Research Article | Open Access
I. Amirali, G. M. Amiraliyev, M. Cakir, E. Cimen, "Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations", The Scientific World Journal, vol. 2014, Article ID 497393, 7 pages, 2014. https://doi.org/10.1155/2014/497393
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.
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 , homogeneous fluid flow in fissured rocks , 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
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
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
Example 2. Now consider the following nonlinear problem: where