Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

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 firstand 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.


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 a r x u x r x r f t, x , x x 1 , . . ., x m ∈ Ω, 0 < t < 1, u 0, x 1 0 α ρ u ρ, x dρ ϕ x , u t 0, x ψ x , x ∈ Ω, u 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 x 1 , . . ., 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.

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

2.1
We introduce the Hilbert space L 2h L 2 Ω h of the grid functions

2.3
To the differential operator A x generated by problem 1.1 , we assign the difference operator A x h by the formula 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 L 2 Ω h .With the help of A x h we arrive at the nonlocal boundary value problem for an infinite system of ordinary differential equations.
In the second step, we replace problem 2.5 by the difference scheme 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: where M 1 does not depend on τ, h, ϕ h x , ψ h x , and 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 problem
the following coercivity inequality holds [29]: Moreover, the second order of accuracy difference schemes and 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 inequalities hold, where M 2 is independent of τ, h, ϕ h x , ψ h x , and 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 M 1 and M 2 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.

The First Order of Accuracy in Time Difference Scheme
In this section, the nonlocal boundary value problem for one dimensional hyperbolic equation is considered.
The exact solution of problem 3.1 is Applying the formulas 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 x for approximate solutions of nonlocal boundary value problem 3.1 .It can be written in the matrix form

3.5
Here and D I N 1 is the identity matrix.

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: 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 × 1 column matrices, α 1 , β 1 are zero matrices, and 3.9

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 x 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 . Here , 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.

The Second Order of Accuracy in Time Difference Scheme
Generated by A 2 Applying 3.3 and formulas 3.12 and using difference scheme 2.11 , we obtain the second order of accuracy difference scheme for approximate solutions of problem 3.1 .One can write the N 1 × M 1 system of linear equations 3.13 as the matrix equation

3.14
Here, A n , B n , C n , D n , E n , and R are N 1 × N 1 square matrices:

3.15
We denote 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:

3.18
For solution of the last difference equation, we need to find u M , u M−1 u M 0, 3.19

Error Analysis
The errors are computed by 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.
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 .