Abstract and Applied Analysis

Volume 2012 (2012), Article ID 824819, 20 pages

doi:10.1155/2012/824819

Research Article

Numerical Solution of Stochastic Hyperbolic Equations

Necmettin Aggez¹ and Maral Ashyralyyewa²

¹Department of Mathematics, Fatih University, 34500 Istanbul, Turkey

²Department of Applied Mathematics and Informatics, Magtymguly Turkmen State University, Ashgabat, Turkmenistan

Received 30 March 2012; Revised 25 May 2012; Accepted 27 May 2012

Academic Editor: Valery  Covachev

Copyright © 2012 Necmettin Aggez and Maral Ashyralyyewa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [1–7] and the references therein). Numerical methods and theory of solutions of initial boundary value problem for stochastic partial differential equations have been studied in [8–16]. 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

Using (5.27) that we get is an identity and is a zero matrices and . The rest are the same as in Example 5.3.

For these examples, the errors of the numerical solution derived by difference scheme (2.2) computed by

and the results are given in Table 1.

Table 1

Example 5.1 (Neumann)
0.3028
0.1219
0.0554

Example 5.2 (Dirichlet)
0.4004
0.2137
0.1342

Example 5.3 (Dirichlet-Neumann)
0.3040
0.1145
0.0494

Example 5.4 (Neumann-Dirichlet)
0.3439
0.1844
0.0957

The numerical solutions are recorded for different values of , where represents the exact solution and represents the numerical solution at . To obtain the results, we simulated the 1000 sample paths of Brownian motion for each level of discretization.

Thus, results show that the error is stable and decreases in an exponential manner.

Acknowledgment

The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for the helpful suggestions to the improvement of this paper.

References

A. Ashyralyev and I. Hasgur, “Linear stochastic diffrential equations in a Hilbert space,” in Proceedings of the Abstracts of the National Statistics Conference, vol. 95, pp. 1–6, Ankara, Turkey, 1999.

R. F. Curtain and P. L. Falb, “Stochastic differential equations in Hilbert space,” Journal of Differential Equations, vol. 10, pp. 412–430, 1971.

A. Ichikawa, “Linear stochastic evolution equations in Hilbert space,” Journal of Differential Equations, vol. 28, no. 2, pp. 266–277, 1978.

G. Da Prato, “Regularity properties of a stochastic convolution integral,” Analisi Matematica, vol. 72, no. 4, pp. 217–219, 1982.

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, UK, 1992.

A. Ashyralyev and G. Michaletsky, “The approximation of solutions of stochastic differential equations in Hilbert space by the difference schemes,” in Trudy Nauchno-Prakticheskoy Konferencii, vol. 1, pp. 85–95, Differencialniye Uravneniya i ih Prilozheniya, Ashgabat, Turkmenistan, 1993.

P. H. Bezandry and T. Diagana, “Existence of quadratic-mean almost periodic solutions to some stochastic hyperbolic differential equations,” Electronic Journal of Differential Equations, vol. 2009, no. 111, pp. 1–14, 2009.

E. J. Allen, S. J. Novosel, and Z. Zhang, “Finite element and difference approximation of some linear stochastic partial differential equations,” Stochastics and Stochastics Reports, vol. 64, no. 1-2, pp. 117–142, 1998.

E. Hausenblas, “Numerical analysis of semilinear stochastic evolution equations in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 147, no. 2, pp. 485–516, 2002.

A. Yurtsever and A. Yazliyev, “High order accuracy difference scheme for stochastic parabolic equation in a Hilbert space,” in Some Problems of Applied Mathematics, pp. 212–220, Fatih University, Istanbul, Turkey, 2000.

T. Shardlow, “Numerical methods for stochastic parabolic PDEs,” Numerical Functional Analysis and Optimization, vol. 20, no. 1-2, pp. 121–145, 1999.

A. Ashyralyev, “On modified Crank-Nicholson [Nicolson] difference schemes for stochastic parabolic equation,” Numerical Functional Analysis and Optimization. An International Journal, vol. 29, no. 3-4, pp. 268–282, 2008.

L. Han, L. G. Han, and X. B. Gong, “Implicit finite-difference plane wave migration in tti media,” Chinese Journal of Geophysics, vol. 54, no. 4, pp. 1090–1097, 2011.

A. Jentzen, “Higher order pathwise numerical approximations of SPDEs with additive noise,” SIAM Journal on Numerical Analysis, vol. 49, no. 2, pp. 642–667, 2011.

A. Jentzen and P. E. Kloeden, “The numerical approximation of stochastic partial differential equations,” Milan Journal of Mathematics, vol. 77, pp. 205–244, 2009.

A. Ashyralyev and E. M. San, “Finite difference method for stochastic parabolic equations,” in AIP Conference Proceedings, vol. 1389, pp. 589–592, 2011.

A. Ashyralyev and M. Akat, “An approximation of stochastic hyperbolic equations,” in AIP Conference Proceedings, vol. 1389, pp. 625–628, 2011.

S. Peszat and J. Zabczyk, “Nonlinear stochastic wave and heat equations,” Probability Theory and Related Fields, vol. 116, no. 3, pp. 421–443, 2000.

V. Barbu, G. Da Prato, and L. Tubaro, “Stochastic wave equations with dissipative damping,” Stochastic Processes and their Applications, vol. 117, no. 8, pp. 1001–1013, 2007.

E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics, vol. 3, no. 2, pp. 127–167, 1979.

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Notas de Matematica, North-Holland Publishing, Amsterdam, The Netherlands, 1985.

S. Piskarev and Y. Shaw, “On certain operator families related to cosine operator function,” Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 3585–3592, 1997.

A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkhäuser Verlag, Basel, Switzerland, 2004.

A. Ashyralyev and H. O. Fattorini, “On uniform difference schemes for second-order singular perturbation problems in Banach spaces,” SIAM Journal on Mathematical Analysis, vol. 23, no. 1, pp. 29–54, 1992.

P. E. Sobolevskii, Difference Methods for Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975.

A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2 of Iterative Methods, Birkhauser Verlag, Basel, Switzerland, 1989.