Solution of the Generalized Abel Integral Equation by Using Almost Bernstein Operational Matrix

Abstract

A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise

Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations

“…The solution methods of the Abel integral equations are collocation methods [9][10][11][12][13], Adomian decomposition methods [14][15][16], homotopy perturbation methods [17], quadrature methods [18][19][20], homotopy analysis methods [21], and Laplace transform methods [22][23][24][25][26]. Besides, Abel integral equations are also solved by using Chebyshev [27][28][29], Legendre [30,31], Taylor [32], Bernstein [33,34], Block-Pulse [35], and Laguerre [36] functions. Moreover, numerical methods on other fractional integral equations are hybrid collocation [37], smoothing technique [38], piecewise constant orthogonal functions approximation [39], the Haar wavelet method [40], the Galerkin method [41], Bernstein's approximation [42,43], the Simpson 3/8 rule method [44], mechanical quadrature [45], Legendre Pseudo spectral [46], and the iterative numerical method [47].…”

Solution of Abel Integral Equation Using Differential Transform Method

“…Pandey and Mandal [6] obtained numerical solution of a system of generalized Abel integral equations using Bernstein polynomials. Dixit et al [7] solved generalized Abel integral equation by using Bernstein operational matrix. Pandey et al [8] used collocation method for the solution of generalized Abel integral equations.…”

An efficient and flexible Abel-inversion method for noisy data

“…(depending on a particular problem) or smoothing the input data u(x), again using some polynomials. For references, see Graig (1979), Knill, Dgani, & Vogel (1993, Bendinelli (1991), Dixit, Pandey, Kumar, & Singh (2011 The key point in this approach is to truncate the degree of the polynomial approximation, so the high-frequency components are cut off thus stabilizing the solution. The main drawbacks are that (i)they do not guarantee convergence of the numeric solution to the true solution as the accuracy of the input data increases, and (ii)the choice of the polynomial degree is rather arbitrary.…”

“…(depending on a particular problem) or smoothing the input data u(x), again using some polynomials. For references, see Graig (1979), Knill, Dgani, & Vogel (1993), Bendinelli (1991), Dixit, Pandey, Kumar, & Singh (2011), etc. The key point in this approach is to truncate the degree of the polynomial approximation, so the high-frequency components are cut off thus stabilizing the solution.…”