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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 840621, 14 pages
An Approximation of Ultra-Parabolic Equations
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.
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.
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.  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 ) 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 .
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.
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  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
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.
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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