An Approximation of Ultra-Parabolic Equations

and Applied Analysis 3 is true for all positive integers n. It is obvious that for n 1, 2 formula 1.7 is true. Assume that for n r


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 ∂u t, s ∂t ∂u t, s ∂s Au t, s f t, s , 0 < t, s < T, u 0, s ψ s , 0 ≤ s ≤ T, u t, 0 ϕ t , 0 ≤ t ≤ T, 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 C is independent of τ, ψ m , ϕ k , and f k,m .
Proof.Using 1.2 , we get From that it follows where R I τA −1 .By the mathematical induction, we will prove that is true for all positive integers n.It is obvious that for n 1, 2 formula 1.7 is true.Assume that for n r is true.In formula 1.6 , replacing k and m with k − r and m − r, respectively, we have 1.9 Then, using 1.8 and 1.9 , we get From that it follows is true for n r 1.So, formula 1.7 is proved.For m > k, replacing n with k in formula 1.7 , we obtain that Using estimate see 13 and triangle inequality, we get for any k and m.For k > m, replacing n with m in formula 1.7 , we get From estimate 1.13 and triangle inequality, it follows that for any k and m.Thus, Theorem 1.1 is proved.
Theorem 1.2.For the solution of 1.3 , we have the following stability inequality: where C is independent of τ, ψ m , ϕ k, and f k,m .
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 B I − τA/2 I τA/2 −1 and C I τA/2 −1 .

Application
Let Ω be the unit open cube in the n-dimensional Euclidean space where α r x > a > 0 x ∈ Ω and f t, s, x t, s ∈ 0, 1 , x ∈ Ω are given smooth functions and δ > 0 is a sufficiently large number.We introduce the Banach spaces . ., n of all continuous functions satisfying a H ölder condition with the indicator β β 1 , . . ., β n , which is equipped with the norm where C Ω stands for the Banach space of all continuous functions defined on Ω, equipped with the norm It is known that the differential expression 01 Ω and satisfying the condition v 0 on S.
The discretization of problem 2.1 is carried out in two steps.In the first step, let us define the grid sets

2.5
We introduce the Banach spaces

2.6
To the differential operator A generated by problem 2.1 , we assign the difference operator A x h by the formula acting in the space of grid functions u h x , satisfying the condition u h x 0 for all x ∈ S h .With the help of A x h , we arrive at the initial boundary-value problem ∂u h t, s, x ∂t 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

2.10
It is known that 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 C 1 is independent of τ, ψ h m , ϕ h k , and f h k,m .
Theorem 2.2.For the solution of difference scheme 2.10 , we have the following stability inequality: where C 1 does not depend on τ, ψ h m , ϕ h k , and f h k,m .

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 u t, s, x e − t s sin πx.

3.2
Using the first order of accuracy in t and s implicit difference scheme 2.9 , we obtain the difference scheme first order of accuracy in t and s and second-order of accuracy in for approximate solutions of initial boundary value problem 3.3 .It can be written in the matrix form

3.6
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 u M 0, α j j 1, . . ., M−1 are N 1 2 × N 1 2 square matrices, β j j 1, . . ., M−1 are N 1 2 × 1 coloumn matrices, α 1 , β 1 are zero matrices, and

3.8
Using the second-order of accuracy in t and s implicit difference scheme 2.10 , we obtain the difference scheme second-order of accuracy in t and s and second-order of accuracy in x n nh, 1 ≤ n ≤ M, Mh π 3.9 for approximate solutions of initial boundary value problem 3.9 .The matrix form 3.4 can be written.Here, . We seek a solution of the matrix equation by the same algorithm 3.7 and 3.8 .

Error Analysis
The errors are computed by  1 that as N, M 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 .
we consider the boundary-value problem for the multidimensional parabolic equation

Table 1
of the numerical solutions, where u t k , s m , x n represents the exact solution and u k,m n represents the numerical solution at t k , s m , x n , and the results are given in Table1.It may be noted from Table