Abstract and Applied Analysis

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

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Volume 2012 |Article ID 684248 | https://doi.org/10.1155/2012/684248

Allaberen Ashyralyev, Mehmet Emin San, "An Approximation of Semigroups Method for Stochastic Parabolic Equations", Abstract and Applied Analysis, vol. 2012, Article ID 684248, 24 pages, 2012. https://doi.org/10.1155/2012/684248

An Approximation of Semigroups Method for Stochastic Parabolic Equations

Academic Editor: Ravshan Ashurov
Received25 May 2012
Accepted08 Jun 2012
Published13 Nov 2012

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 [14] 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 [515] 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