Abstract and Applied Analysis

Volume 2012 (2012), Article ID 684248, 24 pages

doi:10.1155/2012/684248

Research Article

An Approximation of Semigroups Method for Stochastic Parabolic Equations

Allaberen Ashyralyev1,2 and Mehmet Emin San³

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

²Department of Mathematics, ITTU, 74012 Ashgabat, Turkmenistan

³Certified Dental Supply LLC 43 River Road, Nutley, NJ 07031, USA

Received 25 May 2012; Accepted 8 June 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Allaberen Ashyralyev and Mehmet Emin San. 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 single-step difference scheme for the numerical solution of the nonlocal-boundary value problem for stochastic parabolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In application, the convergence estimates for the solution of the difference scheme are obtained for two nonlocal-boundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

1. Introduction

It is known that most problems in heat flow, fusion process, model financial instruments like options, bonds, and interest rates, and other areas which are involved with uncertainty lead to stochastic differential equation with parabolic type. These equations can be derived as models of indeterministic systems and considered as methods for solving boundary value problems.

The method of operators as a tool for investigation of the solution to stochastic partial differential equations in Hilbert and Banach spaces has been systematically developed by several authors (see [1–4] and the references therein). Finite difference method for the solution of initial boundary value problem for stochastic differential equations has been studied extensively by many researchers (see [5–15] and the references therein). However, multipoint nonlocal-boundary value problems were not well investigated.

In the present paper the multipoint nonlocal-boundary value problem

for stochastic parabolic differential equations in a Hilbert space with a self-adjoint positive definite operator is considered. Here

(i) is a standard Wiener process given on the probability space .(ii)For any , is an element of space , where is subspace of .(iii) is element of space of -valued measurable processes, where is a subspace of .

Here, denote the space of -valued measurable processes which satisfy

(a) is measurable, a.e. in ,(b).

The main goal of this study is to construct and investigate the difference schemes for the multipoint nonlocal-boundary value problems (1.1). The outline of the paper is as follows. In Section 2, the exact single-step difference scheme for the solution of the problem (1.1) is presented. In Section 3, the -th order of accuracy Rothe difference scheme is constructed and investigated for the approximate solution of the problem (1.1). The estimate of convergence for the solution of this difference scheme is obtained. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of two multipoint nonlocal-boundary value problems for stochastic parabolic equations are obtained. In Section 4, the numerical application for one-dimensional stochastic parabolic equation is presented.

2. The Exact Single-Step Difference Scheme

Now, let us give some lemmas we need in the sequel. Throughout this paper, let be a Hilbert space, let be a positive definite self-adjoint operator with , where .

Lemma 2.1. The following estimate holds:

Lemma 2.2. Suppose that assumption

holds. Then, the operator

has an inverse

and the following estimate is satisfied:

Proof. The proof follows from the triangle inequality, assumption (2.2), and estimate

Let us now obtain the formula for the mild solution of problem (1.1). It is clear that under the assumptions (i)-(ii) and

the Cauchy problem

and has a unique mild solution, which is represented by the following formula:

Then from this formula and the multipoint nonlocal-boundary condition

we get

By Lemma 2.2 the operator has a bounded inverse . Then

Therefore, we have formulas (2.9) and (2.12) for the solution of problem (1.1).

Now, we will consider the single-step exact difference scheme. On the segment we consider the uniform grid space

with step . Here is a fixed positive integer.

Theorem 2.3. Let be the solution of (1.1) at the grid points . Then is the solution of the multipoint nonlocal-boundary value problem for the following difference equation (see [16]):

Proof. Putting and into the formula (2.9), we can write

Hence, we obtain the following relation between and :

Last relation and equality (2.14) are equivalent. Theorem 2.3 is proved.

Note that problem (2.14) is called the single-step exact difference scheme for the solution of the problem (1.1).

3. Rothe Difference Scheme

In this section, the -th order of accuracy Rothe difference scheme is constructed and investigated for the approximate solution of the problem (1.1). The estimate of convergence for the solution of this difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of two multipoint nonlocal-boundary value problems for stochastic parabolic equations are obtained.

3.1. 1/2-th Order-of-Accuracy Rothe Difference Scheme

Let us give some lemmas we need in the sequel.

Lemma 3.1. The following estimates hold:

where .

Lemma 3.2. Suppose that assumption (2.2) holds. Then, the operator

has a bounded inverse

and the following estimate is satisfied:

Proof. The proof follows from the triangle inequality, assumption (2.2), and estimate

From (2.14) it is clear that for the approximate solution of the multipoint nonlocal-boundary value problem (1.1) it is necessary to approximate the expressions

and multipoint nonlocal-boundary condition

It is possible under stronger assumption than (ii) for . Assume that

Replacing the expressions with , the expression with , and the function with , we get the implicit Rothe difference scheme:

Let us now obtain the formula for the solution of problem (3.10). It is clear that the Rothe difference scheme

for the solution of the Cauchy problem (2.8) has a unique solution, which is represented by the following formula:

Then from this formula and the multipoint nonlocal-boundary condition

we get

By Lemma 3.2 the operator has a bounded inverse . Then

Therefore, we have formulas (3.12) and (3.15) for the solution of problem (3.10). Now, we will study the convergence of difference scheme (3.10).

