Advances in Mathematical Physics
Volume 2016 (2016), Article ID 7854806, 7 pages
http://dx.doi.org/10.1155/2016/7854806

Research Article

Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations

M. Sameeh and A. Elsaid

Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt

Received 29 June 2016; Accepted 27 September 2016

Academic Editor: Nikolai A. Kudryashov

Copyright © 2016 M. Sameeh and A. Elsaid. 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.

An efficient technique for solving parabolic partial integrodifferential equation is presented. This technique is based on Chebyshev polynomials and finite difference method. A priori error estimate for the proposed technique is deduced. Some examples are presented to illustrate the validity and efficiency of the presented method.

1. Introduction

Partial integrodifferential equations (PIDEs) are the equations that combine partial differentiation and integration of the unknown function. They are used in modeling several phenomena where the effect of memory must be considered.

In this work, we consider the parabolic PIDE of the following form

subject to the following initial and boundary conditions:

where , , , and are continuous functions.

This class of equations appears in various fields of physics and engineering such as heat conduction [1], compression of poroviscoelastic media [2], reaction diffusion problems [3], and nuclear reactor dynamics [4].

PIDEs are solved by some numerical methods [5–12]. In this work, the Chebyshev polynomials are applied through finite difference method to obtain an approximate solution for problems (1) and (2).

Chebyshev polynomials were proposed by the Russian mathematician Chebyshev. Many authors depend on Chebyshev collocation method to solve different types of equations such as linear differential equations [13], systems of high-order linear differential equations with variable coefficients [14], systems of high-order linear Fredholm-Volterra integrodifferential equations [15], fourth-order Sturm-Liouville problems [16], Troeschs problem [17], nonlinear differential equations [18], linear partial differential equations [19, 20], and nonlinear Fredholm-Volterra integrodifferential equations [21].

This paper is organized into six sections. Section 2 lists some notations and definitions of Chebyshev polynomials. Section 3 presents a description for the technique of solution to problems (1) and (2). In Section 4, we deduce a priori error estimate for the approximate solution. Section 5 is devoted to present some examples that illustrate the proposed technique of solution. Finally, Section 6 presents the conclusions of this study.

2. Fundamental Relations

On a general interval , the shifted Chebyshev polynomials take the following form:

and the Chebyshev collocation points are given by

The function on is approximated using truncated shifted Chebyshev series in the form

The symbol sum with single prime indicates that the summation involves rather than . Similarly, the derivatives , , are written as

The function and its derivatives can be written in matrix form as

where

Matrix can be obtained from by the following relation [13]:

where

where

Then, can be written in the form

3. Description of Chebyshev Collocation Method

The partial time-derivative in (1) is discretized using finite difference method. Denote the time step by and the value of the function at time by . Then we have

The integral is handled numerically using the composite trapezoidal rule, and (1) is discretized into the following form:

which is simplified to

where

The approximate solution of (15) is computed using truncated Chebyshev series.

Theorem 1. If the assumed approximate solution of problem (15) is described by (5), then the discrete Chebyshev system is given by

where are the Chebyshev collocation points. The fundamental matrix form of the discrete Chebyshev system is given by

where

The boundary conditions are integrated into system of (18) in the following form:

The final form of the system is given by

We construct the two matrices and by replacing the first row and last row of the matrix by the corresponding row of boundary conditions.

4. Error Analysis

In (1), the time is discretized using finite difference method. From Taylor series, we have

Approximation of integral term using composite trapezoidal integration rule leads to an error of order . So (15) can be written in the form

Equation (23) shows that the truncation error due to time discretization of (1) is of order .

If is the exact solution of (1), then is the exact solution of (23). Let and be the Chebyshev series solution of (15) and (23), respectively. If , then the discrete Chebyshev system is given by

By subtracting (18) from (24), the following relation is obtained:

Let be the Lagrange interpolating polynomial of order of on the grid of Chebyshev collocation points. Then is given by

where

Theorem 2 (see [22]). If , then for any grid of nodes and for any the interpolation error at is given by

where and are the nodal polynomial given by

Let . Then is the solution of the following equation:

where

can be written as . So the discrete Chebyshev system of (30) is given by

by subtracting (18) from (32), we obtain

Theorem 3. Let be the Chebyshev series solution of (15) and be the exact solution of (23). If is sufficiently smooth, then

Proof. Adding and subtracting the polynomial yield

The upper bound for the second term can be found using some properties of norm

The upper bound for the second term can be found by adding and subtracting the polynomial following the work in [20].

Summing up these two upper bounds yields (34).

5. Numerical Examples

Example 1 (see [12]). Consider the parabolic PIDE (1) with the initial condition

and with boundary conditions

and with , , , and The exact solution of this example is , where and .
The maximum absolute error is tabulated in Table 1 for Chebyshev polynomial together with the results of [12] at and . The graphs of exact and approximate solutions for are illustrated in Figure 1.
The maximum absolute error is tabulated in Table 2 for Chebyshev polynomial at and .

Table 1: error of Example 1 at , , and , .

Chebyshev collocation method
Backward-Euler scheme [12]

0.31

0.63

0.94

1.26

1.57

Table 2: error of Example 1 at , , , and .

error

0.2

0.4

0.6

0.8

1

Figure 1: Exact and Chebyshev approximated solutions of Example 1 at .

Example 2. Consider the parabolic PIDE (1) with the initial condition

and with boundary conditions

and with , , , and . The exact solution of this example is given by
The maximum absolute error is tabulated in Table 3 for Chebyshev polynomial at different times. The graphs of exact and approximate solutions for and are illustrated in Figure 2.

Table 3: error of Example 2 at , , and .

error

0.2

0.4

0.6

0.8

1

Figure 2: Exact and Chebyshev approximated solutions of Example 2 at .

Example 3 (see [10]). Consider the parabolic PIDE (1) with the initial condition

and with boundary conditions

and with , , , and which can be chosen such that the exact solution in this example is .
The maximum absolute error is tabulated in Table 4 for Chebyshev polynomial together with the results of [10]. The graphs of exact and approximate solutions for are illustrated in Figure 3.

Table 4: error of Example 3 at .

Chebyshev collocation method
Radial basis functions [10]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Exact and Chebyshev approximated solutions of Example 3 at .

Example 4. Consider the parabolic PIDE (1) with the initial condition

and with boundary conditions

and with , , , and which can be chosen such that the exact solution in this example is .
The maximum absolute error is tabulated in Table 5 for Chebyshev polynomial at different values of and . The graphs of exact and approximate solutions for , at , and are illustrated in Figures 4 and 5, respectively.

Table 5: error of Example 4 at .

error
error

, ,  
,  ,

0.2

0.4

0.6

0.8

1

Figure 4: Exact and Chebyshev approximated solutions of Example 4 at .

Figure 5: Exact and Chebyshev approximated solutions of Example 4 at .

6. Conclusions

Chebyshev collocation method is successfully used for solving parabolic partial integrodifferential equation. This method reduced the considered problem into linear system of algebraic equations that can be solved successively to obtain a numerical solution at varied time levels. Numerical examples show that the results of above scheme are in good agreement with the exact ones. Comparisons with the results obtained by using radial basis functions and backward-Euler scheme show that the Chebyshev method yields good results with fewer number of iterations. The method also renders good results for problems where solution exhibits fast changes but with larger number of iterations as anticipated by the error estimate. Moreover, the above scheme can be developed to solve nonlinear parabolic partial integrodifferential equation.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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