1. Introduction
Spectral schemes are invaluable tools for the numerical solution of fractional partial differential equations (FPDEs), ordinary differential equations (ODEs), integral equations (IEs), and integrodifferential equations (IDEs).
Spectral approaches are a class of schemes used in applied mathematics and scientific computing to numerically solve certain differential equations and nonlinear integral equations. In recent years, these approaches have been used in modeling of many problems of physical phenomena, engineering and chemical processes in chemical kinetics [
1], super fluidity biology and economics [
2], axially symmetric problems in the case of an elastic body containing an inclusion [
3], and fluid dynamics [
4], and the Hammerstein integral equation is employed for modeling nonlinear physical phenomena such as electromagnetic fluid dynamics reformulation of boundary value problems with a nonlinear boundary condition [
5].
Various numerical approaches have been presented for solving a class of nonlinear singular integral equations including Abel’s integral equation, Hammerstein integral equation, Volterra integral equation, etc. For example, Noeiaghdam et al. in [
6] applied the Laplace homotopy analysis method to solve Abel’s integral equation, and validation of this method was discussed in [
7]. Also, the numerical studies on the Volterra integral equation with discontinuous kernels can be found in [
8,
9]. Allaei et al. in [
10] presented an analytical and computational method for a class of nonlinear singular integral equations. Maleknejad et al. in [
11] proposed a new numerical approach for solving the nonlinear integral equations of Hammerstein and Volterra–Hammerstein. In [
12], the authors applied the operational Tau method (OTM) to find a numerical solution for weakly singular Volterra integral equations (WSVIEs) and Abel’s equation.
Other researchers have attempted to solve nonlinear integral equations in recent years. Among them, in recent years, Mehdi Dehghan et al. in [
13] solved nonlinear fractional integrodifferential equations (NFIDEs) by using the collocation numerical method. Li Zhu and Qibin Fan in [
14] presented a spectral method based on the second Chebyshev wavelet (SCW) operational matrix for solving the fractional nonlinear Fredholm integrodifferential equation, and the Ferdholm and Volterra integral equations.
Nemati in [
15] applied a numerical approach for solving nonlinear fractional integrodifferential equations with weakly singular kernels by using a modification of hat functions. Somveer et al. [
16] presented an efficient spectral method based on shifted Legendre polynomials for solving nonlinear Volterra singular partial integrodifferential equations (PIDEs) which involve both integrals and derivatives of a function.
Recently, with the effort of other scientists, many of the nonlinear differential and integral equations which appear in different fields of physical phenomena and engineering were solved by using numerical methods, and nonlinear differential and integral equations have also been explored in delayed scaled consensus problems [
17,
18,
19,
20,
21,
22,
23,
24].
In the study of many nonlinear problems in heat conduction, boundary-layer heat transfer, chemical kinetics, and superfluidity, we are often led to singular Volterra integral equations for which real answers are hard to find [
10]. In this article, we use efficient functions such as Genocchi polynomials and their operational matrices to solve nonlinear Volterra integral equations with weakly singular kernels of the following form:
where
is in
on the interval
;
is locally Lipchitz continuous, smooth, and a Hammerstein nonlinear function; and
are real positive numbers.
For future works, we can use other polynomials like Chebyshev, Lagger, etc. for implementation, and by comparing the archived results, we can expand the present method and implement it on the system of nonlinear Volterra integral equations or nonlinear Volterra integral equations of mixed type. Because of important applications of the first kind of Volterra integral equations with discontinuous kernels in load leveling problems and power engineering systems, the proposed method can also be used for future works.
The rest of the article is organized as follows: In
Section 2, we state some necessary basic definitions and properties of Genocchi polynomials. Numerical implementation of the suggested technique based on Genocchi polynomials is shown in
Section 3.
Section 4 estimates the error analysis of our proposed technique. In
Section 5, two examples with tables and graphs are presented to show the efficiency and accuracy of the proposed scheme.
Section 6 provides some discussion and concluding remarks.
3. Implementation of the Genocchi Polynomial Method for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels
In this section, we implement a new spectral approach based on the Genocchi polynomials to solve the following equation:
where
is in
on the interval
;
is locally Lipchitz continuous, smooth, and a Hammerstein nonlinear function; and
are real positive numbers.
Let us assume that function
is arbitrary; then, we can approximate it, as follows:
where
is a vector of unknown coefficient;
; and
, where
is a
matrix of coefficients that can be approximated by
.
Thus, we need to compute the following integral before applying the new approach to solve Equation (1).
