀.(2.6)

Proof. The proof of estimate (2.6) is based on the estimate ‖‖𝐼−𝐵_𝜏‖‖_𝐻↦𝐻≥1−^𝑛_𝑘=1||𝛼_𝑘||1+||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||−^𝑛_𝑘=1||𝛽_𝑘||1+||||𝜆_𝑘−𝜆^𝑘−_𝑛𝜏𝜏||||_𝑘=1||𝛼_𝑘||^𝑛_𝑘=1||𝛽_𝑘||1+||||𝜆_𝑘−𝜆^𝑘−_𝑛𝜏𝜏||||_𝑘=1||𝛼_𝑘||||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||^𝑛_𝑘=1||𝛽_𝑘||||||𝜆_𝑘−𝜆^𝑘𝜏𝜏||||.(2.7) Estimate (2.7) follows from the triangle inequality and estimate (2.3). Lemma 2.2 is proved.
Now, we will obtain the formula for the solution of problem (2.1). It is easy to show that (see, e.g., [18]) there is unique solution of the problem 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+𝐴𝑢_𝑘+𝜏²4𝐴²𝑓_𝑘𝑢_𝑘+1=𝑓_𝑘,𝐼=+𝑓𝜏₂𝑡_𝑘,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝐴2𝜏⁻¹𝑢₁−𝑢₀−𝜏2𝑓⁰−𝐴𝑢₀=𝜔,𝑓₀=𝑓(0),𝑢₀=𝜇,(2.8) and the following formula holds: 𝑢₀=𝜇,𝑢₁=𝐼+𝜏²𝐴𝑢_𝑘2⁻¹𝜇+𝜏𝜔+𝜏2²𝑓⁰,=𝐼+𝜏²𝐴2⁻¹⎛⎜⎜⎝𝑅^𝑘−1𝐼−𝑖𝜏𝐴^1/2+𝑅^𝑘−1𝐼+𝑖𝜏𝐴^1/2+𝐼+𝜏₂2⎞⎟⎟⎠𝜇𝐴2⁻¹𝑖𝐴^1/2⁻¹𝑅^𝑘−1𝑅⁻¹−𝑅^𝑘−1𝑅⁻¹−_𝑘−12𝜔+𝜏2𝑓⁰_𝑠=1𝜏2𝑖𝐴^−1/2𝑅^𝑘−𝑠−𝑅^𝑘−𝑠𝑓^𝑠,2≤𝑘≤𝑁.(2.9) Applying formula (2.9) and the nonlocal boundary conditions in problem (2.1), we get 𝜇=𝑇_𝜏^𝑛_𝑚=1𝛼_𝑚𝜏2𝑓⁰×𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹×𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹+_𝑛2_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−⎛⎜⎜⎝_𝑛2⎞⎟⎟⎠⎞⎟⎟⎠_𝑚₌₁𝛼_𝑚[𝜆^{𝑚/𝜏]−1}_𝑠=1𝜏2𝑖𝐴^−1/2𝑅^{[𝜆𝑚/𝜏]−s}−𝑅^{[𝜆𝑚/𝜏]−s}𝑓^𝑠−^𝑛_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚×_{[𝜆𝑚/𝜏]−2}𝜏𝜏_𝑠=1𝜏2𝑖𝐴^1/2𝑖𝐴^−1/2𝑅^{[𝜆𝑚/𝜏]−s}𝐼−𝑖2𝜏𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−s}𝐼−𝑖2𝜏𝐴^1/2𝑓^𝑠+^𝑛_𝑚₌₁𝛼_𝑚𝜏𝜆_𝑚−𝜆^𝑚𝜏𝜏𝑅𝑅𝑓^{[𝜆𝑚/𝜏]−1}×𝐼+_𝑛−𝜑_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹×𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹2−^𝑛_𝑚=1𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2+_𝑛2⎞⎟⎟⎠⎤⎥⎥⎦_𝑚₌₁𝛼_𝑚𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹+_𝑛2_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2×𝜏2⎞⎟⎟⎠⎤⎥⎥⎦2𝑓⁰^𝑛_𝑘=1𝛽_𝑘𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/22+𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−_𝑛2⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴×𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹×𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹2+^𝑛_𝑘=1𝛽_𝑘[𝜆^{𝑘/𝜏]−2}_𝑠=1𝜏×𝐴_−1/22𝑖𝑖𝐴^1/2𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼+𝑖2𝜏𝐴^1/2+𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼−𝑖2𝜏𝐴^1/2𝑓^𝑠+^𝑛_𝑘=1𝛽_𝑘𝜏𝑅𝑅𝑓_{[𝜆𝑘/𝜏]−1}+^𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴^{[𝜆𝑘/𝜏]−1}_𝑠=1𝜏2𝑖𝐴^−1/2×𝑅^{[𝜆𝑘/𝜏]−𝑠}−𝑅^{[𝜆𝑘/𝜏]−𝑠}𝑓^𝑠+^𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝑓^{[𝜆𝑘/𝜏]−1}𝜔=𝑇_𝜏+𝜓,𝐼−^𝑛_𝑚=1𝛼_𝑚𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝐼−𝑖𝜏𝐴^1/22+𝑅^{[𝜆𝑚/𝜏]−1}𝐼+𝑖𝜏𝐴^1/2−_𝑛2⎞⎟⎟⎠_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴×𝑅_{[𝜆𝑚/𝜏]−1}2⁻¹𝑖𝐴^1/2𝐼−(𝑖𝜏/2)𝐴^1/2𝐼+𝑖𝜏𝐴^1/2−𝑅^{[𝜆𝑚/𝜏]−1}𝐼+(𝑖𝜏/2)𝐴^1/2𝐼−𝑖𝜏𝐴^1/2×𝜏2⎤⎥⎥⎦2𝑓⁰^𝑛_𝑘=1𝛽_𝑘𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/22+𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−_𝑛2⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴×𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹×𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹2+^𝑛_𝑘=1𝛽_𝑘[𝜆^{𝑘/𝜏]−2}_𝑠=1𝜏×𝐴_−1/22𝑖𝑖𝐴^1/2𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼+𝑖2𝜏𝐴^1/2+𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼−𝑖2𝜏𝐴^1/2𝑓^𝑠+^𝑛_𝑘=1𝛽_𝑘𝑅𝑅𝑓_{[𝜆𝑘/𝜏]−1}+^𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴^{[𝜆𝑘/𝜏]−1}_𝑠=1𝜏×𝐴_−1/22𝑖−⎡⎢⎢⎣𝑅^{[𝜆𝑘/𝜏]−𝑠}−𝑅^{[𝜆𝑘/𝜏]−𝑠}𝑓^𝑠+𝜏2+𝜆^𝑘−𝜆𝜏^𝑘𝜏𝑓^{[𝜆𝑘/𝜏]−1}+𝜓_𝑛_𝑘=1𝛽_𝑘𝜏2𝐴𝐼+𝜏²𝐴2⁻¹𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏𝐴^1/2+𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖𝜏𝐴^1/2−122^𝑛_𝑘=1𝛽_𝑘𝐼+𝜏²𝐴2⁻¹−𝑅𝑅^{[𝜆𝑘/𝜏]−1}_{[𝜆𝑘/𝜏]−1}𝐼−𝑖2𝜏𝐴^1/2𝐼+𝑖𝜏𝐴^1/2×𝜏𝐼+𝑖2𝜏𝐴^1/2𝐼−𝑖𝜏𝐴^1/22𝑓⁰^𝑛_𝑚=1𝛼_𝑚𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹+_𝑛2_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−⎛⎜⎜⎝_𝑛2⎞⎟⎟⎠⎞⎟⎟⎠_𝑚₌₁𝛼_𝑚[𝜆^{𝑚/𝜏]−1}_𝑠=1𝜏2𝑖𝐴^−1/2𝑅^{[𝜆𝑚/𝜏]−𝑠}−𝑅^{[𝜆𝑚/𝜏]−𝑠}𝑓^𝑠+^𝑛_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚×_{[𝜆𝑚/𝜏]−2}𝜏𝜏_𝑠=1𝜏2𝑖𝐴^−1/2𝑖𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−𝑠}𝐼+𝑖2𝜏𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−𝑠}𝐼−𝑖2𝜏𝐴^1/2𝑓^𝑠+^𝑛_𝑚₌₁𝛼_𝑚𝜏𝜆_𝑚−𝜆^𝑚𝜏𝜏𝑅𝑅𝑓^{[𝜆𝑚/𝜏]−1}−𝜑.(2.10) Thus, formulas (2.9) and (2.10) give a solution of problem (2.1).

