ESTIMATES FOR DIFFERENTIAL DIFFERENCE SCHEMES TO PSEUDO-PARABOLIC INITIAL-BOUNDARY VALUE PROBLEM WITH DELAY

We consider the one dimensional initial-boundary Sobolev problem with delay. For solving this problem numerically, we construct fourth order differentialdifference scheme and obtain the error estimate for its solution. Further we use the appropriate Runge-Kutta method for the realization of our differential-difference problem. Key WordsSobolev Problem, Delay Difference Scheme, Error Estimate


INTRODUCTION
We consider the initial-boundary value problem for pseudo-parabolic differential equation with delay in the domain ( , ) ( , ) (0, ) ( , ) 0 where 0 , f and  are sufficiently smooth functions satisfying certain regularity conditions to be specified, 0 r  represents the delay parameter.Equations of this type arise in many areas of mechanics and physics.They are used to study heat conduction [7], homogeneous fluid flow in fissured rocks [5], shear in second order fluids [12,19] and other physical models.The important characteristic of these models is that they express the conservation of a certain quantity (mass, momentum, heat, etc.) in any sub-domain.For a discussion of existence and uniqueness results of pseudo-parabolic equations see [6,8,13,18].Various finite difference schemes have been constructed to treat such problems [1][2][3][4] For example in [10] two difference approximation schemes to a nonlinear pseudo-parabolic equation are developed.Each of these schemes possesses a unique solution which can be obtained by an iterative procedure.Further in [17] two difference streamline diffusion schemes for solving linear Sobolev equations with convection-dominated term are given.We can see other numerical methods of this type of equations in [11,15] (see also the references cited in them).In [9] a Crank-Nicolson-Galerkin approximation with extrapolated coefficients is presented for three cases for the nonlinear Sobolev equation along with a conjugate gradient iterative procedure which can be used efficiently to solve the different linear systems of algebraic equations arising at each step from the Galerkin method.In [20] the author study a finite volume element approximation of pseudo-parabolic equations in three spatial dimensions.
In this study, we use the method of lines for the discretization in space variable for the problem (1.1)- (1.3).The method of lines is a general technique for solving partial differential equations by typically using finite difference relationships for the spatial derivatives or the time derivative.Our aim is to get a fourth order accurate differential-difference scheme and to establish the error estimate for its solution.

CONSTRUCTION OF THE SCHEME
On the  , we introduce the uniform mesh for any mesh function i g .To construct the difference scheme, we will use the following relation which is valid for where tT  (5) Using formula (4) in (5), we obtain where Taking into account the following relations For the error function       i i i z t y t u t , from the relations ( 6)-( 8) and ( 9)-( 11), we have the following differential-difference problem

A PRIORI ESTIMATE
In this section, we give a lemma which is used in the next section for establishing the error estimate Lemma 3.1.
. Then the solution of the following initial value problem For the solution of ( 15)-( 16), we can write From this relation, we get After denoting   , the inequality (18) reduces to 01 0 ( ) ( ) Using variable transformation r   in (19), it can be seen clearly that From here, by virtue of Gronwall's inequality, we easily arrive at (17).

THE ERROR ESTIMATE
Now we give the main result of this paper.Then the error of the problem ( 9)- (11) satisfies where C is a constant which is independent of h .
. Then the scheme ( 12)-( 14) can be expressed in vector form as 12 12 where

The matrix
M can be diagonalized as [ 14,16 ]   Multiplying equation ( 21) on the left by B and denoting , the initial-value problem (21)-( 22) is turned into the decomposed system as : If we rewrite (23) in the form Taking into account the following inequality   To solve this problem numerically, we use the appropriate Runge-Kutta method.The spatial and time steps are both taken to be 0.1 .The values for exact and numerical solutions and appropriate pointwise errors are shown in Table 1 and Table 2