Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations

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.


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.
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].
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.

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 (1/2) * 0 rather than  * 0 .Similarly, the derivatives  () (),  = 0, 1, 2, 3, . .., 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

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 W and Z by replacing the first row and last row of the matrix [, ] by the corresponding row of boundary conditions.

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 (() 2 ).So ( 15) can be written in the form Equation ( 23) shows that the truncation error due to time discretization of (1) is of order (() 2 ).
Summing up these two upper bounds yields (34).
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  = 1 are illustrated in Figure 3.  and with  = 0,  = −1, (, ) =  − , and (, ) which can be chosen such that the exact solution in this example is (, ) =  −  − sin().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  = 1, at  = 10,  = 5, and  = 10,  = 50 are illustrated in Figures 4 and  5, respectively.

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 Advances in Mathematical Physics 7 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.