Theorem 2.3. Suppose that assumption (2.2) holds and 𝜑∈𝐷(𝐴),  𝜓∈𝐷(𝐴^1/2). Then, for the solution of difference scheme (2.1) the stability inequalities max_{0≤𝑘≤𝑁}‖‖𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝐴^−1/2𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^−1/2𝜓‖‖_𝐻+‖𝜑‖_𝐻max_{0≤𝑘≤𝑁},(2.11)‖‖𝐴^1/2𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^1/2𝜑‖‖_𝐻+‖𝜓‖_𝐻max_{1≤𝑘≤𝑁−1},(2.12)‖‖𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1‖‖_𝐻+max_{0≤𝑘≤𝑁−1}‖‖‖𝐴𝑢_𝑘+𝜏²4𝐴²𝑢_𝑘+1‖‖‖_𝐻≤𝑀^𝑁−1_𝑘=1‖‖𝑓_𝑘−𝑓_𝑘−1‖‖_𝐻+‖‖𝑓₀‖‖_𝐻+‖‖𝐴^1/2𝜓‖‖_𝐻+‖𝐴𝜑‖_𝐻(2.13) hold, where 𝑀 does not depend on 𝜏,  𝜑,  𝜓, and 𝑓_𝑘,  0≤𝑘≤𝑁−1.

