Generalized Abel Integral Equation Research Articles

Optimal quadrature formulas in Sobolev space for solving the generalized Abel integral equation

In this article, a composite optimal quadrature formula is constructed for an approximate analytical solution of the generalized integral Abel equation in the Sobolev functional space. The optimal coefficients of this quadrature formula have been found. In addition, using the constructed composite optimal quadrature formula, numerical results of examples of the generalized Abel integral equation were obtained and compared with the exact solution.

A novel algorithm and its convergence analysis for solving the generalized Abel integral equations through fractional calculus

This paper concerns a new semi analytical method for approximate solution of the first type generalized Abel integral equations. An especial numerical fractional derivative which is L1 method will be used for solving this kind of equations. Also, existence, uniqueness, convergence and stability of the given scheme will be studied through some theorems and lemmas. Moreover, some numerical examples are presented and the results are compared with their exact solutions and some other numerical methods to illustrate the capability of this algorithm for solving these generalized Abel integral equations.

Singular expansions and collocation methods for generalized Abel integral equations

In this paper, the generalized Abel integral equation with singular solution is studied. First, the finite-term psi-series expansion of the solution about the origin is derived using the method of undetermined coefficients, which gives the complete singular information of the solution. Second, this singular expansion is used to separate the singularity such that the Abel integral equation is converted into a perturbed one with smooth solution on a regular interval. Third, a product trapezoidal method is designed to solve the transformed equation on the regular interval and the convergence analysis is conducted to show that the scheme has second order accuracy combining with a controllable perturbation term. Fourth, a Chebyshev collocation method is further constructed on the regular interval to show that the global orthogonal polynomial interpolation is also suitable for solving the transformed Abel integral equation with high accuracy. Finally, two numerical examples confirm the correctness of the truncated psi-series solution and the effectiveness of the piecewise and global collocation methods with singularity separation for solving this kind of Abel integral equation.

Optimal formulas for the approximate-analytical solution of the general Abel integral equation in the Sobolev space

This article discusses the development of a new algorithm, which is based on optimal quadrature formulas for obtaining solutions to the generalized Abel integral equations. Based on the study, it was found that the proposed scheme is very effective, extremely accurate and can be extended to other special tasks. The quadrature formulas presented in this paper are optimal in the Sobolev space of functions that have square integrable derivatives of order m. Using the quadrature formula and the Maple computer algebra system, exact and approximate values of the Abel integral equations are found, illustrating the effectiveness of the proposed approach.

Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems.

A Generalized Barycentric Rational Interpolation Method for Generalized Abel Integral Equations

The paper is devoted to the numerical solution of generalized Abel integral equation. First, the generalized barycentric rational interpolants have been introduced and their properties investigated thoroughly. Then, a numerical method based on these barycentric rational interpolations and the Legendre–Gauss quadrature rule is developed for solving the generalized Abel integral equation. The main advantages of the presented method is that it provides an infinitely smooth approximate solution with no real poles for the generalized Abel integral equation.

An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases

This work introduces a direct method based on orthonormal Bernstein polynomials wavelet bases, to present a stable algorithm for numerical inversion of a system of generalized Abel integral equations. The application of all the currently existing numerical inversion methods was strictly limited to only one portion of the generalized Abel integral equations. The proposed method is quite accurate, and several numerical illustrations demonstrate the convergence and utilization of the proposed method compared to some of the preexisting numerical solution techniques. The permanence of the numerical result under the effect of small perturbation in input data has been examined, which is depicted with the use of numerical illustrations.

An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics

In this paper, we aim to solve nonlinear system of generalized Abel integral equations arising in astrophysics. The suggested approach is operational matrix technique by using Legendre scaling functions as a basis. Convergence analysis of the suggested technique is provided. Numerical experiments are performed to show the effectiveness of the proposed scheme. The results are shown through figures and tables.

