Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article
Special Issue

Well-Posed and Ill-Posed Boundary Value Problems for PDE

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Research Article | Open Access

Volume 2012 |Article ID 824819 | 20 pages | https://doi.org/10.1155/2012/824819

Numerical Solution of Stochastic Hyperbolic Equations

Academic Editor: Valery Covachev
Received30 Mar 2012
Revised25 May 2012
Accepted27 May 2012
Published02 Aug 2012

Abstract

A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

1. Introduction

Stochastic partial differential equations have been studied extensively by many researchers. For example, the method of operators as a tool for investigation of the solution to stochastic equations in Hilbert and Banach spaces have been used systematically by several authors (see, [17] and the references therein). Numerical methods and theory of solutions of initial boundary value problem for stochastic partial differential equations have been studied in [816]. Moreover, the authors of [17] presented a two-step difference scheme for the numerical solution of the following initial value problem: for stochastic hyperbolic differential equations. We have the following.

(i) is a standard Wiener process given on the probability space .

(ii) For any , is an element of the space , where is a subspace of .

Here, [18] denote the space of -valued measurable processes which satisfy(a),(b).

The convergence estimates for the solution of the difference scheme are established.

In the present work, we consider the following initial value problem: for stochastic hyperbolic equation in a Hilbert space with a self-adjoint positive definite operator with , where . In addition to (i) and (ii), we put the following.

(iii) and are elements of the space of -valued measurable processes, where is a subspace of .

By the solutions provided in [19] (page 423, ) and in [20] (page 1005, ), under the assumptions (i), (ii), and (iii), the initial value problem (1.2) has a unique mild solution given by the following formula: For the theory of cosine and sine operator-function we refer to [21, 22].

Our interest in this study is to construct and investigate the difference scheme for the initial value problem (1.2). The convergence estimate for the solution of the difference scheme is proved. In applications, the theorems on convergence estimates for the solution of difference schemes for the numerical solution of initial-boundary value problems for hyperbolic equations are established. The theoretical statements for the solution of this difference scheme are supported by the result of the numerical experiments.

2. The Exact Difference Scheme

We consider the following uniform grid space: with step . Here, is a fixed positive integer.

Theorem 2.1. Let be the solution of the initial value problem (1.2) at the grid points . Then, is the solution of the initial value problem for the following difference equation:

Proof. Putting into the formula (1.3), we can write Using (2.3), the definition of the sine and cosine operator function, we obtain It follows that Hence, we get the relation between and as This relation and equality (2.2) are equivalent. Theorem 2.1 is proved.

3. Convergence of the Difference Scheme

For the approximate solution of problem (1.2), we need to approximate the following expressions: Using Taylor's formula and Pade approximation of the function at , we get Applying the difference scheme (2.2) and formula (3.2), we can construct the following difference scheme: for the approximate solution of the initial value problem (1.2). Using the definition of and , we can write (3.3) in the following equivalent form: Now, let us give the lemma we need in the sequel from papers [23, 24].

Lemma 3.1. The following estimates hold: where The following Theorem on convergence of difference scheme (3.5) is established.

Theorem 3.2. Assume that then the estimate of convergence holds. Here, does not depend on .

Proof. Using the formula for the solution of second order difference equation and the definition of and , we can write Using (2.4) and (3.13), we obtain where Let us estimate the expected value of for all , separately. We start with and . Using (3.6), (3.7), and (3.8), we obtain Estimates for the expected value of for all , separately, were also used in paper [17]. Combining these estimates, we obtain (3.12). Theorem 3.2 is proved.

4. Applications

First, let be the unit open cube in the -dimensional Euclidean space with boundary . In , the initial-boundary value problem for the following multidimensional hyperbolic equation: with the Dirichlet condition is considered. Here, and are given smooth functions with respect to and .

The discretization of (4.1) is carried out in two steps. In the first step, define the grid space , ,.