Proof. By [18], the following estimates max_{0≤𝑘≤𝑁}‖‖𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝐴^−1/2𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^−1/2𝜔‖‖_𝐻+‖𝜇‖_𝐻max_{0≤𝑘≤𝑁},(2.14)‖‖𝐴^1/2𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^1/2𝜇‖‖_𝐻+‖𝜔‖_𝐻max_{1≤𝑘≤𝑁−1},(2.15)‖‖𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1‖‖_𝐻+max_{0≤𝑘≤𝑁−1}‖‖‖𝐴𝑢_𝑘+𝜏²4𝐴²𝑢_𝑘+1‖‖‖_𝐻≤𝑀^𝑁−1_𝑘=1‖‖𝑓_𝑘−𝑓_𝑘−1‖‖_𝐻+‖‖𝑓₀‖‖_𝐻+‖‖𝐴^1/2𝜔‖‖_𝐻+‖𝐴𝜇‖_𝐻(2.16) hold for the solution of (2.8). Using formulas of 𝜇, 𝜔, and (2.3) and (2.6) the following estimates obtained ‖𝜇‖_𝐻≤𝑀^𝑁−1_𝑠=0‖‖𝐴^−1/2𝑓_𝑠‖‖_𝐻𝜏+‖‖𝐴^−1/2𝜓‖‖_𝐻+‖𝜑‖_𝐻‖‖𝐴,(2.17)_−1/2𝜔‖‖_𝐻≤𝑀^𝑁−1_𝑠=0‖‖𝐴^−1/2𝑓_𝑠‖‖_𝐻𝜏+‖‖𝐴^−1/2𝜓‖‖_𝐻+‖𝜑‖_𝐻.(2.18) Estimate (2.11) follows from (2.14), (2.17), and (2.18). In a similar manner, we obtain max_{0≤𝑘≤𝑁}‖‖𝐴^1/2𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^1/2𝜑‖‖_𝐻+‖𝜓‖_𝐻.(2.19) Now, we obtain the estimates for ‖𝐴𝜇‖_𝐻 and ‖𝐴^1/2𝜔‖_𝐻. First, applying 𝐴 to the formula of 𝜇 and using Abel’s formula, we can write 𝐴𝜇=𝑇_𝜏⎧⎪⎨⎪⎩𝜏2𝑓⁰^𝑛_𝑚=1𝛼_𝑚𝐼+𝜏²𝐴2⁻¹𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹+2𝑖_𝑛𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐴𝐼+𝜏²𝐴2⁻¹×𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−⎛⎜⎜⎝12⎞⎟⎟⎠2^𝑛_𝑚=1𝛼_𝑚[𝜆^{𝑚/𝜏]−1}_𝑠=2𝑅^{[𝜆𝑚/𝜏]−s}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑚/𝜏]−s}𝐼−𝑖𝜏×𝑓_𝑠2𝐴^1/2⁻¹−𝑓_𝑠−1+^𝑛_𝑚₌₁𝛼_𝑚12𝑅^{[𝜆𝑚/𝜏]−1}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑚/𝜏]−1}𝐼−𝑖𝜏2𝐴^1/2⁻¹𝑓¹−^𝑛_𝑚₌₁𝛼_𝑚𝐼+𝜏²𝐴4⁻¹𝑓_{[𝜆𝑚/𝜏]−1}+^𝑛_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚×⎡⎢⎢⎣_{[𝜆𝑚/𝜏]−2}𝜏𝜏_𝑠=212𝑖𝐴^1/2+1×𝑅^{[𝜆𝑚/𝜏]−𝑠}+𝑅^{[𝜆𝑚/𝜏]−𝑠}𝑓^𝑠−𝑓_𝑠−12𝑖𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−1}+𝑅^{[𝜆𝑚/𝜏]−1}𝑓¹+𝑖𝐴^1/2𝑓_{[𝜆𝑚/𝜏]−2}+_𝑛⎤⎥⎥⎦_𝑚₌₁𝛼_𝑚𝜏𝜆_𝑚−𝜆^𝑚𝜏𝜏𝐴𝑅𝑅𝑓^{[𝜆𝑚/𝜏]−1}×𝐼+_𝑛−𝐴𝜑_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴𝐼+𝜏²𝐴2⁻¹𝑖𝐴^1/2⁻¹×𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹2−^𝑛_𝑚=1𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴2⁻¹×𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2+_𝑛2⎤⎥⎥⎦_𝑚₌₁𝛼_𝑚𝐼+𝜏²𝐴2⁻¹𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹2𝑖+^𝑛_𝑚=1𝛼_𝑚𝜆^𝑚−𝜆^𝑚×𝐴_1/2𝜏𝜏𝐼+𝜏²𝐴2⁻¹×𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2×⎡⎢⎢⎣𝜏2⎤⎥⎥⎦2𝑓⁰⎛⎜⎜⎝^𝑛_𝑘=1𝛽_𝑘𝐴^1/2𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/22+𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2+_𝑛2⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝑖𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴×𝐼+𝜏²𝐴2⁻¹×𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−_𝑛2_𝑘=1𝛽_𝑘[𝜆^{𝑘/𝜏]−2}_𝑠=212𝑖𝑅^{[𝜆𝑘/𝜏]−𝑠}−_𝑛+𝑅^{[𝜆𝑘/𝜏]−𝑠}𝑓^𝑠−𝑓_𝑠−1_𝑘=1𝛽_𝑘12𝑖𝑅^{[𝜆𝑘/𝜏]−1}+𝑅^{[𝜆𝑘/𝜏]−1}𝑓¹−^𝑛_𝑘=1𝛽_𝑘𝑖𝑓_{[𝜆𝑚/𝜏]−2}+^𝑛_𝑘=1𝛽_𝑘𝐴^1/2𝜏𝑅𝑅𝑓_{[𝜆𝑘/𝜏]−1}+^𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘×⎡⎢⎢⎣1𝜏𝜏2𝐴^{−1/2[𝜆𝑘/𝜏]−1}_𝑠=2𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼−𝑖𝜏×𝑓_𝑠2𝐴^1/2⁻¹−𝑓_𝑠−1+12𝐴^−1/2𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏2𝐴^1/2⁻¹𝑓¹−𝐴^−1/2𝐼+𝜏²𝐴4⁻¹𝑓_{[𝜆𝑘/𝜏]−1}+_𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴^1/2𝑓_{[𝜆𝑘/𝜏]−1}+𝐴^1/2𝜓⎤⎥⎥⎦⎫⎪⎬⎪⎭.(2.20) Second, applying 𝐴^1/2 to the formula of 𝜔 and using Abel’s formula, we can write 𝐴^1/2𝜔=𝑇_𝜏𝐼−^𝑛_𝑚=1𝛼_𝑚𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝐼−𝑖𝜏𝐴^1/22+𝑅^{[𝜆𝑚/𝜏]−1}𝐼+𝑖𝜏𝐴^1/2−_𝑛2⎞⎟⎟⎠_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐼+𝜏²𝐴×𝑅_{[𝜆𝑚/𝜏]−1}2⁻¹𝑖𝐴^1/2𝐼−(𝑖𝜏/2)𝐴^1/2𝐼+𝑖𝜏𝐴^1/2−𝑅^{[𝜆𝑚/𝜏]−1}𝐼+(𝑖𝜏/2)𝐴^1/2𝐼−𝑖𝜏𝐴^1/2×122𝑓⁰^𝑛_𝑘=1𝛽_𝑘𝐴^1/2𝜏𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/22+𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2+_𝑛2⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐼+𝜏²𝐴2⁻¹×𝐴^1/2𝑖𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑘/𝜏]−1}𝑅⁻¹−_𝑛2_𝑘=1𝛽_𝑘[𝜆^{𝑘/𝜏]−2}_𝑠=212𝑖𝑅^{[𝜆𝑘/𝜏]−s}×𝑓_𝑠+𝑅^{[𝜆𝑘/𝜏]−s}−𝑓_𝑠−1−^𝑛_𝑘=1𝛽_𝑘12𝑖𝑅^{[𝜆𝑘/𝜏]−1}+𝑅^{[𝜆𝑘/𝜏]−1}𝑓¹−^𝑛_𝑘=1𝛽_𝑘𝑖𝑓_{[𝜆𝑘/𝜏]−2}+^𝑛_𝑘=1𝛽_𝑘𝐴^1/2𝜏𝑅𝑅𝑓_{[𝜆𝑘/𝜏]−1}+^𝑛_𝑘=1𝛽_𝑘𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏𝐴^−1/2×⎡⎢⎢⎣12^{[𝜆𝑘/𝜏]−1}_𝑠=2𝑅^{[𝜆𝑘/𝜏]−s}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑘/𝜏]−𝑠}𝐼−𝑖𝜏×𝑓_𝑠2𝐴^1/2⁻¹−𝑓_𝑠−1+12𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏2𝐴^1/2⁻¹𝑓¹−𝐼+𝜏²𝐴4⁻¹𝑓_{[𝜆𝑘/𝜏]−1}+𝐴^1/2−⎡⎢⎢⎣_𝑛𝜓_𝑘=1𝛽_𝑘𝐴^1/2𝜏2𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝐼−𝑖𝜏𝐴^1/2+𝑅^{[𝜆𝑘/𝜏]−1}𝐼+𝑖𝜏𝐴^1/2−_𝑛2⎞⎟⎟⎠_𝑘=1𝛽_𝑘𝐴^−1/2𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑘/𝜏]−1}𝐼−(𝑖𝜏/2)𝐴^1/2𝐼+𝑖𝜏𝐴^1/2−𝑅_{[𝜆𝑘/𝜏]−1}2𝐼+(𝑖𝜏/2)𝐴^1/2𝐼−𝑖𝜏𝐴^1/2×122𝑓⁰^𝑛_𝑚=1𝛼_𝑚𝐼+𝜏²𝐴2⁻¹×𝜏𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹−𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹+2𝑖_𝑛𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚𝜏𝜏𝐴𝐼+𝜏²𝐴2⁻¹×⎛⎜⎜⎝𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼−(𝑖𝜏/2)𝐴^1/2+𝑅^{[𝜆𝑚/𝜏]−1}𝑅⁻¹𝐼+(𝑖𝜏/2)𝐴^1/2−12⎞⎟⎟⎠⎞⎟⎟⎠2^𝑛_𝑚=1𝛼_𝑚𝐴^−1/2×^{[𝜆𝑚/𝜏]−1}_𝑠=2𝑅^{[𝜆𝑚/𝜏]−𝑠}𝐼+𝑖2𝜏𝐴^1/2⁻¹+𝑅^{[𝜆𝑚/𝜏]−𝑠}𝐼−𝑖𝜏×𝑓_𝑠2𝐴^1/2⁻¹−𝑓_𝑠−1+12^𝑛_𝑚=1𝛼_𝑚𝐴^−1/2×𝑅^{[𝜆𝑚/𝜏]−1}𝐼+𝑖𝜏2𝐴^1/2⁻¹+𝑅^{[𝜆𝑚/𝜏]−1}𝐼−𝑖𝜏2𝐴^1/2⁻¹𝑓¹−^𝑛_𝑚₌₁𝛼_𝑚𝐴^−1/2𝐼+𝜏²𝐴4⁻¹𝑓_{[𝜆𝑚/𝜏]−1}−^𝑛_𝑚₌₁𝛼_𝑚𝜆^𝑚−𝜆^𝑚×⎡⎢⎢⎣_{[𝜆𝑚/𝜏]−2}𝜏𝜏_𝑠=212𝑖𝐴^1/2+1×𝑅^{[𝜆𝑚/𝜏]−𝑠}+𝑅^{[𝜆𝑚/𝜏]−𝑠}𝑓^𝑠−𝑓_𝑠−12𝑖𝐴^1/2𝑅^{[𝜆𝑚/𝜏]−1}+𝑅^{[𝜆𝑚/𝜏]−1}𝑓¹+𝑖𝐴^1/2𝑓_{[𝜆𝑚/𝜏]−2}+_𝑛⎤⎥⎥⎦_𝑚₌₁𝛼_𝑚𝜏𝜆_𝑚−𝜆^𝑚𝜏𝜏𝐴𝑅𝑅𝑓^{[𝜆𝑚/𝜏]−1}−𝐴𝜑.(2.21) The following estimates ‖𝐴𝜇‖_𝐻≤𝑀^𝑁−1_𝑠=1‖‖𝑓_𝑠−𝑓_𝑠−1‖‖_𝐻+‖‖𝑓₀‖‖_𝐻+‖‖𝐴^1/2𝜓‖‖_𝐻+‖𝐴𝜑‖_𝐻‖‖𝐴_1/2,𝜔‖‖_𝐻≤𝑀^𝑁−1_𝑠=1‖‖𝑓_𝑠−𝑓_𝑠−1‖‖_𝐻+‖‖𝑓₀‖‖_𝐻+‖‖𝐴^1/2𝜓‖‖_𝐻+‖𝐴𝜑‖_𝐻(2.22) are obtained by using formulas (2.20), (2.21), (2.3), and (2.6).
Estimate (2.13) follows from (2.16), and (2.22). Theorem 2.3 is proved.
Now, let us consider the applications of Theorem 2.3. First, the nonlocal the mixed boundary value problem for hyperbolic equation 𝑢𝑢(_𝑡𝑡0−,𝑥)𝑎=(𝑥)_𝑛𝑢_𝑥_𝑥+𝛿𝑢=𝑓(𝑡,𝑥),0<𝑡<1,0<𝑥<1,_𝑚₌₁𝑢𝛼_𝑚𝑢𝜆_𝑚,𝑥+𝜑(𝑥),0≤𝑥≤1,_𝑡(0,𝑥)=^𝑛_𝑘=1𝑢(𝑡,0)=𝑢(𝑡,1𝛽)_𝑘𝑢,_𝑡𝑢𝜆_𝑘𝑥,𝑥+𝜓(𝑥),0≤𝑥≤1,(𝑡,0)=𝑢_𝑥(𝑡,1),0≤𝑡≤1,(2.23) under assumption (2.2), is considered. Here 𝑎_𝑟(𝑥),  (𝑥∈(0,1)),  𝜑(𝑥),𝜓(𝑥)(𝑥∈[0,1]) and 𝑓(𝑡,𝑥)(𝑡∈(0,1),𝑥∈(0,1)) are given smooth functions and 𝑎_𝑟(𝑥)≥𝑎>0,  𝛿>0. The discretization of problem (2.23) is carried out in two steps.
In the first step, the grid space is defined as follows: [0,1]_ℎ=𝑥∶𝑥_𝑟=𝑟ℎ,0≤𝑟≤𝐾,𝐾ℎ=1.(2.24) We introduce the Hilbert space 𝐿_2ℎ=𝐿₂([0,1]_ℎ),  𝑊^12ℎ=𝑊^12ℎ([0,1]_ℎ) and 𝑊^22ℎ=𝑊^22ℎ([0,1]_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑_𝑟}^1𝐾−1 defined on [0,1]_ℎ, equipped with the norms ‖‖𝜑^ℎ‖‖_𝐿2ℎ=^𝐾−1_𝑟=1‖‖𝜑_ℎ||𝜑^ℎ(𝑥)||²ℎ^1/2,‖‖_𝑊12ℎ=‖‖𝜑^ℎ‖‖_𝐿2ℎ+^𝐾−1_𝑟=1|||𝜑_ℎ_𝑥,𝑗|||²‖‖𝜑_ℎℎ^1/2,‖‖_𝑊22ℎ=‖‖𝜑^ℎ‖‖_𝐿2ℎ+^𝐾−1_𝑟=1|||𝜑_ℎ_𝑥,𝑗|||²ℎ^1/2+^𝐾−1_𝑟=1|||𝜑_ℎ_{𝑥𝑥,𝑗}|||²ℎ^1/2,(2.25) respectively. To the differential operator 𝐴 generated by problem (2.23), we assign the difference operator 𝐴^𝑥ℎ by the formula 𝐴^𝑥ℎ𝜑^ℎ(𝑥)=−(𝑎(𝑥)𝜑_𝑥)_𝑥,𝑟+𝛿𝜑_𝑟^1𝐾−1,(2.26) acting in the space of grid functions 𝜑^ℎ(𝑥)={𝜑_𝑟}^𝐾0 satisfying the conditions 𝜑₀=𝜑_𝐾,  𝜑₁−𝜑₀=𝜑_𝐾−𝜑_𝐾−1. With the help of 𝐴^𝑥ℎ, we arrive at the nonlocal boundary value problem 𝑑²𝑣^ℎ(𝑡,𝑥)𝑑𝑡²𝑣_ℎ+𝐴^𝑥ℎ𝑣^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0≤𝑡≤1,𝑥∈[0,1]_ℎ,(0,𝑥)=^𝑛_𝑗=1𝑣_ℎ𝑡𝛼_𝑗𝑣^ℎ𝜆_𝑗,𝑥+𝜑^ℎ(𝑥),𝑥∈[0,1]_ℎ,(0,𝑥)=^𝑛_𝑗=1𝛽_𝑗𝑣^ℎ𝑡𝜆_𝑗,𝑥+𝜓^ℎ(𝑥),𝑥∈[0,1]_ℎ,(2.27) for an infinite system of ordinary differential equations.
In the second step, we replace problem (2.27) by difference scheme (2.28) 𝑢^ℎ𝑘+1(𝑥)−2𝑢^ℎ𝑘(𝑥)+𝑢^ℎ𝑘−1(𝑥)𝜏²+𝐴^𝑥ℎ𝑢^ℎ𝑘(𝑥)+𝜏²4𝐴^𝑥ℎ²𝑓_ℎ𝑘𝑢^ℎ𝑘+1(𝑥)=𝑓^ℎ𝑘(𝑥),𝑥∈[0,1]_ℎ,𝑢(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,_ℎ0(𝑥)=^𝑛_𝑗=1𝛼_𝑗𝑢^ℎ𝜆_𝑗^/𝜏(𝑥)+𝜏⁻¹𝑢^ℎ[𝜆_𝑗^/𝜏](𝑥)−𝑢^ℎ[𝜆_{𝑗/𝜏]−1}(𝑥)𝜆_𝑗−𝜆^𝑗𝜏𝜏+𝜑^ℎ𝐼+𝜏₂(𝑥),𝑥∈[0,1]_ℎ,𝐴^𝑥ℎ2𝑢^ℎ1(𝑥)−𝑢^ℎ0(𝑥)𝜏−𝜏2𝑓^ℎ0=(𝑥)−𝐴^𝑥ℎ𝑢^ℎ0(𝑥)_𝑛_𝑗=1𝛽_𝑗⎧⎪⎨⎪⎩𝑢^ℎ𝜆_𝑗/𝜏(𝑥)−𝑢^ℎ𝜆_{𝑗/𝜏−1}(𝑥)𝜏+𝜏2+𝜆^𝑗−𝜆^𝑗𝜏𝜏×𝑓^{ℎ[𝜆𝑗/𝜏]}+𝜓_ℎ(𝑥)−𝐴^𝑥ℎ𝑢^ℎ[𝜆_𝑗/𝜏](𝑥)⎫⎪⎬⎪⎭(𝑥),𝑓_ℎ0(𝑥)=𝑓^ℎ(0,𝑥),𝑥∈[0,1]_ℎ.(2.28)

