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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 682752, 31 pages
http://dx.doi.org/10.1155/2012/682752
Research Article

Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

Department of Mathematics, Trakya University, 22030 Edirne, Turkey

Received 21 February 2012; Accepted 28 March 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Deniz Agirseven. 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 and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

1. Introduction

Increase in interest in the theoretical aspects of numerical methods for delay differential equations points out that delay differential equations are capable of generating extensive and conceivable models for phenomena in many branches of sciences. Numerical solutions of the delay ordinary differential equations have been studied mostly for ordinary differential equations (cf., e.g., [114] and the references therein). Nevertheless, delay partial differential equations are less in demand than delay ordinary differential equations. Different kinds of problems for delay partial differential equations are solved by using operator approach (see, e.g., [1517]).

In recent years, Ashyralyev and Sobolevskii considered the initial-value problem for linear delay partial differential equations of parabolic type in the spaces of functions defined on the segment with values in a Banach space and the stability inequalities were established under stronger assumption than the necessary condition of the stability of the differential problem. The stability estimates for the solutions of difference schemes of the first- and second-order accuracy difference schemes for approximately solving this initial-value problem for delay differential equations of parabolic type were presented. They obtained the stability estimates in Hölder norms for solutions of the initial-value problem of the delay differential and difference equations of the parabolic type [15, 16]. Gabriella used extrapolation spaces to solve Banach spaces valued delay differential equations with unbounded delay operators. The author proved regularity properties of various types of solutions and investigated the existence of strong and weak solutions for a class of abstract semilinear delay equations [17].

In this paper, finite difference (see, e.g., [1828]) and homotopy analysis methods (HAM) (see, e.g., [2937]) for the approximate solutions of the delay differential equation of the parabolic type are studied. Here are given smooth bounded functions and .

Difference schemes which are accurate to first and second orders for the approximate solution of problem (1.1) are presented. The convergence estimates for the solution of these difference schemes are obtained. For the numerical study, procedure of modified Gauss elimination method is used to solve these difference schemes. Homotopy analysis method is applied to find the solution of problem (1.1). The numerical results are obtained at the same points for each method. Comparison of finite difference and homotopy analysis methods is given on the problem.

2. The Finite Difference Method

In this section, the first and second orders of accuracy in for the approximate solution of problem (1.1) are considered. The convergence estimates for the solution of these difference schemes are established. A procedure of modified Gauss elimination method is used to solve these difference schemes.

2.1. The Difference Scheme, Convergence Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, we define the grid space To formulate our results, we introduce the Banach space, , of all grid functions defined on with equipped with the norm Moreover, is the Banach space of all grid functions defined on with values in equipped with the norm To the differential operator generated by problem (1.1), we assign the difference operators by the formulas acting in the space of grid functions satisfying the conditions . It is well known that is a strongly positive operator in . With the help of and , we arrive at the initial value problem In the second step, we consider difference schemes of first and second orders of accuracy

Theorem 2.1. Assume that Suppose that problem (1.1) has a smooth solution and Then, for the solution of difference scheme (2.7), the following convergence estimate holds: with being a real number independent of , , and .

Proof. Using notations of and , we can obtain the following formula for the solution: where . The proof of Theorem 2.1 is based on the formulas (2.12), on the convergence theorem, on the difference schemes in (see [38]), on the estimate and on the fact that in the norms are equivalent to the norms in uniformly in for (see, [18]).

Theorem 2.2. Assume that assumption (2.9) of Theorem 2.1 and the following conditions hold: Then for the solution of difference scheme (2.8), the following convergence estimate is satisfied: with being a real number independent of , , and .

Proof. Using notations of and again, we can obtain the following formula for the solution: where . The proof of Theorem 2.2 is based on the formulas (2.16), on the convergence theorem, on the difference schemes in (see, [38]), on the estimate (2.13), and on the equivalence of the norms as in Theorem 2.1.
Finally, the numerical methods are given in the following section for the solution of delay parabolic differential equation with the Dirichlet condition. The method is illustrated by numerical examples.

2.2. Numerical Results

We consider the initial-boundary-value problem for the delay parabolic differential equation.

