1. Introduction
Fractional calculus is an old topic; it was started from some fractional order derivative questions raised by Leibniz in 1695 and Euler 1730 but a yet novel one. It has been developed through extensive work to date. Many mathematicians have been involved and contributed dramatically to the field, such as Fourier, Laplace, Riesz and many more. Most recently, numerous scientists provided new definitions of fractional order derivatives and integrals that opened a new era in the history of fractional derivatives, such as the Atangana–Baleanu fractional integral [
1], the Caputo fractional derivative [
2] and the Caputo–Fabrizio fractional derivative [
3]. There is a series of new lines of research that is devoted to fractional calculus and its applications in many disciplines, such as physics, engineering and modeling [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38].
In the literature, there are plenty of contributions on the use of wavelets and their generalizations to model and solve several problems of differential and integral equations of different types and applications in pure mathematics, engineering and physics; see, for example [
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55]. In this paper, we use framelets with three generators generated via set of B-splines in order to solve fractional Volterra integral equations (FVIEs). Usually, it is difficult and sometimes impossible to find exact solutions for such types of integral equations. Therefore, developing numerical algorithms aimed to find a numerical approximation is essential.
In this paper, we consider the following form of fractional Volterra integral equation (FVIE)
with the following initial conditions (ICs)
where
is the known Caputo fractional order derivative given by
The purpose here is to provide an approximate solution of the fractioal Volterra initial value problem (FVIVP) given in Equations (
1) and (
2) in the form of the truncated expansions of a framelet system, where a set of functions
is called a framelet for
if there exists a positive number
such that the inequality
holds for any function
Note that according to the inequality (
3), for a function
it is obvious to obtain the following associated framelet representation
The framelets are constructed using B-spline functions. The B-splines
of order
M are defined recursively by the following equation
where
is the indicator function over
.
B-splines are non-negative refinable functions in the sense that
where
such that
is a polynomial of trigonometric functions with
, and
is
-periodic function in the frequency domain and called the low mask of
.
The framelet system
is constructed via the oblique extension principle (OEP) [
39] and has the form
and satisfies the following equations
where
are the low and high masks of the
respectively.
The representation in Equation (
4) is truncated by the series
, such that
Let us present some examples of framelet systems.
Example 1. Consider the refinable function, . Then, based on the OEP presented in [39] we are able to construct the following framelets explicitly,Then, the system where forms a framelet system for . The graphs of the framelets are plotted in Figure 1. Example 2. Consider the refinable function, . Then, again based on the OEP we haveThen, the system where forms a framelet system for . The graphs of the framelets are plotted in Figure 2. 2. Matrix Formulation Using Framelets
In this section, we provide the general framework of the aforementioned numerical scheme based on the collocation discretization of the domain. We also provide two results related to the existence and uniqueness of the solution.
Consider the FVIE defined in Equation (
1). Based on the truncated expansion obtained in Equation (
5), we have
where the
nth derivative is approximated by the truncated framelet expansion as follows:
and
is the Riemann–Liouville fractional-integral operator defined by
Therefore, using the Caputo derivative, we then get
With a little algebra, Equation (
6) can be simplified to the following
Now, based on a dyadic discretization points of the domain of the framelet system being used, say,
, and by plugging these point into the equations above, we have
The above equation yields a system of equations that can be easily solved to obtain the unknown coefficients in order to get the approximate solution of order m.
We now provide two main results with regard to the existence and uniqueness of the FVIVP defined in Equations (
1) and (
2).
Theorem 1 (Existence).
Assume that a, b and are continuous functions on . Then there exists a real-valued function u defined on solving the FVIVP given in Equations (1) and (2) such thatwhere and .
Proof. Apply the Riemann–Liouville integral operator of both sides of Equation (
1), and using the ICs we have,
The idea is to show that
is a self mapping operator on the non-empty set
where
and has a fixed point in
. Hence
Which means is a self mapping function and this completing the proof. □
Theorem 2 (Uniqueness).
Assume that a, b and are continuous functions on . Let and are upper bounds for a, b and , respectively. Then, the FVIVP defined in Equations (1) and (2) has a unique solution if Proof. Assume that the FVIVP has two solutions
and
. Then, we have
and
By taking the Riemann–Liouville integral, we get
and
Therefore, as
where
, the result is concluded. □
4. Conclusions
The framelet system we used in this paper was generated using three wavelet frame functions with compact support and constructed based on using the non-negative functions, B-splines.
We have also established two important results on the existence and uniqueness of the Equations (
1) and (
2) considered in this paper. The proposed method was tested by numerically solving many important examples of fractional Volterra integral equations. This work is an extension of the work published in [
47] by involving the fractional order derivative, namely, the Caputo fractional derivative sense.
The approximate solutions are supported by numerical evidence given in
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6, and graphical illustrations in
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8,
Figure 9 and
Figure 10, wherein excellent agreement with the exact solutions was accomplished with only a few framelet truncated partial sums.
Based on the graphical and numerical evidence, we conclude that the accuracy of the method is increased by two important factors:
- 1.
Number of terms of the partial sum of the framelet truncated expansion being used;
- 2.
The vanishing moments order of the framelet system being used, where increasing these terms will result an increase in the accuracy as well as the efficiency of the algorithm.