Theorem 2.4. Let 𝜏 and ℎ be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.28) satisfies the following stability estimates: max_{0≤𝑘≤𝑁}‖‖𝑢^ℎ𝑘‖‖_𝐿2ℎ+max_{0≤𝑘≤𝑁}‖‖𝑢^ℎ𝑘_𝑥‖‖_𝐿2ℎ≤𝑀₁max_{0≤𝑘≤𝑁−1}‖‖𝑓^ℎ𝑘‖‖_𝐿2ℎ+‖‖𝜓^ℎ‖‖_𝐿2ℎ+‖‖𝜑^ℎ_𝑥‖‖_𝐿2ℎmax_{1≤𝑘≤𝑁−1},‖‖𝜏⁻²𝑢^ℎ𝑘+1−2𝑢^ℎ𝑘+𝑢^ℎ𝑘−1‖‖_𝐿2ℎ+max_{0≤𝑘≤𝑁}‖‖𝑢^ℎ𝑘_𝑥𝑥‖‖_𝐿2ℎ≤𝑀₁‖‖𝑓^ℎ0‖‖_𝐿2ℎ+max_{1≤𝑘≤𝑁−1}‖‖𝜏⁻¹𝑓^ℎ𝑘−𝑓^ℎ𝑘−1‖‖_𝐿2ℎ+‖‖𝜓^ℎ_𝑥‖‖_𝐿2ℎ+‖‖𝜑^ℎ_𝑥_𝑥‖‖_𝐿2ℎ.(2.29) Here, 𝑀₁ does not depend on 𝜏,  ℎ,  𝜑^ℎ(𝑥),  𝜓^ℎ(𝑥) and 𝑓^ℎ𝑘,  0≤𝑘<𝑁.

The proof of Theorem 2.4 is based on abstract Theorem 2.3 and symmetry properties of the operator 𝐴^𝑥ℎ defined by (2.26).

Second, for the 𝑚-dimensional hyperbolic equation under assumption (2.2) is considered. Let Ω be 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. In [0,1]×Ω, the mixed boundary value problem for the multidimensional hyperbolic equation 𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−^𝑚_𝑟=1𝑎_𝑟𝑥=𝑥₁(𝑥)𝑢_𝑥𝑟_𝑥𝑟=𝑓(𝑡,𝑥),𝑢(0,𝑥)=,…_𝑛,𝑥_𝑚∈Ω,0<𝑡<1,_𝑗=1𝑢𝛼_𝑗𝑢𝜆_𝑗,𝑥+𝜑(𝑥),𝑥∈Ω;_𝑡(0,𝑥)=^𝑛_𝑘=1𝑢(𝑡,𝑥)=0,𝑥𝛽_𝑘∈𝑢_𝑡𝑆𝜆_𝑘,𝑥+𝜓(𝑥),𝑥∈Ω;(2.30) is considered.

