Abstract
Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.
1. Introduction
In recent years, the topic of fractional calculus has attracted many scientists because of its several applications in many areas, such as physics, chemistry, biology, and engineering. For a detailed survey with collections of applications in various fields, see, for example, [1–6].
The numerical solution of differential equations of integer order has been a hot topic in numerical and computational mathematics for a long time. There are many different methods and different basis functions have been used to estimate the solution of fractional integro-differential equations or Abel’s integral equations, such as Adomian decomposition method [7, 8], fractional differential transform method [9, 10], collocation method [11, 12], homotopy perturbation method [13, 14], homotopy analysis method [15, 16], variational iteration method [17], discrete Galerkin method [18], and Haar wavelet method [19].
Spectral methods provide a computational approach that has achieved substantial popularity over the last four decades. They have gained new popularity in automatic computations for a wide class of physical problems in fluid and heat flow. Their fascinating merit is the high accuracy. So, they have been applied successfully to numerical simulations of many problems in science and engineering; see [20–24].
The operational matrix of fractional derivatives has been determined for some types of orthogonal polynomials, such as Chebyshev polynomials [25] and Legendre polynomials [26], and for integration has been determined for several types of orthogonal polynomials, such as Chebyshev polynomials [27], Laguerre series [28], and Legendre polynomials [29]. Recently, the Bernstein operational matrix approach is developed for solving a system of high order linear Volterra–Fredholm integro-differential equations in [30].
In the present paper, we use Chebyshev spectral method based on operational matrix to solve system of fractional integro-differential equations:with initial conditionswhere .
And we use Abel’s integral equation:where or , is a continuous function, and is constant.
2. Basic Definitions
In this section, we summarize some basic definitions and properties of fractional calculus theory.
Definition 1. A real function , is said to be in the space , if there exists a real number , such that , where Clearly if .
Definition 2. A function , is said to be in space , if .
Definition 3. The Riemann-Liouville fractional integral operator of order , of a function , is defined as
Definition 4. The Caputo fractional derivatives of order are defined aswhere and is the classical differential operator of order
For Caputo derivative, we haveWe use the ceiling function denoting the smallest integer greater than or equal to and the floor function denoting the largest integer less than or equal to Also and . Recall that, for , the Caputo differential operator coincides with the usual differential operator of integer order.
More properties of the fractional derivatives and the fractional integral can be found in [3, 4].
3. Some Properties of the Shifted Chebyshev Polynomials
The well-known Chebyshev polynomials are defined on the interval and can be determined with the aid of the following recurrence formula:where and The Chebyshev polynomials are orthogonal on the interval with respect to the weight function These polynomials satisfy the relationwhere
The analytic form of the Chebyshev polynomial of degree is given bywhere denotes the integer part of The zeros of are denoted byIn order to use these polynomials on the interval , we defined the so-called shifted Chebyshev polynomials by introducing the change of variable Let the shifted Chebyshev polynomials be denoted by , satisfying the orthogonality relationwhere .
The shifted Chebyshev polynomials are defined as and the analytic form is given bywhere and
In this form, may be generated with the aid of the following recurrence formula:where and The zeros of are denoted byA function , square integrable in , may be expressed in terms of the shifted Chebyshev polynomials aswhere the coefficients are given byIn practice, only the first -terms shifted Chebyshev polynomials are considered. Hence, if we writewhere the shifted Chebyshev coefficient vector and the shifted Chebyshev vector are given bythen the derivative of the vector can be expressed bywhere is the operational matrix of derivative given byfor example, for even , we have
4. The Shifted Chebyshev Operational Matrix (COM) Fractional Derivatives
The main objective of this subsection is to generalize the COM of derivatives for the fractional calculus. By using (19), it is clear thatwhere and the superscript, in , denotes matrix powers. Thus
Lemma 5. Let be a shifted Chebyshev polynomial; then
Theorem 6. Let be the shifted Chebyshev vector defined in (18) and suppose ; thenwhere is the COM of derivatives of order in the Caputo sense and is defined as follows:whereNote that, in , the first rows are all zero.
Proof. See [25].
Remark 7. If , then Theorem 6 gives the same result as (22).
5. System of Fractional Integro-Differential Equations
In order to use COM for system of fractional integro-differential equations of the form (1), we first approximate , and by the shifted Chebyshev polynomials asBy substituting these equations in (1), we getAlso, by substituting (22) and (28) in (2), we obtainThen we have to collocate (31) at the shifted Chebyshev roots . These equations together with (32) generate of algebraic equations which can then be solved for the unknown coefficients of the vectors , using a suitable method. Consequently, the approximate solution , can be obtained.