The exact solution of this problem for , is For the approximate solution of delay parabolic equation (2.17), consider the set of grid points Using difference scheme accurate to first order for the approximate solutions of the initial-boundary-value problem for the delay parabolic equation (2.17), we get the following system of equations: In this first step, applying difference scheme accurate to first order, we obtain a system of equations in matrix form where , , are matrices defined by, is identity matrix and , are column vectors as where is given for any , Here, we denote So, we have second-order difference equation (2.21) with matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method. Hence, we obtain a solution of the matrix equation in the following form: where are square matrices and are column matrices defined by where , is the zero matrix, and is the zero matrix.

Second, using the second order of accuracy difference scheme for the approximate solutions of problem (2.17) and applying formulae we obtain the following system of equations: In the second step, we apply second-order difference scheme to get the system of linear equations in matrix form where , , are matrices defined by, , is identity matrix, and , are column vectors as where is given for any ,

Here, we denote Hence, we have second-order difference equation (2.30) with matrix coefficients. For the solution of this matrix equation, we use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form: where and are square matrices and are column matrices defined by where , is zero matrix, and is zero matrix, and are zero matrices.

We give the results of the numerical analysis. The numerical solutions are recorded for different values of and and represent the numerical solutions of these difference schemes at . Tables 1, 2, 3, and 4 are constructed for in , , , , respectively, and the error is computed by the following formula: Thus, by using the second order of accuracy difference scheme, the accuracy of solution increases faster than the first order of accuracy difference scheme.

tab1
Table 1: Comparison of the errors of different difference schemes in .
tab2
Table 2: Comparison of the errors of different difference schemes in .
tab3
Table 3: Comparison of the errors of different difference schemes in .
tab4
Table 4: Comparison of the errors of different difference schemes in .

3. Homotopy Analysis Method

In this section, we consider homotopy analysis method for the solution of problem (1.1). We study the initial-boundary-value problem for the delay parabolic equation (1.1). To illustrate the basic idea of homotopy analysis method (HAM) developed by Liao (see, e.g., [2935]), the following differential equation is considered: where is a linear operator for problem (1.1), and denote independent variables, is an unknown function, and is a known analytical function. Liao constructs the so-called zero-order deformation equation where is an embedding parameter, is a nonzero auxiliary parameter, is an auxiliary linear operator, is an initial guess of , and is an unknown function. When and , it holds respectively. As increases from 0 to 1, the solution varies from the initial guess to the solution . Expanding in Taylor series with respect to , we get where when the initial guess , the auxiliary linear operator and the auxiliary parameter are chosen properly, the series (3.4) converges at . We get Then define the vectors Differentiating the zero-order deformation equation (3.2) times with respect to the embedding parameter and dividing them by , we obtain the th-order deformation equation where with the initial condition High-order deformation equation (3.8) is governed by the linear operator . can be represented by and high-order deformation equation can be solved consecutively. The th-order approximation of is given by

3.1. Homotopy Analysis Solution

For the approximate solution of the delay parabolic differential equation with the Dirichlet condition, we consider the delay parabolic equation (2.17) and rewrite the equation for in the following form: To solve the initial-boundary-value problem (3.12) by means of HAM, we choose the initial approximation and the linear operator with the property where is constant of integration. From (3.12), we define a linear operator as Firstly, we construct the zero-order deformation equation when and , Then, we get th-order deformation equations (3.8) for with the initial conditions where The solution of the th-order deformation equations (3.20) for is From (3.12) and (3.21), we obtain and so on. Then for , we get and so on.

From (3.6), when we take , the solution of (3.12) can be obtained as Equation (3.24) has the closed form which is the exact solution of (3.12).

Second, we consider the solution of the delay parabolic equation (2.17) for and rewrite this equation in the following form: Now, we choose the initial approximation We take the linear operator (3.14) with the property (3.15), and we define the operator (3.16) from (3.26).

Firstly, we construct the zero-order deformation equation (3.2) and then, we obtain th-order deformation equations (3.8) for with the initial conditions where The solution of the th-order deformation equations (3.29) for is From (3.26) and the th-order deformation equations (3.30), we get and so on. When we choose , we obtain and so on.

From (3.6), the solution of (3.26) for can be obtained as Equation (3.33) has the closed form which is the exact solution of the (3.26).

Now, we consider the solution of the delay parabolic equation (2.17) for and rewrite the equation in the following form: The initial approximation is

We take the linear operator (3.14) with the property (3.15). From parabolic equation (3.35), we define a linear operator (3.16) and obtain the zero-order deformation equation (3.2). Thus, we get the th-order deformation equations (3.8) for with the initial conditions where The solution of the th-order deformation equations (3.38) for is From (3.35) and (3.39), we obtain