Here, 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 discretization of problem (2.30) is carried out in two steps. In the first step, let us define the grid sets Ω_ℎ=𝑥=𝑥_𝑟=ℎ₁𝑟₁,…,ℎ_𝑚Ω𝑟_𝑚,𝑟=𝑟₁,…,𝑟_𝑚,0≤𝑟_𝑗≤𝑁_𝑗,ℎ_𝑗𝑁_𝑗=1,𝑗=1,…,𝑚_ℎ,=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆.(2.31) We introduce the Banach space 𝐿_2ℎ=𝐿₂(Ω_ℎ),  𝑊^12ℎ=𝑊^12ℎ(Ω_ℎ) and 𝑊^22ℎ=𝑊^22ℎ(Ω_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑(ℎ₁𝑟₁,…,ℎ_𝑚𝑟_𝑚)} defined on Ω_ℎ, equipped with the norms ‖‖𝜑^ℎ‖‖_𝐿2(Ω_ℎ)=⎛⎜⎜⎝_𝑥∈Ωℎ||𝜑^ℎ(𝑥)||²ℎ₁⋯ℎ_𝑚⎞⎟⎟⎠^1/2‖‖𝜑_ℎ,‖‖_𝑊12ℎ=‖‖𝜑^ℎ‖‖_𝐿2ℎ+⎛⎜⎜⎝_𝑥∈Ωℎ^𝑚_𝑟=1|||𝜑_ℎ_{𝑥𝑟,𝑗𝑟}|||²ℎ₁⋯ℎ_𝑚⎞⎟⎟⎠^1/2‖‖𝜑_ℎ,‖‖_𝑊22ℎ=‖‖𝜑^ℎ‖‖_𝐿2ℎ+⎛⎜⎜⎝_𝑥∈Ωℎ^𝑚_𝑟=1|||𝜑_ℎ_𝑥𝑟|||²ℎ₁⋯ℎ_𝑚⎞⎟⎟⎠^1/2+⎛⎜⎜⎝_𝑥∈Ωℎ^𝑚_𝑟=1|||𝜑_ℎ_{𝑥𝑟𝑥𝑟,𝑗𝑟}|||²ℎ₁⋯ℎ_𝑚⎞⎟⎟⎠^1/2,(2.32) respectively. To the differential operator 𝐴 generated by problem (2.27), we assign the difference operator 𝐴^𝑥ℎ by the formula 𝐴^𝑥ℎ𝑢^ℎ=−^𝑚_𝑟=1𝑎^𝑟(𝑥)𝑢^ℎ_𝑥𝑟_{𝑥𝑟,𝑗𝑟},(2.33) acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the conditions 𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆_ℎ. Note that 𝐴^𝑥ℎ is a self-adjoint positive definite operator in 𝐿₂(Ω_ℎ). With the help of 𝐴^𝑥ℎ we arrive at the nonlocal boundary value problem 𝑑²𝑣^ℎ(𝑡,𝑥)𝑑𝑡²𝑣_ℎ+𝐴^𝑥ℎ𝑣^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,(0,𝑥)=^𝑛_𝑙=1𝑑𝑣_ℎ𝛼_𝑙𝑣^ℎ𝜆_𝑙,𝑥+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,(0,𝑥)𝑑𝑡=^𝑛_𝑙=1𝛽_𝑙𝑣^ℎ𝑡𝜆_𝑙,𝑥+𝜓^ℎ(𝑥),𝑥∈Ω,(2.34) for an infinite system of ordinary differential equations.

In the second step, we replace problem (2.34) by difference scheme (2.35) 𝑢^ℎ𝑘+1(𝑥)−2𝑢^ℎ𝑘(𝑥)+𝑢^ℎ𝑘−1(𝑥)𝜏²+𝐴^𝑥ℎ𝑢^ℎ𝑘(𝑥)+𝜏²4𝐴^𝑥ℎ²𝑓_ℎ𝑘𝑢^ℎ𝑘+1(𝑥)=𝑓^ℎ𝑘(𝑥),𝑥∈Ω_ℎ,𝑢(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,_ℎ0(𝑥)=^𝑛_𝑙=1𝐼+𝜏₂𝛼_𝑙𝑢^ℎ[𝜆_𝑙^/𝜏](𝑥)+𝜏⁻¹𝑢^ℎ[𝜆_𝑙^/𝜏](𝑥)−𝑢^ℎ[𝜆_{𝑙/𝜏]−1}(𝑥)𝜆_𝑙−𝜆_𝑙/𝜏𝜏+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝐴^𝑥ℎ2𝑢^ℎ1(𝑥)−𝑢^ℎ0(𝑥)𝜏−𝜏2𝑓^ℎ0=(𝑥)−𝐴^𝑥ℎ𝑢^ℎ0(𝑥)_𝑛_𝑙=1𝛽_𝑙𝑢^{ℎ[𝜆𝑙/𝜏]}(𝑥)−𝑢^ℎ[𝜆_{𝑙/𝜏]−1}(𝑥)𝜏+𝜏2+𝜆^𝑙−𝜆^𝑙𝜏𝜏×𝑓^{ℎ[𝜆𝑙/𝜏]}+𝜓_ℎ(𝑥)−𝐴^𝑥ℎ𝑢^ℎ[𝜆_𝑙/𝜏](𝑥)(𝑥),𝑓_ℎ0(𝑥)=𝑓^ℎ(0,𝑥),𝑥∈Ω_ℎ.(2.35)

Theorem 2.5. Let 𝜏 and ℎ be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.35) satisfies the following stability estimates: max_{0≤𝑘≤𝑁}‖‖𝑢^ℎ𝑘‖‖_𝐿2ℎ+max_{𝑚0≤𝑘≤𝑁}_𝑟=1‖‖𝑢^ℎ𝑘_{𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎ≤𝑀₁max^{0≤𝑘≤𝑁−1}‖‖𝑓^ℎ𝑘‖‖_𝐿2ℎ+‖‖𝜓^ℎ‖‖_𝐿2ℎ+^𝑚_𝑟=1‖‖𝜑^ℎ_{𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎmax_{1≤𝑘≤𝑁−1},‖‖𝜏⁻²𝑢^ℎ𝑘+1−2𝑢^ℎ𝑘+𝑢^ℎ𝑘−1‖‖_𝐿2ℎ+max_{𝑚0≤𝑘≤𝑁}_𝑟=1‖‖𝑢^ℎ𝑘_{𝑥𝑟𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎ≤𝑀₁‖‖𝑓^ℎ0‖‖_𝐿2ℎ+max_{1≤𝑘≤𝑁−1}‖‖𝜏⁻¹𝑓^ℎ𝑘−𝑓^ℎ𝑘−1‖‖_𝐿2ℎ+^𝑚_𝑟=1‖‖𝜓^ℎ_{𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎ+^𝑚_𝑟=1‖‖𝜑^ℎ_{𝑥𝑟𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎ.(2.36) Here, 𝑀₁ does not depend on 𝜏,  ℎ,  𝜑^ℎ(𝑥),  𝜓^ℎ(𝑥) and 𝑓^ℎ𝑘,  0≤𝑘<𝑁.

The proof of Theorem 2.5 is based on abstract Theorem 2.3, symmetry properties of the operator 𝐴^𝑥ℎ defined by formula (2.33), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in 𝐿_2ℎ.

Theorem 2.6. For the solutions of the elliptic difference problem 𝐴^𝑥ℎ𝑢^ℎ𝑢_ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,(𝑥)=0,𝑥∈𝑆_ℎ,(2.37) the following coercivity inequality holds [21]: ^𝑚_𝑟=1‖‖𝑢^ℎ𝑥_{𝑟𝑥𝑟,𝑗𝑟}‖‖_𝐿2ℎ≤𝑀‖‖𝜔^ℎ‖‖_𝐿2ℎ.(2.38)

3. The Second Order of Accuracy Difference Scheme Generated by 𝐴

Note that the difference scheme (2.1) generated by 𝐴² is based on the second order approximation formula for the differential equation 𝑢^(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡) and second order of approximation formulas for nonlocal boundary conditions. Using the differential equation above, one can construct the second order of approximation formula for this differential equation. Thus, we obtain the following difference scheme: 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+12𝐴𝑢^𝑘+14𝐴𝑢^𝑘+1𝑓_𝑘+𝑢_𝑘−1=𝑓_𝑘,𝑢=𝑓𝑡_𝑘,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₀=^𝑛_𝑚₌₁𝛼_𝑚𝑢^{[𝜆𝑚/𝜏]}+𝜏⁻¹𝑢_{[𝜆𝑚/𝜏]}−𝑢_{[𝜆𝑚/𝜏]−1}×𝜆_𝑚−𝜆^𝑚𝐼+𝜏₂𝜏𝜏+𝜑,𝐴4𝐼+𝜏²𝐴4𝜏⁻¹𝑢₁−𝑢₀−𝜏2𝑓⁰=_𝑛−𝐴𝑢₀_𝑘=1𝛽_𝑘𝜏⁻¹𝑢_{[𝜆𝑘/𝜏]}−𝑢_{[𝜆𝑘/𝜏]−1}+𝜏2+𝜆^𝑘−𝜆^𝑘𝜏𝜏×𝑓^{[𝜆𝑘/𝜏]}𝑓−𝐴𝑢_{[𝜆𝑘/𝜏]}+𝜓,₀=𝑓(0).(3.1) In exactly the same manner using the method of Section 2 one proves

