Block-pulse functions approach to numerical solution of Abel’s integral equation

Abstract: This study aims to present a computational method for solving Abel’s integral equation of the second kind. The introduced method is based on the use of Block-pulse functions (BPFs) via collocation method. Abel’s integral equations as singular Volterra integral equations are hard and heavy in computation, but because of the properties of BPFs, as is reported in examples, this method is more efficient and more accurate than some other methods for solving this class of integral equations. On the other hand, the benefit of this method is low cost of computing operations. The applied method transforms the singular integral equation into triangular linear algebraic system that can be solved easily. An error analysis is worked out and applications are demonstrated through illustrative examples.


Introduction
The past two decades have been witnessing a strong interest among physicists, engineers, and mathematicians for the theory and numerical modeling of integral and integro-differential equations. These equations are solved analytically; see, for example, the excellent book by Mushkelishvili (1953), and the references therein. The generalized Abel's integral equation on a finite segment

PUBLIC INTEREST STATEMENT
Abel's equation is one of the integral equations derived directly from a concrete problem of mechanics or physics (without passing through a differential equation). Historically, Abel's problem is the first one that led to the study of integral equations. The construction of high-order methods for the equations is, however, not an easy task because of the singularity in the weakly singular kernel. This class of equations (linear or nonlinear, of first kind or second kind) has applications in many fields of physics and experimental sciences: problems in mechanics, scattering theory, spectroscopy, stereology, seismology, and plasma physics lead to such equations.
appeared for the first time in the paper of Zeilon (1922). A comprehensive reference on Abel-type equations, including an extensive list of applications, can be found in Gorenflo and Vessella (1991) and Wazwaz (1997). The construction of high-order methods for the equations is, however, not an easy task because of the singularity in the weakly singular kernel. In fact, in this case the solution y is generally not differentiable at the endpoints of the interval, see Schneider (1979) and Vainikko and Pedas (1981), and due to this, to the best of the authors? knowledge the best convergence rate ever achieved remains only at polynomial order. For example, if we set uniform meshes with n + 1 grid points and apply the spline method so for order m, then the convergence rate is only O(n −2p ) at most (Schneider, 1981;Graham, 1982), and it cannot be improved by increasing m. One way of remedying this is to introduce graded meshes. Then the rate is improved to O(n −m ) (Vainikko & Uba, 1981) which now depends on m, but still at polynomial order. Fettis (1964) proposed a numerical form of the solution to Abel equation by using the Gauss-Jacobi quadrature rule. Piessens and Verbaeten (1973) and Piessens (2000) developed an approximate solution to Abel equation by means of the Chebyshev polynomials of the first kind. Numerical solutions of weakly singular Volterra integral equations were introduced by many researchers, such as Baratella and Orsi (2004) and Abdou and Nasr (2003). Yanzhao, Min, Liping, and Yuesheng (2007) applied hybrid collocation method for solving these equations. Ishik, Sezer, and Gney (2011) used Bernstein series solution for solving linear integro-differential equations with weakly singular kernels. Wazwaz (2002) studied on singular initial value problems in the second-order ordinary differential equations. Yousefi (2006) applied Legendre wavelets to transform Abel integral equation to a algebraic system.
The aim of this work is to present a computational method for solving special case of singular Volterra integral equations of the second kind as named Abel's integral equation defined as where f ∈ L 2 ([0, T]). Our method consists of reducing the given weakly singular integral equation to a set of algebraic equations by expanding the unknown function using block-pulse functions (BPFs) with unknown coefficients. Collocation method is utilized to evaluate the unknown coefficients. Because of orthogonality, disjointness and having completeness properties of these functions, the operational matrix of method is a sparse lower triangular matrix. Without loss of generality, we may The paper is organized as follows: In the next section, we discuss BPFs, their key properties and function approximation by them. In Section 3, we give the description and development of the method for solving singular integral equation. Error analysis and convergence of the method is discussed in Section 4. In Section 5 for showing the efficiency of this method, some numerical examples are presented. Section 6 is devoted to the conclusion of this paper.

Block-pulse functions
BPFs are studied by many authors and applied for solving different problems, for example see Steffens (2006).
A k-set of BPFs over the interval [0, T) is defined as with a positive integer value for k. In this paper, it is assumed that T = 1, so BPFs are defined over [0, 1). BPFs have some main properties, the most important of these properties are disjointness, orthogonality, and completeness. (1) (1) The disjointness property can be clearly obtained from the definition of BPFs (2) The orthogonality property of these functions is (3) The third property is completeness. For every y ∈ L 2 [0, 1), when k approaches to the infinity, Parseval's identity holds, that is where c i are samples of y, for example c i = y( i k ) for i = 0, 1, … , k − 1, and as a result there is no need for integration. The vector C is called the 1D-BPFs coefficient vector.

Numerical implementation
Consider the Abel's integral equation where f ∈ L 2 (ℝ) is known and y(t) is the unknown function to be determined. Now by substituting by considering B(x j ) = e j (the j-th column of identity matrix), we can rewrite Equation 8 as follows Now for computing the unknown coefficients c j , we introduce a recursive formula.
The unknown coefficient c j satisfies the following recursive relation: where

Error analysis
In this section, we give an upper bound for the error of BPFs expansion. Assume that y � (t) ∈ C Remark It is clear that the approximation will be more accurate if k (the number of BPFs) is increased, thus for acheiving better results, using larger k is recommended.

Numerical examples
In . The solution for y(x) is obtained by the method in Section 3 at the scales k = 16, 32, and 64. In Table 1, the exact and approximate solutions of Example 1 both by our method and by Legendre wavelets method (Yousefi, 2006) in some arbitrary points are reported. Comparison with Legendre wavelets expansion, the method of this paper has more accuracy. As proved in Section 4, the error at the level k = 64 is smaller than the error at the levels k = 16 and k = 32.
L. W: Results by Legendre wavelets expansion.
In Figures 1 and 2, pointwise plots are the approximated solution of Example 1 (for k = 32 and k = 64, respectively) in some arbitrary points and the others are the plot of exact solution. Example 2 Consider the equation with the exact solution y(x) = ln(x). The solution for y(x) is obtained by the method in Section 3 at the scales k = 16, 32, and 64. In Table 2, we present the exact and approximate solutions of Example 2 both by our method and by Legendre wavelets method (Yousefi, 2006) in some arbitrary points. Comparison with Legendre wavelets expansion, the method of this paper has more accuracy. As proved in Section 4, the error at the level k = 64 is smaller than the error at the levels k = 16 and k = 32.  In Figures 3 and 4, pointwise plots are the approximated solution of Example 2 (for k = 32 and k = 64, respectively) in some arbitrary points and the others are the plot of exact solution. Obviously, there is less error between two plot for k = 64 rather than k = 32.

Conclusions
In the present paper, BPFs with the collocation method are applied to solve the Abel's integral equation. As for Legendre polynomials, Chebyshev polynomials and other generalized orthogonal polynomials, the calculation procedures are usually too tedious, although some recursive formula are available. The benefits of this method are low cost of setting up the equations and more accuracy and more efficiency of the solution than exiting methods for solving this class of equations. Also, the linear system of Equation 11 is a lower triangular system which can be easily solved by forward substitution with O(k 2 ) operations, therefore, the count of operations is very low. The approach can be extended to nonlinear Abel integral equation and integro differential equation with little additional work. Further research along these lines is under progress and will be reported in due time.