The Scientific World Journal

Volume 2014 (2014), Article ID 497393, 7 pages

http://dx.doi.org/10.1155/2014/497393

Research Article

Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

I. Amirali,¹ G. M. Amiraliyev,¹ M. Cakir,² and E. Cimen³

¹Department of Mathematics, Faculty of Art and Science, Sinop University, 57000 Sinop, Turkey

²Department of Mathematics, Faculty of Science, Yüzüncü Yil University, 65080 Van, Turkey

³Department of Mathematics, Faculty of Education, Yüzüncü Yil University, 65080 Van, Turkey

Received 21 August 2013; Accepted 2 December 2013; Published 4 February 2014

Academic Editors: C.-s. Liu and N. I. Mahmudov

Copyright © 2014 I. Amirali et al. 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.

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

1. Introduction

We consider the initial-boundary value problem for pseudoparabolic equation with delay in the domain ; , , :

where represents the delay parameter (for simplicity we assume that is an integer; i.e., for some integer ), , , and and are given sufficiently smooth functions satisfying certain regularity conditions in and and , respectively, to be specified, and furthermore

Equations of this type arise in many areas of mechanics and physics. Such equations are encountered, for example, as a model for two-phase porous media flows when dynamic effects in the capillary pressure are included [1–3]. They are used also to study heat conduction [4], homogeneous fluid flow in fissured rocks [5], shear in second order fluids [6–8], and other physical models. For a discussion of existence and uniqueness results of pseudoparabolic equations see [1, 9–11]. Various numerical treatments of equations of this type without delay have been considered in [2, 12–19] (see also the references cited in them).

In the present paper finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities with use of the piecewise linear basis functions in space and interpolating quadrature rules with weight and remainder term in integral form, two-level difference scheme is constructed (see also [12–14]) for singular perturbation cases without delay. For the time integration we use the implicit rule. The finite difference discretization is shown to be absolutely stable and convergent of order two in space and of order one in time. Based on the method of energy estimates the error analysis for approximate solution is presented. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

2. Discretization and Mesh

Notation. Let a set of mesh nodes that discretises be given by

with

Define the following finite differences for any mesh function given on by

Introduce the inner products for the mesh functions and defined on as follows:

For any mesh function , vanishing for and we introduce the norms

and “negative” norm for any function  

Given a function , defined on , we will also use the notation

2.1. Difference Scheme

The approach of generating difference scheme is through the integral identity

with the usual piecewise linear basis functions for the space

Using the appropriate interpolating quadrature rules with weight and remainder term in integral form, consistent with [12–14], we obtain the precise relation

where

The remainder term has the form

with

Based on (12), we propose the following difference scheme for approximating (1):

where is defined by (12).

3. The Error Estimates and Convergence

To estimate the convergence of this method, note that the error function is the solution of the discrete problem,

Before obtaining the estimate for the solution (18) we give the following Lemma.

Lemma 1. Let the mesh function , defined on , satisfy

where given, is an integer. Then

where

Proof. For and inequality (19) reduces to

Applying now the difference analogue of the Gronwall's inequality we get

For , after replacing in (19) , we have

which by virtue of difference analogue of the Gronwall's inequality leads to (20), immediately.

Theorem 2. Let the derivatives , ,   and   be continuous and bounded on , , and , , ,  and  . Then for the discrete problem (17) the following error estimate holds:

Proof. Consider identity

After some manipulations, we get

Multiplying this inequality by and summing it up from to , also, using here the inequality

we obtain

Denoting

we have

where

Applying now Lemma 1 we obtain

Further, in view of the fact that

we obtain

where and are given by (15). From (35), under the assumed smoothness, we have

which together with (33) completes the proof of the theorem.

Remark 3. Under sufficiently smoothness of and for calculations of and appropriate numerical quadrature formulae can be applied; for example, , , and so forth.

4. Numerical Results

In this section, we present numerical results obtained by applying the numerical method (17) to the particular problems.

Example 1. Consider the following linear problems:

where

The exact solution of this problem is . The computational results are presented in Tables 1 and 2.

Table 1: The numerical results on .

Nodes

Exact solution
Numerical solution

Pointwise error

Numerical solution

Pointwise error

Table 2: The numerical results on .

Nodes

Exact solution
Numerical solution  

Pointwise error  

Numerical solution  

Pointwise error  

Example 2. Now consider the following nonlinear problem:

where

The exact solution of this problem is . The computational results are presented in Tables 3 and 4.

Table 3: The numerical results on .

Nodes

Exact solution
Numerical solution

Pointwise error  

Numerical solution

Pointwise error  

Table 4: The numerical results on .

Nodes

Exact solution
Numerical solution

Pointwise error  

Numerical solution  

Pointwise error  

It can be observed that the obtained results are essentially in agreement with the theoretical analysis described above.

5. Conclusion

In this paper, we proposed an efficient numerical method for solving initial-boundary value problem for the semilinear pseudoparabolic equation. The proposed finite difference method was constructed, and based on the method of energy estimates the fully discrete scheme was shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates were obtained in discrete norm. Numerical results were presented, which numerically validate this theoretical result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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