Theorem 3.1. Suppose that assumption (2.2) holds and 𝜑∈𝐷(𝐴),  𝜓∈𝐷(𝐴^1/2). Then, for the solution of difference scheme (3.1) the following stability inequalities max_{0≤𝑘≤𝑁}‖‖𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝐴^−1/2𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^−1/2𝜓‖‖_𝐻+‖𝜑‖_𝐻max_{0≤𝑘≤𝑁},(3.2)‖‖𝐴^1/2𝑢_𝑘‖‖_𝐻≤𝑀^𝑁−1_𝑘=0‖‖𝑓_𝑘‖‖_𝐻𝜏+‖‖𝐴^1/2𝜑‖‖_𝐻+‖𝜓‖_𝐻max_{1≤𝑘≤𝑁−1},(3.3)‖‖𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1‖‖_𝐻+max_{1≤𝑘≤𝑁−1}‖‖‖12𝐴𝑢^𝑘+14𝐴(𝑢^𝑘+1+𝑢_𝑘−1)‖‖‖_𝐻≤𝑀^𝑁−1_𝑘=1‖‖𝑓_𝑘−𝑓_𝑘−1‖‖_𝐻+‖‖𝑓₀‖‖_𝐻+‖‖𝐴^1/2𝜓‖‖_𝐻+‖𝐴𝜑‖_𝐻(3.4) hold, where 𝑀 does not depend on 𝜏,  𝜑,  𝜓 and 𝑓_𝑘,  0≤𝑘≤𝑁−1.

Moreover, we can construct the difference schemes of a second order of accuracy with respect to one variable for approximate solution of nonlocal boundary value problems (2.30) and (2.34). Therefore, abstract Theorem 3.1 permits us to obtain the stability estimates for the solution of these difference schemes.

4. Numerical Analysis

In this section, several numerical examples are presented for the solution of problem (4.1), demonstrating the accuracy of the difference schemes. In general, we have not been able to determine a sharp estimate for the constants figuring in the stability inequalities. However, the numerical examples are presented for the following nonlocal boundary value problem 𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−𝜕²𝑢(𝑡,𝑥)𝜕𝑥²𝑢(0,𝑥)=1+𝑢(𝑡,𝑥)=6𝑡+2𝑡³sin𝑥,0<𝑡<1,0<𝑥<𝜋,𝑢4𝑢(1,𝑥)−14𝑢21,𝑥−372sin𝑥,0≤𝑥≤𝜋,_𝑡(0,𝑥)=14𝑢^𝑡(1,𝑥)−1𝑢(𝑡,0)=𝑢4(𝑡𝑢,^𝑡𝜋)21=,𝑥0,−019≤6s𝑡in≤𝑥1,.0≤𝑥≤𝜋,(4.1) The exact solution of this problem is 𝑢(𝑡,𝑥)=𝑡³sin𝑥.

We 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_𝑘,𝑥_𝑛)∶𝑡_𝑘=𝑘𝜏,0≤𝑘≤𝑁,𝑁𝜏=1,𝑥_𝑛=𝑛ℎ,0≤𝑛≤𝑀,𝑀ℎ=𝜋}. For the approximate solution of problem (4.1), we have applied the first order and the two different types of second orders of accuracy difference schemes, respectively. Solving these difference equations we have applied a procedure of modified Gauss elimination method with respect to 𝑛 with matrix coefficients.

First let us consider the first order of accuracy in time difference scheme (1.4) for the approximate solution of the Cauchy problem (4.1). Using difference scheme (1.4), we obtain 𝑢^𝑛𝑘+1−2𝑢^𝑘𝑛+𝑢^𝑛𝑘−1𝜏²−𝑢^𝑘+1𝑛+1−2𝑢^𝑛𝑘+1+𝑢^{𝑘+1𝑛−1}ℎ²1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥_𝑛+𝑢^𝑛𝑘+1=𝑓𝑡_𝑘+1,𝑥_𝑛,𝑡_𝑘=𝑘𝜏,𝑢=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=𝜋,_0𝑛=14𝑢^𝑁𝑛−14𝑢^{𝑛(𝑁/2)+1}−7𝜏32sin𝑥,1≤𝑛≤𝑀−1,₋₁𝑢^1𝑛𝑥−𝑢^0𝑛=14𝜏⁻¹𝑢^𝑁𝑛−𝑢^𝑛𝑁−1−41𝜏⁻¹𝑢^{𝑛(𝑁/2)+1}−𝑢^𝑛𝑁/2−196sin𝑥,_𝑛=𝑛ℎ,1≤𝑛≤𝑀−1,𝑢^𝑘0=𝑢^𝑘𝑀=0,0≤𝑘≤𝑁,(4.2) for the approximate solution of Cauchy problem (4.1). We have (𝑁+1)×(𝑁+1) system of linear equation in (4.2) and we can write them in the matrix form as 𝐴𝑈_𝑛+1+𝐵𝑈_𝑛𝑈+𝐶𝑈_𝑛−1=𝐷𝜑_𝑛,1≤𝑛≤𝑀−1,₀=0̃,𝑈_𝑀=0̃,(4.3) where 𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1𝐴0=0⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣000000⋅000000⋅𝑎0000⋅⋅𝑎0000⋅1⋅⋅⋅⋅⋅⋅00000⋅𝑎0000⋅00000⋅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},−00000𝑏⋅110000𝑏𝑐⋅𝑑0000𝑏𝑐⋅𝑑00000𝑐⋅𝑑000000⋅⋅⋅⋅⋅⋅⋅⋅⋅−0000004⋅10000000⋅⋅⋅⋅⋅⋅⋅⋅⋅000000𝑏⋅00000𝑏𝑐⋅𝑑00000𝑐⋅−𝑑00000⋅41414⋅0041−41⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)}𝐶=𝐴,𝐷=⎡⎢⎢⎢⎢⎢⎢⎣100⋅010⋅⋅⋅⋅⋅001⋅⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝜑_𝑘𝑛,=⎡⎢⎢⎢⎢⎢⎢⎣𝜑^0𝑛𝜑^1𝑛𝜑⋮_𝑁𝑛⎤⎥⎥⎥⎥⎥⎥⎦_(𝑁+1)×1𝜑,0≤𝑘≤𝑁,_0𝑛=−732sin𝑥^𝑛,𝜑^𝑁𝑛=−9𝜑_𝑘𝑛16sin𝑥^𝑛,=𝑓6𝑡𝑡^𝑘+1_𝑘+1+2𝑡_𝑘+1³sin𝑥^𝑛,𝑈,𝑥_𝑛,1≤𝑘≤𝑁−1,_𝑘𝑠=⎡⎢⎢⎢⎢⎢⎢⎣𝑢^0𝑠𝑢^1𝑠𝑢⋮_𝑁𝑠⎤⎥⎥⎥⎥⎥⎥⎦_(𝑁+1)×1,0≤𝑘≤𝑁,𝑠=𝑛±1,𝑛.(4.4) Here, 𝑎=−1ℎ²,𝑏=1𝜏²,𝑐=−2𝜏²,𝑑=1𝜏²+2ℎ²+1.(4.5) For the solution of difference equation (4.2), we have applied the modified Gauss elimination method. Therefore, we seek a solution of the matrix equation by using the following iteration formula: 𝑢_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1,𝑛=𝑀−1,…,2,1,0,(4.6) where 𝛼_𝑗,  𝛽_𝑗(𝑗=1,…,𝑀) are (𝑁+1)×(𝑁+1) square matrices and𝛾_𝑗 are (𝑁+1)×1 column matrices. Now, we obtain formula of 𝛼_𝑛+1,  𝛽_𝑛+1, 𝛽𝛼_𝑛+1=−𝐵+𝐶𝛼_𝑛⁻¹𝐴,_𝑛+1=𝐵+𝐶𝛼_𝑛⁻¹𝐷𝜑_𝑛−𝐶𝛽_𝑛,𝑛=1,2,3,…𝑀−1.(4.7) Note that 𝛼₁=⎡⎢⎢⎢⎢⎣00⋅⋅⋅⋅00⋅⎤⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝛽₁=⎡⎢⎢⎢⎢⎣00⋅⋅⋅⋅00⋅⎤⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},(4.8) and 𝑈_𝑀=0̃.

