Abstract
This paper deals with the numerical solution of the Abel integral equation based on Müntz–Legendre wavelets. To this end, the Abel integral operator is represented by Müntz–Legendre wavelets as an operational matrix. To find this matrix, we use the similarity between the Abel integral operator and the fractional integral operator. The proposed method can be easily used to solve weakly singular Volterra integral equations. We have proved the convergence of the proposed method. To demonstrate the ability and accuracy of the method, some numerical examples are presented.
1. Introduction
In this paper, we focus our attention on constructing and applying Müntz–Legendre (M-L) wavelets that will be used as the basis in the pseudospectral method to solve the famous Abel integral equationin which Abel’s integral operator of order is defined in [1] as follows:
Here, given , , and are assumed to be continuous functions on and with . Further, we suppose that the kernel function is equal to the form . In other words, the desired equation is assumed to be linear.
The Abel equation is a special case of the integral equations with the weakly singular kernel that was first introduced by Abel. In investigating the generalization of the tautochrone problem, he introduced this equation [2]. This equation appears widely in modeling many physical problems, such as nuclear physics, X-ray radiography, fluid flow [3], scattering theory, plasma diagnostics, semiconductors, physical electronics, and nonlinear diffusion[1, 4]. Given this equation’s wide application, solving this equation is very important. But one cannot always solve the equation analytically, and we need to use numerical methods for it.
Among the many papers that have considered the numerical solution of this equation, we can mention some of them. Saadatmandi and Dehghan [5] utilized the collocation method based on shifted Legendre polynomials. Piessens and Verbaeten [6] introduced a numerical method based on Chebyshev polynomials, and after approximating the unknown solution based on these bases, they obtained the solution as a sum of hypergeometric functions. Using the Bernstein operational matrix, Singh et al. [7] introduced a stable numerical method to solve this problem. In [8], we can find the integrable solution of the Abel integral equation under certain conditions, and also the sufficient and necessary conditions for the existence of this solution are presented. In [9], the authors proposed the Laplace transform method to solve the problem, where they assumed that the solution would be differentiable and continuous. Saray [10] introduced a novel and efficient method based on Alpert’s multiwavelets. In this work, after introducing the sparse representation of the Abel integral operator, the Abel integral equation is reduced to a sparse system of linear algebraic equations in the linear form, and this causes a reduction in time and computational costs. In [11], the unbounded solutions of the nonlinear Abel integral equations are investigated. Li and Zhao [12] used Mikusinski’s operator of fractional order to solve the Abel integral equation.
The outline of this article is as follows: the M-L wavelets are constructed in Section 2, and then the Abel integral operator is represented based on these bases. In Section 3, the Abel integral equation is solved by using the pseudospectral method based on M-L wavelets. This section contains the error analysis and the conditions for convergence are investigated. To demonstrate the efficiency and accuracy of the presented method, some numerical examples are given in Section 4.
2. Müntz–Legendre Wavelets
As we know, multiresolution analysis (MRA) is a significant procedure for constructing wavelets. According to MRA, a family of nested subspaces exists such that they satisfy certain circumstances [13]where is equal to or any bounded interval.
Recently, the M-L wavelets have been used to solve some equations, such as fractional optimal control problems [14], fractional pantograph differential equations [15], fractional differential equations [16], and multiorder fractional differential equations [17]. To solve the Abel equation, we first briefly introduce the M-L wavelets as follows.
Given , assume that the subspace is spanned by the scaled and translated version of a set of bases, which are called multiscaling functions, viz.,where and with . The parameter is called refinement level and is the multiplicity parameter. In the sequel, we intend to introduce the functions .
Motivated by [17], we denote the M-L polynomials aswhere and the coefficient is obtained by
It can be easily shown that these polynomials satisfy the orthogonality requirements and form an orthogonal system, viawhere denotes the Kronecker symbol and is given by
Considering the definition of , one can introduce the M-L wavelets [17], via
Due to the definition of M-L wavelets, one can introduce the projection operator that maps any function onto as follows:where and throughout the paper, the superscript is used for the matrix transpose. Here is a vector function of dimension whose -th element is , and the -th element of the vector is evaluated by
It follows from [15] that one may be able to bound the projection error in the sense of Sobolev norms.
Lemma 1 (see [15]). Given , assume that . If , thenand for , we havein which is the Sobolev space and the related norm is determined by
2.1. Representation of Abel Integral Operator in Müntz–Legendre Wavelets
In this subsection, we consider the Abel operator as a fractional integral operator, and after representing the fractional integral operator in M-L wavelets as an operational matrix, we find a representation of the Abel integral operator in M-L wavelets. To this end, it is necessary to define some concepts about fraction calculation.
Definition 1 (see [18]). Let . The Riemann–Liouville fractional integral operator of order is determined bywhere represents the gamma function.
Remark 1. It can be verified that the fractional integral of the functions is given by
Lemma 2 (cf Lemma 2.1(a). see [19]). The fractional integration operators are bound in for as follows:
It follows from [19] that if , then the function itself belongs to . We recall that the Abel integral operator of order is determined by
There is a similarity between the Abel integral operator and the Riemann–Liouville fractional integral operator , viz,
Thus, we can use the Riemann–Liouville fractional integral operator instead of the Abel integral operator .