In our computations we used the Gaussian elimination method to solve the resulting linear system of algebraic equations and Newton’s iteration method to solve the resulting nonlinear system of algebraic equations.
Now, we can present the following problems.
Example 8. Consider the following system of fractional integro-differential equations [7, 15]:with the initial conditionsThe exact solutions, when , are
We use Chebyshev spectral method; we may write the approximate solutionwhereSubstituting (36) in (33) and (34), for , we getThe roots of the shifted Chebyshev polynomial are given byNow, for calculating the shifted Chebyshev coefficient for , substitute (40) in (38), and solving the resulting linear system of equations and (39), we getThus using (36), we get
Table 1 shows the comparison between the exact solution and the approximate solution with the absolute error at , and Table 2 shows the maximum of absolute error between exact solutions and approximate solutions for various choices of Figure 1 shows the graph of the exact solution and the approximate solution at , . Figure 2 shows the graph of the exact solutions and the approximate solutions at , , and .
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Example 9. Consider the following nonlinear fractional system of integro-differential equations [15]:with the initial conditionsThe exact solutions, when , are
We use Chebyshev spectral method; we may write the approximate solutionwhereSubstituting (46) in (43) and (44), for , we getThe roots of the shifted Chebyshev polynomial areNow, for calculating the shifted Chebyshev coefficient for , substitute (50) in (48), and solving the resulting nonlinear system of equations and (49), we getThus using (46), we getTable 3 shows the comparison between the exact solution and the approximate solution with the absolute error at . Figure 3 shows the graph of the exact solution and the approximate solution at , . Figure 4 shows the graph of the exact solutions and the approximate solutions at , , and .
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6. Abel’s Integral Equation
In order to use COM for Abel’s integral equation of the form (3), we first approximate by the shifted Chebyshev polynomials asBy substituting (53) in (3), we getThen we have to collocate (54) at the shifted Chebyshev roots . These equations generate linear algebraic equations which can be solved for the unknown coefficients of the vector , using a suitable method. Consequently, given in (53) can be calculated, which gives a solution of (3).
Example 10. Consider Abel’s integral equation of the first kind [31, 32]which has the exact solution .
By applying the Chebyshev spectral method, we may write the approximate solutionSubstituting (56) in (55), we getThe roots of the shifted Chebyshev polynomial areNow, calculating the shifted Chebyshev coefficient for by substituting (58) in (57) and solving four equations yieldsTherefore, we havewhich is the exact solution.
Example 11. Consider Abel’s integral equation of the first kind [32] which has the exact solution
Table 4 shows the comparison between the exact solution and the approximate solution with the absolute error at , and Table 5 shows the maximum of absolute error between exact solution and approximate solution for various choices of Figure 5 shows the graph of the exact solution and the approximate solution at .
Example 12. Consider Abel’s integral equation of the second kind [32, 33]which has the exact solution
By applying the Chebyshev spectral method, we may write the approximate solutionSubstituting (63) in (62), we getThe roots of the shifted Chebyshev polynomial areNow, calculating the shifted Chebyshev coefficient for by substituting (65) in (64) and solving three equations yieldsTherefore, we havewhich is the exact solution.
Example 13. Consider Abel’s integral equation of the second kind [32, 34]which has the exact solution
We use Chebyshev spectral method; we may write the approximate solutionSubstituting (69) in (68), we getThe roots of the shifted Chebyshev polynomial areNow, calculating the shifted Chebyshev coefficient for by substituting (71) in (70) and solving three equations yields Therefore, we have which is the exact solution.
Example 14. Consider Abel’s integral equation of the second kind [32, 34]which has the exact solution
Table 6 shows the comparison between the exact solution and the approximate solution with the absolute error at , and Table 7 shows the maximum of absolute error between exact solution and approximate solution for various choices of Figure 6 shows the graph of the exact solution and the approximate solution at .
Example 15. Consider the linear system of singular Volterra integral equations [10]The exact solutions are
Table 8 shows the comparison between the exact solution and the approximate solution with the absolute error at , and Table 9 shows the maximum of absolute error between exact solution and approximate solution for various choices of Figure 7 shows the graph of the exact solution and the approximate solution at .
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7. Conclusion
In this article, we develop the Chebyshev spectral method based on operational matrix for solving linear and nonlinear system of fractional integro-differential equations and Abel’s integral equations. Our approach was based on the shifted Chebyshev collocation methods. It can be concluded that the Chebyshev spectral method is very powerful and efficient technique for finding exact solutions for wide classes of problems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.