Thus, using the formulas and matrices above we obtain the difference scheme first order of accuracy in 𝑡 and second order of accuracy in 𝑥 for approximate solution of nonlocal boundary value problem (4.1).

Second, let us consider the second order of accuracy in time implicit difference scheme (3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain 𝑈^𝑛𝑘+1−2𝑈^𝑘𝑛+𝑈^𝑛𝑘−1𝜏²−𝑈^𝑘𝑛+1−2𝑈^𝑘𝑛+𝑈^𝑘𝑛−12ℎ²−𝑈^𝑘+1𝑛+1−2𝑈^𝑛𝑘+1+𝑈^{𝑘+1𝑛−1}4ℎ²−𝑈^{𝑘−1𝑛+1}−2𝑈^𝑛𝑘−1+𝑈^{𝑘−1𝑛−1}4ℎ²=6𝑡_𝑘+2𝑡_𝑘³sin𝑥𝑥^𝑛,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,_𝑛𝑢=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=𝜋,_0𝑛=14𝑢^𝑁−14𝑢^(𝑁/2)+1−7𝐼+𝜏₂32sin𝑥,1≤𝑛≤𝑀−1,𝐴^𝑥ℎ4𝐼+𝜏²𝐴^𝑥ℎ4𝑢^ℎ1𝑥_𝑛−𝑢^ℎ0𝑥_𝑛+𝜏2𝑓^ℎ0,𝑥_𝑛−𝐴^𝑥ℎ=1𝑢^ℎ0𝑥_𝑛−4(12𝜏)⁻¹−𝑢^ℎ𝑁−2𝑥_𝑛+4𝑢^ℎ𝑁−1𝑥_𝑛−3𝑢^ℎ𝑁𝑥_𝑛4(2𝜏)⁻¹−3𝑢^{ℎ(𝑁/2)+1}𝑥_𝑛+𝑢4𝑢^{ℎ(𝑁/2)+1}𝑥_𝑛−𝑢^{ℎ(𝑁/2)+3}𝑥_𝑛−196sin𝑥^𝑛,1≤𝑛≤𝑀−1,_𝑘0=𝑢^𝑘𝑀=0,0≤𝑘≤𝑁,(4.9) for the approximate solution of problem (4.1). We have again the same linear system (4.3); therefore, we use exactly the same method that we have used for the solution of difference equation (4.2), but matrices are different as follows: 𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣10𝐴0=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣00𝑧000⋅⋅𝑤0𝑧00⋅1𝑤0𝑧00⋅00𝑧00⋅⋅⋅⋅⋅⋅⋅000𝑧0⋅𝑤0000⋅000𝑧0⋅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},−0000𝑏3⋅000𝑏𝑐⋅𝑑000𝑐⋅𝑑0000⋅⋅⋅⋅⋅⋅⋅000004⋅000000⋅000000⋅⋅⋅⋅⋅⋅⋅⋅0000𝑏𝑐⋅𝑑0000𝑐⋅−𝑑0000⋅142𝜏42𝜏−12𝜏0⋅−38𝜏84𝜏−18𝜏⋅−18𝜏48𝜏−38𝜏⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)}𝐶=𝐴,𝐷=⎡⎢⎢⎢⎢⎢⎢⎣100⋅010⋅⋅⋅⋅⋅001⋅⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},.(4.10) Here, 𝑥=1𝜏²+12ℎ²+14,𝑦=−𝜏2²+1ℎ²+12,𝑧=4−ℎ1²,𝑤=−12ℎ².(4.11) Thus, using the above matrices we obtain the difference scheme second order of accuracy in 𝑡 and 𝑥 for approximate solution of nonlocal boundary value problem (4.1).

Next, let us consider the second order of accuracy in time implicit difference scheme (3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain 𝑢^𝑛𝑘+1−2𝑢^𝑘𝑛+𝑢^𝑛𝑘−1𝜏²−𝑢^𝑘𝑛+1−2𝑢^𝑘𝑛+𝑢^𝑘𝑛−1ℎ²+𝑢^𝑘𝑛+𝜏²4𝑢^𝑘+1𝑛+2−4𝑢^𝑘+1𝑛+1+6𝑢^𝑛𝑘+1−4𝑢^{𝑘+1𝑛−1}+𝑢^{𝑘+1𝑛−2}ℎ⁴−2𝑢^𝑘+1𝑛+1−2𝑢^𝑛𝑘+1+𝑢^{𝑘+1𝑛−1}=6𝑡_𝑘ℎ²𝑡_𝑘+2𝑡_𝑘³sin𝑥^𝑛,=𝑘𝜏,1≤𝑢𝑘≤𝑁−1,𝑁𝜏=1,𝑥_𝑛=𝑛ℎ,1≤𝑛≤𝑀−1,𝑀ℎ=𝜋,_0𝑛=14𝑢^𝑁−14𝑢^(𝑁/2)+1−7𝐼+𝜏₂32sin𝑥,1≤𝑛≤𝑀−1,𝐴^𝑥ℎ4𝐼+𝜏²𝐴^𝑥ℎ4𝑢^ℎ1𝑥_𝑛−𝑢^ℎ0𝑥_𝑛+𝜏2𝑓^ℎ0,𝑥_𝑛−𝐴^𝑥ℎ=1𝑢^ℎ0𝑥_𝑛−4(12𝜏)⁻¹−𝑢^ℎ𝑁−2𝑥_𝑛+4𝑢^ℎ𝑁−1𝑥_𝑛−3𝑢^ℎ𝑁𝑥_𝑛−49(2𝜏)⁻¹−3𝑢^{ℎ(𝑁/2)+1}𝑥_𝑛+4𝑢^{ℎ(𝑁/2)+1}𝑥_𝑛−𝑢^{ℎ(𝑁/2)+3}𝑥_𝑛16sin𝑥^𝑛,1≤𝑢𝑛≤𝑀−1,_𝑘0𝑢=𝑢^𝑘𝑀=0,0≤𝑘≤𝑁._𝑘1=45𝑢^𝑘2−15𝑢^𝑘3,𝑢^𝑘𝑀−1=45𝑢^𝑘𝑀−2−15𝑢^𝑘𝑀−3,0≤𝑘≤𝑁,(4.12) for the approximate solution of problem (4.1). We have (𝑁+1)×(𝑁+1) system of linear equation in (4.12) and we can write in the matrix form as 𝑈𝐴𝑈_𝑛+2+𝐵𝑈_𝑛+1+𝐶𝑈_𝑛+𝐷𝑈_𝑛−1+𝐸𝑈_𝑛−2=𝑅𝜑_𝑛,2≤𝑛≤𝑀−2,₀=𝑈_𝑀=0̃,𝑈^𝑘3=4𝑈^𝑘2−5𝑈^𝑘1,𝑈^𝑘𝑀−3=4𝑈^𝑘𝑀−2−5𝑈^𝑘𝑀−1,0≤𝑘≤𝑁,(4.13) where 𝐵𝐴==⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣00000⋅00000⋅𝑤0000⋅00000⋅𝑤𝑥0000000𝑦⋅⋅𝑥0000⋅0000𝑦⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000𝑤⋅0000⋅𝑥0000⋅0000𝑦⋅00000⋅00000⋅⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},_{(𝑁+1)×(𝑁+1)}𝐶=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1000⋅1,−𝑣00003⋅𝑣000𝑡⋅000𝑧𝑡⋅0𝑧000⋅⋅⋅⋅⋅⋅⋅000004⋅000000⋅000000⋅⋅⋅⋅⋅⋅⋅⋅𝑣0000𝑡⋅0000𝑧𝑡⋅−0000𝑧⋅142𝜏42𝜏−12𝜏0⋅−38𝜏84𝜏−18𝜏⋅−18𝜏48𝜏−38𝜏⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)}𝐷=𝐵,𝐸=𝐴,𝑅=⎡⎢⎢⎢⎢⎢⎢⎣100⋅010⋅⋅⋅⋅⋅001⋅⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝑈_𝑘𝑠,=⎡⎢⎢⎢⎢⎢⎢⎣𝑢^0𝑠𝑢^1𝑠𝑢⋮_𝑁𝑠⎤⎥⎥⎥⎥⎥⎥⎦_(𝑁+1)×1,0≤𝑘≤𝑁,𝑠=𝑛±2,𝑛±1,𝑛.(4.14) Here, 𝑥=𝜏²4ℎ⁴,𝑦=−𝜏²ℎ⁴,𝑧=1𝜏²+3𝜏²2ℎ⁴,𝑡=−2𝜏²𝑑=1,𝜏²+2ℎ²+1,𝑣=1𝜏²,𝑤=−1ℎ².(4.15) For the solution of the linear system (4.13), we use the modified variant Gauss elimination method and seek a solution of the matrix equation by the following form: 𝑢_𝑛=𝛼_𝑛+1𝑢_𝑛+1+𝛽_𝑛+1𝑢_𝑛+2+𝛾_𝑛+1,𝑛=𝑀−2,…,2,1,0,(4.16) where 𝛼_𝑗,  𝛽_𝑗  (𝑗=1∶𝑀−1) are (𝑁+1)×(𝑁+1) square matrices and 𝛾_𝑗-𝑠 are (𝑁+1)×1 column matrices. We obtain the following formulas of 𝛼_𝑛+1,  𝛽_𝑛+1,  𝛾_𝑛+1 from linear system (4.13) by using formula (4.16): 𝛼_𝑛+1=𝐶+𝐷𝛼_𝑛+𝐸𝛼_𝑛−1𝛼_𝑛+𝐸𝛽_𝑛−1⁻¹𝛽_𝑛+1×−𝐵−𝐷𝛽_𝑛−𝐸𝛼_𝑛−1𝛽_𝑛,𝛾=−𝐶+𝐷𝛼_𝑛+𝐸𝛼_𝑛−1𝛼_𝑛+𝐸𝛽_𝑛−1⁻¹𝐴,_𝑛+1=𝐶+𝐷𝛼_𝑛+𝐸𝛼_𝑛−1𝛼_𝑛+𝐸𝛽_𝑛−1⁻¹×𝑅𝜑_𝑛−𝐷𝛾_𝑛+𝐸𝛼_𝑛−1𝛾_𝑛+𝐸𝛾_𝑛−1.(4.17) We have 𝛼₁,  𝛽₁,  𝛾₁,  𝛼₂,  𝛽₂,  𝛾₂∶𝛼₁=⎡⎢⎢⎢⎢⎣𝛾00⋅₁⋅⋅⋅00⋅⎤⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝛽₁=⎡⎢⎢⎢⎢⎣00⋅⋅⋅⋅00⋅⎤⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝛼₂=⎡⎢⎢⎢⎢⎢⎢⎣⋮000⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(1)},𝛾₂=⎡⎢⎢⎢⎢⎢⎢⎣⋮000⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(1)},=⎡⎢⎢⎢⎢⎢⎢⎢⎣45040⋅00⋅05⋅⋅⋅⋅04⋅5⎤⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)},𝛽₂=⎡⎢⎢⎢⎢⎢⎢⎢⎣−105−01⋅00⋅0⋅5⋅⋅⋅−0⋅15⎤⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁+1)×(𝑁+1)}𝑈_𝑀,=0̃,𝑈_𝑀−1=𝛽_𝑀−2+5𝐼−4𝐼−𝛼_𝑀−2𝛼_𝑀−1⁻¹4𝐼−𝛼_𝑀−2𝛾_𝑀−1−𝛾_𝑀−2.(4.18) Thus, using the above matrices we obtain the difference scheme second order of accuracy in 𝑡 and 𝑥 for approximate solution of nonlocal boundary value problem (4.1).

