1. Introduction
Fractional calculus (FC) has explored the concept of differentiation and integration to non-integer order. FC is a more generalized version of classical calculus. FC is as old as classical calculus, but it is gaining popularity these days because of the implementations in many domains like science and engineering. Fractional partial differential equations (FPDEs) have gained much popularity because of their exceptional simulation properties in various scientific areas. It has been used to represent physical and technical phenomena that are described tremendously by fractional differential equations. The fractional derivative models are used to recognize better those systems, which required accurate modeling of damping, non-Fourier heat conduction, acoustic dissipation, geophysics, relaxation, creep, viscoelasticity, rheology, and fluid dynamics, Malaria and COVID-19 [
1,
2,
3,
4].
Integro-differential equations of fractional order are the type of models that include both integro-differential equations and fractional derivatives. The analysis of partial integro-differential equations with fractional specifications is an important element of the theory and implementations of FC, which have considered crucial mathematical methods for describing and analysing a broad variety of actual challenges in natural science, technology, and engineering [
5,
6,
7,
8]. The mathematical models of physical phenomena and their implementations in heat conduction [
9], reactor dynamics [
10], flow in fractured bio-materials [
11], electricity swaption [
12], visco-elasticity [
13], population dynamics, convection diffusion [
14] and grain growth [
15]. In this study, we will look at the nonlinear FPIDE [
16]:
with initial condition
and boundary conditions
where
,
L are positive constants, and
,
are given functions.
denotes the Caputo fractional derivative (CFD) and is interpreted as
where
is the Euler’s Gamma function.
At various stages of real systems, fractional derivatives and integral operators are more suitable than standard derivatives and integration, which provide a more precise explanation of structural and genetic features of several dynamical and physical procedures. As a result, accurate computational methods are used to approximately cope with the complexities of fractional derivatives contained in such equations. These complexities are due to the possibility of the singularities of the kernels causing drastic fluctuations in the solution. Consequently, it isn’t easy to acquire a closed-form solution in several implementations, particularly in nonlinear scenarios, so an approximation of physical description is needed. Alternatively, many studies have been conducted to investigate the presence of a unique solution to fractional order integro-differential equations, such as Hu et al. [
17], Li et al. [
18], Karthikeyan and Trujillo [
19], Chuong et al. [
20].
The majority of FPIDE cannot be addressed exact analytically, finding more effective approximate approaches using computational methods would be extremely beneficial. Several authors have focused their attention on searching and exploring solutions of the Fractional partial intego-differential equations (FPIDEs) using various analytical and numerical strategies. Awawdeh et al. [
21] utilized the homotopy analysis approximation to solve the linear FPIDE analytically. Hussain et al. [
22] solved the FPIDE analytically by the variation iteration method. Mittal and Nigam [
23] implemented the Adomian decomposition method to handle the FPIDE. Rawashdeh [
24] suggested a collocation approach for solving the FPIDE numerically by the polynomial spline. Eslahchi et al. [
25] developed the jacobi technique to solve nonlinear FPIDE, also analyzed stability and convergence. Zhao et al. [
26] employed piecewise polynomial collocation approaches to tackle FPIDEs containing weakly singular kernels. Arshed [
27] demonstrated the B-spline technique for solving linear FPIDE. Unhale and Kendre [
28] presented collocation technique to solve the nonlinear FPIDE utilizing the Chebyshev polynomials and the shifted Legendre polynomials. Avazzadeh et al. [
29] established a hybrid technique by blending the Legendre wavelets, and operational matrix of fractional integration. A numerical technique based on Legendre-Laguerre and the collocation method has been considered by Dehestani et al. [
30].
The B-spline was proposed by many authors to solve fractional partial differential models [
31,
32,
33,
34,
35,
36,
37,
38]. These functions can adjust every point in the domain and approximate the solution with maximum frequency accuracy. For solving FPDEs, a variety of numerical approaches have already been developed. However, so far as we know, no such research on the utilization of B-splines exists in solving the nonlinear FPIDE. Therefore, we intend to fill this gap. We aim to extend the ECBS technique for the solution of the nonlinear FPIDE model with a weakly singular kernel. The article is partitioned as follows: In
Section 2, the basis functions and the time approximation are presented. In
Section 3, a derivation of the method is described. The stability and convergence of recommended technique are analyzed in
Section 4. Two applications and discussions are shown in
Section 5. Finally, the results of the recommended technique are displayed in
Section 6.
3. Derivation of the Procedure
We employ the ECBS and the CFD to address the proposed model. By plugging Equations (
8) and (
10) in (
1), we obtain
Linearize the non-linear term as [
41]:
By plugging Equation
into
, we have
Using expression
in the above equation, we obtain
After some computation in the above expression, we get
where
,
The above equation can be rewritten as:
The Equation
can be converted into matrix form as:
The order of the system
is
. Two linear equations from the boundary conditions are required for a unique solution. To initiate the iteration on
, the corresponding initial conditions are applied:
As a result,
transforms to a matrix system:
where
and
.
5. Numerical Implementation
Here, we include the simulation results of the problem (
1)–(
3) by employing the suggested algorithm. Maximum errors,
errors and order of convergence between exact and computed solutions are employed to demonstrate the reliability of the applied model. The following formula can be utilized in order to calculate the convergence order numerically.
where
and
are the maximum errors at
and
respectively.
Example 1. Consider the modeled problem (1)–(3) with the initial condition and source term are shown below:withwhereand analytical solution is . Table 1 shows the
,
errors and order of the recommended technique for
,
,
at various
.
Table 2 presents the comparison of the
errors with the results given by [
16] and the order of convergence at different
h. In
Table 3, the absolute errors are demonstrated for the
and different
at
.
Figure 1 illustrates the 3D error plot of (1) for
,
, when
and
at
.
Figure 2 shows the compatibility of exact and computed values for
,
. In
Figure 3, we present the approximated and exact graph of Example 1, for
,
,
, and
at
. Graphs show that the close relationship between the exact and calculated values.
Table 4 demonstrates the
and order in time side
,
and various
at
. In
Table 5, the comparison of
errors of 2 for
,
,
at different
h. In
Table 6, we present the absolute errors for
and several values of
at some knots, when
,
, and
. Therefore, we achieve the convergence order in time direction is
, and space direction is 2. A comparison plot of computed and exact values has been displayed in
Figure 4 for
,
at
.
Figure 5 depicts the errors of Example 2 for
,
,
for
and
at
The exact and calculated values are presented in
Figure 6 for
,
,
,
.