Therefore, by considering Relation (27), we let
since we have
By substituting Equation (29) into Equation (28), we have
We approximate Equation (30) as follows:
and
Thus, we need to convert the integral part of Equation (32) to the matrix form. Therefore, by assuming
, we can write the following:
and using Equation (27), we have
Therefore, by using Relation (34), we can rewrite Equation (33) as follows:
If we consider
, then, we can reconstruct Equation (35) in the matrix form as follows:
where
is an infinite diagonal matrix and
Now, each element of infinite vector
can be approximated by using the Genocchi polynomials as follows:
and we obtain
Substituting (39) in (32), we have
By using Equations (40) and (39), we get
We select
nodal points of the Newton–Cotes rule for finding vector
as follows:
By collocating Equation (41) at the points
, we have
We can solve the nonlinear system (43) by using the Newton iteration scheme to calculate unknown vector
After calculating unknown vector
by solving the nonlinear Equation (43), we use Equations (29), (31), and (32) to obtain the approximate solution of Equation (1), as follows:
4. Error Analysis
In this section, we perform error estimation of the approximation solution to find the error boundaries of the new numerical approach by applying the Genocchi polynomials. Consider the nonlinear Volterra integral equations with weakly singular kernels of the form Equation (1),
We suppose that
,
, and
. Here, we let
be an arbitrary function of
, so, it has the best approximation of
. Let
, that is,
where
.
is approximated by using the truncated Genocchi polynomials:
where
and
.
In the following study, we present an upper bound for the error of Equation (45). Let be the error function of Equation (1), where are the exact and approximate solutions
Therefore, the mean error bound is presented as follows:
On the other hand,
is continuous on the interval
and locally Lipchitz continuous in
; therefore, there is a constant
such that
Then, by using Equations (47) and (48), we have
By substituting (4) into (49), we get
where
is defined by
On the other hand,
is the beta function that is usually defined by
Therefore, by using Inequality (51) and Equation (52), we get
and
5. Illustrative Examples
In this section, two numerical examples are performed to check the perfection of the proposed method as well as the accuracy and efficiency of the Genocchi polynomials scheme.
In order to demonstrate the error of a new numerical approach based on Genocchi polynomials, we define the notations as follows:
where
is the exact solution and
is the approximate function to the proposed method and we have
. In our implementation, the calculations are done on a personal computer with core-i5 processor, 2.67 GHZ frequency, and 4 GB memory, and the codes were written in Mathematica 11 software.
Example 1. We consider the following nonlinear Volterra integral equation
which was proposed in [
10]:
The exact solution of this equation is We solved this equation with the proposed numerical method by using different values of N. The diagonal matrix
with elements
and vector
for
are obtain in the following forms:
Also, the unknown vector elements C are as follows:
After numerical computations, a system of algebraic nonlinear equations is obtained under the proposed method. Therefore, by solving this system, we obtain the approximate solution for
as follows:
According to the error boundaries in Relation (55), we have
Figure 2 is devoted to comparing the exact solution with the approximate solution obtained from the proposed method for
. Observing
Figure 2, overlap of the exact and approximate solutions shows the exactness and correctness of the proposed method. The absolute error functions with
are shown in
Figure 3,
Figure 4,
Figure 5,
Figure 6. Therefore, these plots quickly explain that the proposed approach has small absolute errors.
We reported the numerical results of the exact and approximate solutions for various values
on the interval
in
Table 1. On the other hand, numerical results are showed for different values
in
Table 2. The absolute error functions are displayed for various values of
on the interval
for this problem in
Table 3. Also,
Table 4 compares the numerical results of a new proposed numerical approach with Euler’s method [
10] for different values of
. Also,
Table 4 indicates that the new numerical method has better accuracy and efficiency compared to the old method.
Example 2. Next, we discuss the following Lighthill’s equation which was proposed in [
10]
and extensively studied in [
10,
29,
30]
. The authors employed the iterative method and schemes to solve this integral equation. The numerical results for this example are obtained by the presented approach for different values of
and are given in
Table 5 and
Table 6. Also, in
Table 7, the maximum absolute errors can be compared with those that were achieved by Euler’s method in [
10] by different values of
on the interval
. We can see that our proposed method is very fast compared to Euler’s method.
Figure 7 displays the convergence approximate solutions using our method (Genocchi polynomials) and the Picard iteration
with different values of
on the interval
with
for this problem.
6. Conclusions and Future Work
In the study of many nonlinear problems in heat conduction, boundary-layer heat transfer, chemical kinetics, and superfluidity, we are often led to singular Volterra integral equations that are difficult to solve analytically. In this article, a spectral method based on Genocchi polynomials is presented for solving nonlinear Volterra integral equations with weakly singular kernels. An error analysis of the spectral approach has been done. Two numerical examples are provided to confirm the applicability and accuracy of the scheme. Also, the proposed method results have been compared with Euler’s method to show the superiority of the present method with better results in smaller N. For future works, we can use other polynomials like Chebyshev, Lagger, etc. for implementation, and by comparing the archived results, we can expand the present method and implement it on the system of nonlinear Volterra integral equations and nonlinear Volterra integral equations of mix type or the first kind of Volterra integral equations with discontinuous kernels.