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 formsubject 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 byThe 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 asThe 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 towhere 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 bywhere The boundary conditions are integrated into system of (18) in the following form: The final form of the system is given byWe 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 formEquation (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 byBy 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 bywhere
Theorem 2 (see [22]). If , then for any grid of nodes and for any the interpolation error at is given bywhere and are the nodal polynomial given byLet . Then is the solution of the following equation:where can be written as . So the discrete Chebyshev system of (30) is given byby 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 normThe 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 .
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.
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.
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.
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.