Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

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.

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][13][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 2 The Scientific World Journal are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

Discretization and Mesh
Notation. Let a set of mesh nodes that discretises be given by with = { = ℎ, = 1, 2, . . . , − 1, ℎ = } , Define the following finite differences for any mesh function V = V( ) given on by Introduce the inner products for the mesh functions V and defined on as follows: For any mesh function V , vanishing for = 0 and = we introduce the norms and "negative" norm for any function (1 ≤ ≤ − 1) Given a function ≡ ≡ ( , );̆= ( −1) , 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][13][14], we obtain the precise relation where The remainder term has the form with The Scientific World Journal Based on (12), we propose the following difference scheme for approximating (1): where ℓ is defined by (12).

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.
After some manipulations, we get Multiplying this inequality by and summing it up from = 1 to = , also, using here the inequality we obtain we have Applying now Lemma 1 we obtain Further, in view of the fact that we obtain where (0) and (1) are given by (15). From (35), under the assumed smoothness, we have which together with (33) completes the proof of the theorem.

Numerical Results
In this section, we present numerical results obtained by applying the numerical method (17) to the particular problems.
The exact solution of this problem is ( , ) = − (cosh( ) − cosh(1) + − 1). The computational results are presented in Tables 3 and 4.   6 The Scientific World Journal  It can be observed that the obtained results are essentially in agreement with the theoretical analysis described above.

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.