Abstract and Applied Analysis

Volume 2012 (2012), Article ID 682752, 31 pages

doi:10.1155/2012/682752

Research Article

Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

Deniz Agirseven

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., [1–14] 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., [15–17]).

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., [18–28]) and homotopy analysis methods (HAM) (see, e.g., [29–37]) 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.

Table 1: Comparison of the errors of different difference schemes in .

Method

Difference scheme (2.20)
0.00599088
0.00286092
0.00139629

Difference scheme (2.29)
0.00076265
0.00020317
0.00005148

Table 2: Comparison of the errors of different difference schemes in .

Method

Difference scheme (2.20)
0.07324151
0.03693061
0.01845008

Difference scheme (2.29)
0.00077055
0.00020541
0.00005206

Table 3: Comparison of the errors of different difference schemes in .

Method

Difference scheme (2.20)
0.03520081
0.01749862
0.00872690

Difference scheme (2.29)
0.00067084
0.00017845
0.00004521

Table 4: Comparison of the errors of different difference schemes in .

Method

Difference scheme (2.20)
0.01687166
0.00844577
0.00421761

Difference scheme (2.29)
0.00045527
0.00012107
0.00003067

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., [29–35]), 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

and so on. When we choose , we obtain

and so on.

From (3.6), the solution of (3.35) for is

This series has the closed form

which is the exact solution of the (3.35).

Finally, we consider the solution of the delay parabolic equation (2.17) for and rewrite this equation in the following form:

We take the initial approximation

and the linear operator (3.14) with the property (3.15). From (3.44), we define linear operator (3.16).

We construct the zero-order deformation equation (3.2) and the th-order deformation equations (3.8) for with the initial conditions

where

The solution of the th-order deformation equations (3.47) for is

From (3.44) and (3.48), we get

and so on. For , we get

and so on.

From (3.6), the solution of (3.44) is obtained as

This series has the closed form

which is the exact solution of (3.44).

We give the HAM solutions in , , , . We use four terms for evaluating the approximate solution . According to the -curve of , the solution series is convergent when , , , and , respectively, in , , , . We take to determine how much the approximate solution is accurate and compute the absolute errors at the points , , , in Table 5.

Table 5: The absolute error at when .

0.5
0.68896672324764
0.68718517400418
0.00178154924346

1.5
0.31617066069370
0.31528438731741
0.00088627337628

2.5
0.14472535694258
0.14432199982347
0.00040335711910

3.5
0.06624155329591
0.06619299448877
0.00004855880714

4. Conclusion

The numerical solutions of first order of difference scheme (2.20) and second order of difference scheme (2.29) for different values of and and the approximate solutions obtained by HAM for in (3.11) when are given at the same points , , , in Tables 6, 7, 8, and 9, respectively. The absolute errors computed show that, with homotopy analysis method, the results are more accurate for the parabolic delay equation (2.17).

Table 6: Comparison of the absolute error in HAM when and the errors of different difference schemes at ().

Method

HAM for
0.68718517400418
0.00178154924346

Difference scheme (2.20) for
0.69069236574147
0.00172564249383

Difference scheme (2.29) for
0.69067805057383
0.00171132732619

Table 7: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
0.31528438731741
0.00088627337628

Difference scheme (2.20) for
0.31705389317070
0.00088323247700

Difference scheme (2.29) for
0.31707957467449
0.00090891398079

Table 8: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
0.14432199982347
0.00040335711910

Difference scheme (2.20) for
0.14512731459568
0.00040195765309

Difference scheme (2.29) for
0.14512723912249
0.00047857754134

Table 9: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
0.06619299448877
0.00004855880714

Difference scheme (2.20) for
0.06640694878963
0.00016953954937

Difference scheme (2.29) for
0.06628727563690
0.00004572234990

Although HAM seems to be more rapid than finite difference method, the series solutions obtained by HAM are convergence only for the regions determined by convergence control parameter . So, convergence region is limited for HAM. The comparison of two methods of finite difference and homotopy analysis shows that latter is more rapid and more accurate in the cases that series solutions are convergence. When we take out of the convergence region determined by curves, it is shown that finite difference method is faster and more accurate than HAM. The approximate solutions obtained by HAM for different values of chosen from out of the convergence region of the series solutions and the numerical solutions of first and second order of difference schemes (2.20) and (2.29) for are given in Tables 10, 11, 12, and 13, respectively at the same points , , , .

Table 10: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
1.82382103940268
1.13485431615504

Difference scheme (2.20) for
0.71754584429923
0.02857912105159

Difference scheme (2.29) for
0.64396829164838
0.04499843159926

Table 11: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
−0.71640688651704
1.0325774721074

Difference scheme (2.20) for
0.26625468031687
0.04991598037683

Difference scheme (2.29) for
0.26908021797566
0.04709044271804

Table 12: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
1.85743968284120
1.71271432589862

Difference scheme (2.20) for
0.09250455294144
0.05222080400114

Difference scheme (2.29) for
0.11418378483060
0.03054157211198

Table 13: Comparison of the absolute error in HAM when and the errors of different difference schemes at .

Method

HAM for
1.33954603011737
1.27330447682146

Difference scheme (2.20) for
0.02904127716967
0.03720027612623

Difference scheme (2.29) for
0.04842804202239
0.01781351127352

Despite HAM, by finite difference method, we can guarantee the convergence in the whole domain that (2.17) is defined in. Therefore finite difference method is more efficient than HAM.

Acknowledgments

This work is supported by Trakya University Scientific Research Projects Unit (Project number: 2010-91). The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for his valuable suggestions for the improvement of this paper.

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