The Lamb-Bateman integral equation and the fractional derivatives

TL;DR

This paper demonstrates that the Lamb-Bateman integral equation, used in solitary wave diffraction studies, is fundamentally an Abel integral equation and can be solved using fractional calculus techniques.

Contribution

It establishes the equivalence of the Lamb-Bateman integral equation to the Abel integral equation and applies fractional calculus for its solution.

Findings

The Lamb-Bateman integral equation is equivalent to the Abel integral equation.

Fractional calculus provides a method to solve the Lamb-Bateman equation.

The approach offers new insights into wave diffraction analysis.

Abstract

The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.