Matlab is a programming language for numeric scientific computation. One of its characteristic features is the use of matrices as the only data type. Therefore, the implementations of numerical examples are carried out by Matlab.

Now, we will give some numerical results for the solutions of (4.2), (4.9), and (4.12) for different 𝑀=𝑁 values where 𝑁 and 𝑀 are the step numbers for the time and space variables, respectively. Note that the grid step numbers 𝑁 and 𝑀 in the given examples are chosen equal for clarity and this is not necessary for the stability and solutions of the difference schemes.

The errors are computed by the following formula: 𝐸^𝑁𝑀=max_{1≤𝑘≤𝑀−1}^𝑁−1_𝑘=1||𝑢𝑡_𝑘,𝑥_𝑛−𝑢^𝑘𝑛||²ℎ^1/2.(4.19) Here, 𝑢(𝑡_𝑘,𝑥_𝑛) represents the exact solution and 𝑢^𝑘𝑛 represents the approximate solution at (𝑡_𝑘,𝑥_𝑛). The errors and the related CPU times are presented in Tables 1 and 2, respectively, for 𝑁=𝑀=20,  30,  40,  80,  300, and 500. The implementations are carried out by MATLAB 7.1 software package and obtained by a PC System 64 bit, Pentium (R) Core(TM) i5 CPU, 3.20 6 Hz, 3.19 Hz, 4000 Mb of RAM.

Table 1 

Comparison of errors for approximate solutions.

Table 2 

CPU times.

Let us denote the first order of accuracy difference scheme (4.2) as 𝐹, the second order of accuracy difference scheme generated by three points (4.9) as 𝑆1, and the second order of accuracy difference scheme generated by five points (4.12) as 𝑆2.

In order to get accurate results, CPU times are recorded by running each program 100 times for small values 𝑁=𝑀=20,  30,  40,  80 and taking the average of the elapsed time.

The following conclusions can be noted for the comparison of the numerical results presented in the tables above. (1)(i) In the tables, it is noted that almost the same accuracy is achieved by 𝑆1 with data error=0.0020,𝑁=20 and by 𝐹 with data error=0.0019,𝑁=500 in different CPU times; 545s and 0.0086s, respectively. This means the use of the difference scheme 𝑆1 accelerates the computation with a ratio of more than 545/0.0086≅63372 times, that is, 𝑆1 is considerably faster than 𝐹.(ii) It is also noted that almost the same accuracy is achieved by the difference scheme 𝑆2 with data error =0.0031,𝑁=20, and by the difference scheme 𝐹 with data error=0.0032,𝑁=300 in different CPU times; 0.0112s and 69s, respectively. Thus, the use of the difference scheme 𝑆2 accelerates the computation with a ratio of more than 69/0.0112≅6160 times, that is, 𝑆2 is considerably faster than 𝐹.(iii) When we consider almost the same CPU times for 𝐹 and 𝑆1 as 0.0080s ≅0.0086s, 𝐹 computes the solution with an error=0.0494, for 𝑁=20, where the difference scheme 𝑆1 computes the solution with an error=0.0020, which is almost 25 times smaller error than the computation error of 𝐹. Namely, 𝑆1 yields 25 times more accurate results than 𝐹. For 𝑆2, this ratio reduces to 0.0112/0.0080≅1.4 times.(iv) While both types of the second order difference schemes reach approximately the same accuracy, the CPU time for the difference scheme 𝑆2 is always greater than 𝑆1.(v) The CPU times of difference schemes 𝐹,  𝑆1, and 𝑆2 recorder for 𝑁≥100 exceed ones. Matlab gives “out of memory” error for 𝑆2 with values 𝑁=500 which means the memory of the computer is not enough. It is not necessary to have results for 𝑆2 when 𝑁=500, since it is obvious that CPU time of 𝑆2 is more than that of 𝐹 and 𝑆1.(vi) It is observed from the tables that for larger 𝑁 values the numerical results become approximately the same for each difference scheme in the reliable range of the CPU times. This indicates that the approximation made for the solution of problem (4.1) is valid.

In conclusion, the second order difference schemes are much more accurate than the first order difference scheme, and the second order difference scheme generated by three points is more convenient than the second order difference scheme generated by five points when considering the CPU times and the error levels. Comparing with many other numerical methods, our method is not based on the relationship between the grid step sizes of time and space variables (see [25–30] and the references therein).

Acknowledgment

The authors would like to thank Professor P. E. Sobolevskii for his helpful suggestions to the improvement of this paper.