About Solution of the Nonlinear Generalized Abel Integral Equation

As is known, many problems of electronics, nuclear physics, optics and astrophysics, etc. are described by the Abel integral equation of the first kind. In this paper we consider the nonlinear generalized Abel equation and show that its solution can be represented as an integral of a power function. It is shown that the constructed analytical solution and the symbolic solution obtained by means of the computer mathematics system Maple coincides, and their planar and spatial graphs are presented.

The Construction of Operational Matrix of Fractional Integration Using the Fractional Chebyshev Polynomials

In this paper we present a computational method for solving generalized Abel integral equations. To this end, first, a new operational matrix of fractional order integration for fractional Chebyshev polynomials is derived. The error analysis of the proposed method is studied theoretically. Then, by applying this matrix, collocation and Galerkin methods, we reduce the original problem to a linear system of algebraic equations. Finally, comparison of numerical results with the exact solution shows the validity and efficiency of the presented method.

The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such as emission tomography using Compton cameras. In this paper, we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and where the symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Since the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel integral equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to efficiently implement our inversion method and present numerical results.

Generalized Abel Inversion Using Extended Hat Functions Operational Matrix

Abel type integral equations play a vital role in the study of compressible flows around axially symmetric bodies. The relationship between emissivity and the measured intensity, as measured from the outside cylindrically symmetric, optically thin extended radiation source, is given by this equation as well. The aim of the present paper is to propose a stable algorithm for the numerical inversion of the following generalized Abel integral equation:I(y)=a(y)∫αy((rμ-1ε(r))/(yμ-rμ)γ)dr+b(y)∫yβ((rμ-1ε(r)) /(rμ-yμ)γ)dr,α≤y≤β,0<γ<1, using our newly constructed extended hat functions operational matrix of integration, and give an error analysis of the algorithm. The earlier numerical inversions available for the above equation assumed eithera(y)=0orb(y)=0.

On Solution Representation of Generalized Abel Integral Equation

An explicit representation for solution of the generalized Abel integral equation , where is the Riemann-Liouville fractional integral, in terms of the Wright function, is constructed.

A stable numerical inversion of generalized Abelʼs integral equation

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.

Laguerre Polynomials Based Numerical Method to Solve a System of Generalized Abel Integral Equations

In this paper, a numerical method is presented to solve a system of generalized Abel integral equations which arise frequently while solving a variety of mixed boundary value problems of classical theory of elasticity. In the method, Laguerre polynomials are used for the approximation of the solution of the system of integral equations. Some test examples are given to show the validity of the method presented. Finally, the effectiveness of the proposed method is concluded in terms of the accuracy achieved for the analyzed test examples.

Generalized Abel Inversion Using Homotopy Perturbation Method

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. Obtaining the physically relevant quantity from the measured one requires, therefore the inversion of the Abel’s integral equation. The aim of this letter is to present a user friendly algorithm to invert generalized Abel integral equation by using homotopy perturbation method. The stability of the algorithm is analysed. The validity and applicability of this powerful technique is illustrated through various particular cases which demonstrate its efficiency and simplicity in solving these types of integral equations.

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

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.

Solution of the Generalized Abel Integral Equation

A direct function theoretic method is employed to determine the closed form solution of the generalized Abel integral equation. The present form of the solution involves only weakly singular integrals to be evaluated finally as opposed to the known form that requires evaluation of strongly singular integrals of the Cauchy type.

Operationl method for solving generalized abel integral equation of second kind

A modification of the mikusinski operational caculus is used to obtain explicit solution of some classes of integral equations of the second kind of abel type .inparticular the solution of the generalized abel equation with the riemann-liouville fractional integral opereator is expressed in terms of the mittag-leffler function of several variables.

Solution of a system of generalized Abel integral equations using fractional calculus

A method is presented for obtaining useful closed form solution of a system of generalized Abel integral equations by using the ideas of fractional integral operators and their applications. This system appears in solving certain mixed boundary value problems arising in the classical theory of elasticity.