1. Introduction
In this study, we propose an efficient scheme for solving the fractional Fredholm integro-differential equations (FFIDEs) of order
(
) on the finite interval
with the Caputo fractional derivative
and initial conditions
in which
,
are constants,
, for
and
for
. Here, the function
f is assumed to be a sufficiently smooth linear or nonlinear function on
with
,
k is a continuous function on
, and the linear or nonlinear function
g is assumed to be continuous and satisfies the Lipschitz condition
where
is the Lipschitz constant.
These types of equations have a very valuable role in modeling some physical phenomena, such as glass-forming process [
1], epidemic processes [
2], and viscoelasticity [
3]. There exist several papers that offer analytical methods for solving such equations. However, when the problem is complicated, the existing analytical methods no longer work and we cannot find the exact solution. Therefore, numerical methods are often suggested to solve this problem. In [
4], the Spline collocation method is applied to solve the problem. Momani et al. [
5,
6] used the Adomian decomposition method for solving the fourth-order and systems of FFIDEs. To solve a special type of these equations, i.e.,
with the initial conditions
Zhu et al. [
7] proposed the Galerkin method based on the Chebyshev wavelet. After introducing the Chebyshev wavelet and the operational matrix of the Riemann–Liouville fractional integral for this basis, they used the Galerkin method to reduce (4) to a system of algebraic equations. The fractional differential transform scheme is used to solve the equation [
8]. Saeedi et al. [
9] used the same procedure based on CAS wavelets. Shahmorad et al. [
10] proposed the Tau–like numerical algorithm to solve the delay fractional integro-differential equation. To read more about the methods provided, please refer to [
11,
12].
Recently, the Müntz–Legendre wavelets have been applied to find the numerical solution of some equations, such as fractional pantograph differential equations [
13], fractional optimal control problems [
14], fractional differential equations [
15] and multi-order fractional differential equations [
16].
The outline of this article is as follows: In
Section 2, we provide an introduction to fractional calculation and introduce the Müntz–Legendre wavelets.
Section 3 is dedicated to the application of the wavelet collocation method for solving FFIDEs. In this section, an a posteriori error estimate is also surveyed. In
Section 4, some numerical implementations are performed to demonstrate the accuracy and efficiency of the method.
3. Wavelet Collocation Method
In the present section, we utilize the collocation method based on ML wavelets to obtain an approximate solution of the fractional Fredholm integro-differential Equation (1). In the operator form, Equation (1) may be written as
in which the operator
is denoted by
If y is a sufficiently smooth function on , it can be proved that Equation (1) has a unique solution on .
Lemma 4. Given , let . Assume that f, k, and u are continuous functions. Then is the solution of (1) if, and only if, satisfies the integral equation Proof. The proof is similar to the proof of Theorem 3.24 in [
17]. □
To obtain the discretization of (29), the numerical solution may be approximated by the operator
, i.e.,
where
U is a
N-dimension vector whose elements should be found. Replacing (30) into (29), we get
Now, we transfer all terms in (31) onto via the projection operator as follows
Let us put
, then we can write
where the
j-th element of the
N dimensional vector
is obtained by
.
After putting the approximate solution
into
and then approximating it and the kernel function
using operator
, we have
where
G is an
N-dimensional vector whose
j-th element is
, and
K is a square matrix of dimension
whose
-element is
Replacing (33) into
, we obtain
To give rise to the discretized form of
, using the operational matrix
and (35), we obtain
In the same way as the previous item, we can use the projection
for the term
, as
where
F is a
N-dimension vector whose
j-th element is
.
Now we refer to (29) and rewrite it using (32), (35) and (36) as follows.
where
is the residual function that our goal is to reduce to zero. By choosing the collocation points
which satisfy
, we obtain a system of nonlinear or linear algebraic equations. After solving this system, we can find the unknown coefficients
U. Here, the collocation points are chosen so that they are the roots of the shifted Chebyshev and Legendre polynomials.
Error Analysis
Theorem 1. Assume that is a sufficiently smooth function on and the functions g and f are continuous and satisfy the Lipschitz conditions (3) andrespectively. If with , and , then the a posteriori error estimate can be found as, Proof. Motivated by Lemma 3, we have
where
, and to approximate it, we can consider two situations.
- 1.
if
, then we have
and it follows from Lemma 1 that
Since the function g is continuous, then is bounded.
- 2.
Let
. Motivated by the Lemma 2.21 [
17], it is easy to write
Taking the norm from both sides of (42) and using Lemma 2, we have
As a result, we can bound this case according to the previous one.
Further, we can obtain
and
in which
, and
. Similar to the process used to calculate
, it can be used to approximate
and
.
If
k is a continuous function and
g satisfies the Lipschitz condition (3), then we can write
It is easy to find a bound for
according to the Lemma 1, via
Subtracting (29) from
and taking the norm from both sides, it follows from (39), (44)–(47) that
in which
and
. If
, then we can bound the error as follows.
□