Abstract and Applied Analysis

Volume 2012 (2012), Article ID 840621, 14 pages

doi:10.1155/2012/840621

Research Article

An Approximation of Ultra-Parabolic Equations

Allaberen Ashyralyev1,2 and Serhat Yılmaz¹

¹Department of Mathematics, Fatih University, Istanbul, Turkey

²Department of Mathematics, ITTU, Ashgabat, Turkmenistan, Turkey

Received 7 February 2012; Accepted 3 April 2012

Academic Editor: Hasan Ali Yurtsever

Copyright © 2012 Allaberen Ashyralyev and Serhat Yılmaz. 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.

The first and second order of accuracy difference schemes for the approximate solution of the initial boundary value problem for ultra-parabolic equations are presented. Stability of these difference schemes is established. Theoretical results are supported by the result of numerical examples.

1. Introduction

Mathematical models that are formulated in terms of ultraparabolic equations are of great importance in various problems for instance in age-dependent population model, in the mathematical model of Brownian motion, in the theory of boundary layers, and so forth, see [1–5]. We refer also to [6–9] and the references therein for existence and uniqueness results and other properties of these models. On the other hand, Akrivis et al. [10] and Ashyralyev and Yılmaz [11, 12] developed numerical methods for ultraparabolic equations. In this paper, our interest is studying the stability of first- and second-order difference schemes for the approximate solution of the initial boundary value problem for ultraparabolic equations

in an arbitrary Banach space with a strongly positive operator . For approximately solving problem (1.1), the first-order of accuracy difference scheme

and second-order of accuracy difference scheme

are presented. The stability estimates for the solution of difference schemes (1.2) and (1.3) are established. In applications, the stability in maximum norm of difference shemes for multidimensional ultraparabolic equations with Dirichlet condition is established. Applying the difference schemes, the numerical methods are proposed for solving one-dimensional ultraparabolic equations.

Theorem 1.1. For the solution of (1.2), we have the following stability inequality:

where is independent of , , , and .

Proof. Using (1.2), we get

From that it follows

where . By the mathematical induction, we will prove that

is true for all positive integers . It is obvious that for formula (1.7) is true. Assume that for

is true. In formula (1.6), replacing and with and , respectively, we have

Then, using (1.8) and (1.9), we get

From that it follows

is true for . So, formula (1.7) is proved. For , replacing with in formula (1.7), we obtain that

Using estimate (see [13])

and triangle inequality, we get

for any and . For , replacing with in formula (1.7), we get

From estimate (1.13) and triangle inequality, it follows that

for any and . Thus, Theorem 1.1 is proved.

Theorem 1.2. For the solution of (1.3), we have the following stability inequality:

where is independent of , , and .

The proof of Theorem 1.2 is based on the following formulas:

for the solution of difference scheme (1.3) and the following estimate [14]:

where and C=.

2. Application

Let be the unit open cube in the -dimensional Euclidean space    with boundary . In , we consider the boundary-value problem for the multidimensional parabolic equation

where and are given smooth functions and is a sufficiently large number.

We introduce the Banach spaces , , of all continuous functions satisfying a Hölder condition with the indicator , , , and with weight , , , which is equipped with the norm

where stands for the Banach space of all continuous functions defined on , equipped with the norm

It is known that the differential expression

defines a positive operator acting on with domain and satisfying the condition on .

The discretization of problem (2.1) is carried out in two steps. In the first step, let us define the grid sets

We introduce the Banach spaces , of grid functions defined on , equipped with the norms

To the differential operator generated by problem (2.1), we assign the difference operator by the formula

acting in the space of grid functions , satisfying the condition for all .

With the help of , we arrive at the initial boundary-value problem

for an infinite system of ordinary differential equations.

In the second step, we replace problem (2.8) by difference scheme(1.2)

and by difference scheme(1.3)

It is known that is a positive operator in () and . Let us give a number of corollaries of Theorems 1.1 and 1.2.

Theorem 2.1. For the solution of difference scheme (2.9), we have the following stability inequality:

where is independent of , , , and .

Theorem 2.2. For the solution of difference scheme (2.10), we have the following stability inequality:

where does not depend on , , , and .

3. Numerical Analysis

In this section, the initial boundary value problem

for one-dimensional ultraparabolic equations is considered.

The exact solution of problem (3.1) is

Using the first order of accuracy in and implicit difference scheme (2.9), we obtain the difference scheme first order of accuracy in and and second-order of accuracy in

for approximate solutions of initial boundary value problem (3.3). It can be written in the matrix form

Here

where

This type system was used by Samarskii and Nikolaev [15] for difference equations. For the solution of matrix equation (3.4), we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form:

where , are square matrices, are coloumn matrices, , are zero matrices, and

Using the second-order of accuracy in and implicit difference scheme (2.10), we obtain the difference scheme second-order of accuracy in and and second-order of accuracy in

for approximate solutions of initial boundary value problem (3.9). The matrix form (3.4) can be written. Here,

where

We seek a solution of the matrix equation by the same algorithm (3.7) and (3.8).

4. Error Analysis

The errors are computed by

of the numerical solutions, where represents the exact solution and represents the numerical solution at , and the results are given in Table 1.

Table 1

Difference schemes    

Difference scheme (3.3)
0.028100
0.014400
0.006811
0.003225

Difference scheme (3.9)
0.000511
0.000121
0.000028
0.000006

It may be noted from Table 1 that as , increase, the value of the errors associated with difference scheme (3.3) decreases by a factor of approximately 1/2 and the errors associated with difference scheme (3.9) decrease by a factor of approximately 1/4. This confirms that difference scheme (3.3) is first order and difference scheme (3.9) is second-order as stated in Section 1. Moreover, the results show that the second-order of accuracy difference scheme (3.9) are more accurate comparing with the first order of accuracy difference scheme (3.3).

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