Let denote the Hilbert space as The differential operator in (4.1) is replaced with where the difference operator is defined on these grid functions , for all . As it is proved in [25], is a self-adjoint positive definite operator in . Using (4.1) and (4.3), we get In the second step, we replace (4.4) with the difference scheme (3.5) as

Theorem 4.1. Let and be sufficiently small numbers. Then, the solution of difference scheme (4.5) satisfies the convergence estimate as where does not depend on and .

The proof of Theorem 4.1 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator defined by (4.3).

Second, in , the initial-boundary value problem for the following multidimensional hyperbolic equation: with the Neumann condition is considered. Here, is the normal vector to , , , , and are given smooth functions with respect to and .

The discretization of (4.7) is carried out in two steps. In the first step, the differential operator in (4.7) is replaced with where the difference operator is defined on those grid functions , for all , where is the second order of approximation of . As it is proved in [25], is a self-adjoint positive definite operator in . Using (4.7) and (4.8), we get In the second step, we replace (4.9) with the difference scheme (3.5) as

Theorem 4.2. Let and be sufficiently small numbers. Then, the solution of difference scheme (4.10) satisfies the convergence estimate as where does not depend on and .

The proof of Theorem 4.2 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator defined by (4.8).

Third, in , the mixed boundary value problem for the following multidimensional hyperbolic equation: with the Dirichlet-Neumann condition is considered. Here, is the normal vector to ,,, and are given smooth functions with respect to and .

The discretization of (4.12) is carried out in two steps. In the first step, the differential operator in (4.12) is replaced with where the difference operator is defined on those grid functions , for all and , for all , , where is the second order of approximation of . By [25], we can conclude that is a self-adjoint positive definite operator in . Using (4.12) and (4.13), we get In the second step, we replace (4.14) with the difference scheme (3.5) as

Theorem 4.3. Let and be sufficiently small positive numbers. Then, the solution of difference scheme (4.15) satisfies the convergence estimate as where does not depend on and .

The proof of Theorem 4.3 is based on the abstract Theorem 3.2 and the symmetry properties of the difference operator defined by (4.13).

5. Numerical Examples

In this section, we apply finite difference scheme (2.2) to four examples which are stochastic hyperbolic equation with Neumann, Dirichlet, Dirichlet-Neumann, and Neumann-Dirichlet conditions.

Example 5.1. The following initial-boundary value problem: for a stochastic hyperbolic equation is considered. The exact solution of this problem is For the approximate solution of the (5.1), we apply the finite difference scheme (2.2) and we get The system can be written in the following matrix form: Here, the matrix is , and is the identity matrix, This type of system was used by [26] for difference equations. For the solution of matrix equation (5.4), we will use modified Gauss elimination method. We seek a solution of the matrix equation by the following form: where are square matrices, are column matrices is an identity and is a zero matrices, and

Example 5.2. The following initial-boundary value problem: for a stochastic hyperbolic equation is considered. We use the same procedure as in the first example. The exact solution of this problem is For the approximate solution of the (5.11), we can construct the following difference scheme: and it can be written in the following matrix form: Here, the matrices , , , are given in the previous example, and For the solution of matrix equation (5.14), we will use modified Gauss elimination method.
We seek a solution of the matrix equation in the following form: where are square matrices,
are column matrices and are zero matrices, and

Example 5.3. The following initial-boundary value problem: for a stochastic hyperbolic equation is considered. The exact solution of this problem is We get the following difference scheme: for the approximate solutions of (5.18), and we obtain the following matrix equation: Here, the matrices , , , are same as in the first example, and For the solution of matrix equation (5.21), we use the same procedure as in the previous examples. Moreover, , is an identity and is a zero matrices, and

Example 5.4. The following initial boundary value problem: for a stochastic hyperbolic equation is considered. The exact solution of this problem is The following difference scheme: is obtained for the approximate solutions of (5.24), and we obtain the following matrix equation: Here, the matrices , , , are same as in the first example, and