According to the definition of M-L wavelets, the fractional integral operator acting on the vector function may be written as an expansion of M-L wavelets , i.e.,where is an matrix and is famous as the operational matrix of fractional integral for the M-L wavelets.
Before we look at how to calculate the elements of the aforementioned matrix, it is intransitive to introduce the piecewise fractional-order Taylor functions. Let be a fixed number, the piecewise fractional-order Taylor functions can be defined in the following form:
By introducing the square matrix of dimension , whose -th element is computed by
We can expand any elements of vector function (M-L wavelets) by the piecewise fractional-order Taylor functions , viz,
Here the matrix is a matrix and it is called the transformation matrix. In the sequel, we assume that is a vector of dimension whose -th element is . Thus, it is easy to show that
It follows from (16) that one can find the -th element of , via
This gives rise to introduce a diagonal matrix , such that
It is worth noting that this matrix expresses the fractional integral of functions as a combination of themselves and has the following form:where and is a diagonal matrix of the form
Now, we are able to introduce the operational matrix of fractional integral for the M-L wavelets via
Thus, we get
3. Pseudospectral Method
To derive the numerical solution of the second kind of Abel integral equation based on the pseudospectral method, we can approximate the unknown solution with the projection operator , as follows:where is a vector of dimension , whose elements should be found. Note that the function and the kernel function can be approximated in the same manner, i.e.,where and . By substituting (32) into the integral part of the Abel integral equation (1), we obtain
We now substitute equations (31)–(33) into Abel integral equation (1) and simplify to getwhere is the residual function that our goal is to reduce to zero. Let be a number of points in , we select the solution that satisfies the collocation condition , where are called the collocation points. In this paper, we use the shifted Chebyshev and Legendre polynomials zeros as collocation points. The collocation method gives rise to a system of linear or nonlinear algebraic equations. One can derive the unknown coefficients after solving this system.
3.1. Error Analysis
We write the Abel integral equation (1) in the formwhere is a compact operator that maps any continuous function onto . As we said, our goal is to reduce the residual function to zero. Symbolically, we havewhere . We note that if and only if or equivalently,
Let is a solution of (37), then by applying , equation (37) can be written as follows:
Since both the original equations (1) and (38) are defined on , then for the error analysis, we compare them.
Theorem 1. Let us assume that is a bijections operator. Further, let us assume
Then for all sufficiently large , the operator exists as a bounded operator from to . Moreover, it is uniformly bounded
For the solutions of (35) and (38)
Proof. For details, refer to [20].
Since is a compact operator and is a bounded projection, such that as . Then motivated by [20] (Lemma 3.1.2), we haveTherefore, the condition of Theorem 1 is held.
4. Numerical Examples
In this section, some numerical examples are solved to show the validity and efficiency of the method. To do this, we carry out the Maple and MATLAB software simultaneously.
Example 1. For the first example, let us consider the linear Abel integral equation of the second kind with the kernel function , and . The exact solution is given in [10] as follows:where is the Mittag-Leffler functionTable 1 shows the absolute value of errors at different times when the collocation points are chosen to be the Legendre polynomial nodes. As we expected, when the increases (the degree of the bases as well as the number of collocation points increases) the error will decrease. To show the effect of the parameter in the -error, we plot Figure 1 and report Table 2. Figure 2 illustrates the effect of the choosing nodes on the -error and also absolute value of errors at the Chebyshev nodes using different multiplicity . Also, we can see the effect of the multiplicity parameter on the -error and absolute error in Figure 2.
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Example 2. The second example is devoted to the Abel integral equation (1) withThe exact solution is given by [4].
To show the effect of the multiplicity parameter and choosing the collocation points, we report Table 3. This Table also illustrates the effect of parameter on the -error. Due to Table 3, it is obvious that these three parameters have a direct effect on the -error such that when increases, the error decreases. Also, choosing the Chebyshev nodes gives us a better result than Legendre nodes. In Figure 3, we demonstrate the absolute error when multiplicity parameter increases taking and different collocation points.
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Example 3. Let us consider the following Abel integral equation:in which is the hypergeometric function defined in [21]. Also, the exact solution is .
Table 4 shows the absolute value of errors at different times when the collocation points are chosen to be the Chebyshev polynomials nodes. Figure 4 illustrates the effect of selecting the Chebyshev and Legendre nodes. We can also see the effect of increasing the parameter . It is observed that by increasing the parameter , the error decreases.
5. Conclusion
In this paper, we utilize an efficient algorithm based on the wavelet pseudospectral method to solve the well-known Abel integral equation. This method can easily be used to solve weakly singular Volterra integral equations, and this shows the ability of the proposed method. We have compared the method with other methods and shown that this method offers better results. We have proved the convergence of the proposed method. Given the construction of these bases and the role of the parameter , which can be polynomials with fractional powers, compared to other bases, if the exact solution or the known functions in the equation are of the fractional type, the proposed method will provide better results.
Data Availability
The data used to support this study are included within this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest with this work.
Authors’ Contributions
All authors read and approved the final manuscript. All authors contributed equally and significantly to the writing of this paper.
Acknowledgments
This work is supported by the Researchers Supporting Project Number (RSP-2021/401), King Saud University, Riyadh, Saudi Arabia.