Theorem 3.3. Assume that

Then the estimate of convergence

holds. Here and do not depend on .

Proof. Using formulas (2.12) and (3.15), we can write

where

Let us estimate for all , separately. We start with . Using formulas (2.4) and (3.4), we obtain

and also the expression in the above sum can be written in the following formula:

Here . Using formulas (3.26), (3.27), and (3.19), we can write

Let us estimate expected value of . Since

we have that

In the same manner by using the triangle inequality and estimates (3.2) and (3.1), we get

Now, let us estimate . Using formula (3.20), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Let us estimate . Using formula (3.21), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Next, let us estimate . Using formula (3.22), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Next, let us estimate . Using formula (3.23), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Next, let us estimate . Using formula (3.24), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Finally, let us estimate . Using formula (3.25), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get

Since is a Wiener process and

we have that

Applying estimates for , , we get the estimate:

To prove the Theorem 3.3 it suffices to establish the following estimate:

Using formulas (2.9) and (3.12), we can write

where

Let us estimate for all , separately. We start with . Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

Now, we estimate . Using estimate (3.1), we get

Applying the estimate (3.40), we obtain

Now, we estimate . Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

Now, we estimate . We denote that

Then

Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

Since

we have that

Finally, we estimate . We denote that

Therefore,

Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

Since

we have that

Combining estimates , and , we obtain (3.41). Theorem 3.3 is proved.

3.2. Applications

Now, we consider applications of Theorem 3.3. First, let us consider the nonlocal-boundary value problem for one-dimensional stochastic parabolic equation:

where and are smooth functions with respect to .

The discretization of problem (3.58) is carried out in two steps. In the first step, we define the grid space

Let us introduce the Hilbert space of the grid functions defined on , equipped with the norm

To the differential operator generated by problem (3.58), we assign the difference operator by the formula

acting in the space of grid functions satisfying the conditions , . It is well known that is a self-adjoint positive definite operator in . With the help of , we arrive at the nonlocal-boundary value problem:

In the second step, we replace (3.62) with the difference scheme (3.10):

Theorem 3.4. Let and be sufficiently small positive numbers. Then, the solutions of difference scheme (3.63) satisfy the following convergence estimate:

where do not depend on and . Here, one puts as the grid function of exact solution of problem (3.58) at the grid points and .

Proof. Let us introduce the Banach space of abstract mesh functions defined on with values in . Then, difference scheme (3.63) can be reduced to the abstract difference scheme:

in a Hilbert space with the operator by formula (3.62). It is clear that and in . Hence, is a self-adjoint positive definite operator in . Therefore, Theorem 3.3 applies to this case, and Theorem 3.4 is proved.

Second, let be the unit open cube in the -dimensional Euclidean space with boundary . In , the nonlocal boundary value problem for the multidimensional parabolic equation

with the Dirichlet condition is considered. Here , and , are given smooth functions with respect to and .

The discretization of problem (3.66) is carried out in two steps. In the first step, define the grid space .

Let denote the Hilbert space

The differential operator in (3.66) is replaced with

where the difference operator is defined on those grid functions for all . It is well known that is a self-adjoint positive definite operator in .

Using (3.66) and (3.68), we get

In the second step, we replace (3.69) with the difference scheme (3.10):

Theorem 3.5. Let and be sufficiently small positive numbers. Then, the solution of difference scheme (3.70) satisfies the following convergence estimate:

where do not depend on and . Here, one puts as the grid function of exact solution of problem (3.66) at the grid points and .

The proof of Theorem 3.5 is based on the abstract Theorem 3.3 and the symmetry properties of the difference operator defined by formula (3.68).

4. Numerical Application

Now, we consider the numerical application of nonlocal boundary value problem:

for one-dimensional stochastic parabolic equation. For numerical solution of (4.1), we consider the difference scheme -th order of accuracy in and second order of accuracy in for the approximate solution of the nonlocal boundary value problem (4.1):

We will write it in the matrix form

Here

For the solution of the last matrix equation, we use the modified Gauss elimination method (see [17]). We seek a solution of the matrix equation by the following form:

where , are square matrices and , are column matrices and defined by formulas

Here

The error between the exact solution and the solutions derived by difference schemes is shown in Table 1. To obtain the results we simulated the 1,000 sample paths of Brownian motion for each level of discretization. The estimate (3.71) in Theorem 3.5 suggests that if we double the number of nodes, then the error should be decreased by a factor of . The theoretical statement for the solution of this difference scheme is supported by the results of the numerical experiment. In fact, we double and ; the error is even less than half of the previous error.

Table 1: Error analysis.

N/M
10/30
20/60
40/120

Difference scheme (4.2)
0.0929
0.0401
0.0187

Acknowledgment

The authors wish to thank Professor A. Lukashev (Istanbul, Turkey), for his valuable suggestions which helped us to improve the present paper.

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