1. Introduction
Fractional calculus (FC) deals with derivatives and integrals of an arbitrary order (real or complex order). The history of FC started from 30 September 1695, when Leibnitz described a derivative of order
(see [
1]). In the 19th century, Riemann and Liouville defined the concept of differentiation to an arbitrary order (fractional differentiation). However, there were few specific models based on this type of derivative at that time due to which the study of fractional order systems attracted little interest. Nowadays, a growing number of researchers are focusing their attention on FC, and they have shown that fractional systems can retain information that is missing in integral order systems. Scientists pay more attention to FC due to its numerous applications in many areas such as solid mechanics [
2], oscillation of earthquakes [
3], signal processing [
4], economics [
5], electrode-electrolyte polarization [
6], control theory [
7], visco-elastic materials [
8], and continuum and statistical mechanics [
9]. Fractional derivatives can be described in different ways, e.g., Grünwald Letnikow, Caputo, and Generalized Functions Approach. In the present article, we focus on Caputo’s derivative, which is more useful in real-life applications [
10,
11,
12].
In recent years, much attention has been given to the solutions of fractional differential and integro-differential equations. FIDEs have many applications in mechanical, nuclear engineering, chemistry, astronomy, biology, economics, potential theory, and electrostatics. In particular cases, the exact solution of such FIDEs may be found only. In many cases, analytical solutions of integro-differential equations are an unwieldy task, so it is required to obtain an efficient approximate solution. Recently, many effective techniques have been presented to solve integro-differential equations having fractional-order. such as Homotopy perturbation transform method [
12], Reproducing Kernal Hilbert Space Method (RKHSM) [
13,
14], Haar Wavelet Method (HWM) [
15], Taylor Expansion Method (TEM) [
16], Homotopy Analysis Method (HAM) [
17], Euler Wavelet Method (EWM) [
18], Spline Collocation Method (SCM) [
19], Variational Iteration Method (VIM) [
20], Laplace Adomian Decomposition Method (LADM) [
21], Homotopy Perturbation Method (HPM) [
22] and much more [
23,
24,
25,
26].
As Chebyshev polynomials [
27] are considered to be a known family of orthogonal polynomials with many applications on the interval
, they are commonly used in function approximation because of their good properties. S. Nemati et al. [
28] implement a spectral method based on operational matrices of the second kind of Chebyshev polynomials to solve FIDEs having weakly singular kernels. M.S. Mahdy et al. [
29] used the least squares method aid of third kind shifted Chebyshev polynomials to solve a linear system of fractional integro-differential equations. However, to the best of our knowledge, the solution of FIDEs has done little to adapt these polynomials. In the present study, we solve FIDEs by implementing CPM. By implementing the proposed method, we compared our results with other techniques. We solve FIDEs of the form
having initial and boundary sources;
where
and
are real constants,
represents the fractional derivative in Caputo manner for
,
G and
H are well-defined functions.
The paper is structured as follows. In
Section 2, we provide some basic definitions which are further used in current work. In
Section 3, the concept of approximation Chebyshev series expansion by means of Caputo derivative is given. The implementation of the Chebyshev collocation approach to solve Equation (
1) is presented in
Section 4. In
Section 5, we solve some problems to clarify the technique’s effectiveness.
3. Approximation of Chebyshev Series Expansion by Means of Caputo Derivative
The Chebyshev polynomials are explained on the
interval, and can be calculated by means of recurrence formulae as [
31,
32]
where
and
The Chebyshev polynomial having degree
analytical form is defined by [
32]
To define Chebyshev shifted polynomials
, we take the Chebyshev polynomials over the interval
The Chebyshev shifted polynomials are determined by means of the following relation as [
32]
also by means of the below recurrence formula:
where
and
The orthogonality condition is (see [
33])
Now, by using the relation,
and Equation (
8) to obtain Chebyshev shifted polynomials analytical form for order
as:
A function
, described in Chebyshev shifted polynomials form as
where the factors
, for
are determined by
Thus, in practice first
-terms are taken and for few
,
is calculated as
Theorem 1.
The sum of the absolute values of all the omitted coefficients constrains the inaccuracy in approximating by the sum of its first terms. If, however [34]Thus, for all , m and , we get Theorem 2.
Assume that and is calculated by the Chebyshev shifted polynomials as in Equation (16). Then [35]where is defined by 4. Chebyshev Collocation Method
In this part, we implement Chebyshev’s collocation method to solve FIDE (1) having initial and boundary conditions (2) to achieve this goal, we calculated
as
By means of Equations (1) and (21) and Theorem 2 we get
Now we collocate (22) at points
The roots of the shifted Chebyshev polynomial are used to find suitable collocation points
. To apply the Gaussian integration formula for (23), we use the transformation to convert the
to t-interval
Equation (
23), for
may be rewritten as
On applying the Gaussian integration formula, for
we have
where
represents the
zeros of the Chebyshev polynomial
and
are the appropriate weights as in [
36]. For polynomials with a degree of less than
, the correctness of the Gaussian integration formula provides the basis for the above approximation.
We can generate more
equations by substituting (19) into initial or boundary conditions. By putting (19) into the boundary conditions (2), we obtain
Equation (
25), when combined with
equations of initial sources or boundary sources, yields
nonlinear algebraic equations that may be solved using Newton’s iterative approach for
applying Newton’s iterative method. As a result, the function
from (1) can be computed.