Solving Abel integral equations of first kind via fractional calculus

TL;DR

This paper introduces a novel numerical method for solving Abel integral equations of the first kind using fractional calculus, specifically fractional integrals and Caputo derivatives, with demonstrated accuracy via computational examples.

Contribution

The paper presents a new fractional calculus-based approach for numerically solving Abel integral equations, including error estimation and practical implementation.

Findings

Effective numerical solutions for three Abel integral equations

Error bounds for the proposed method

Implementation using Maple demonstrates accuracy

Abstract

We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.