Index transforms with the square of Bessel functions

TL;DR

This paper introduces new index transforms with Bessel function kernels, explores their mathematical properties in Lebesgue spaces, proves inversion theorems, and applies them to solve a third-order PDE involving the Laplacian.

Contribution

It presents novel index transforms with Bessel functions, analyzes their properties, and demonstrates their application to solving specific PDEs.

Findings

Transform boundedness and invertibility in Lebesgue spaces established

Inversion theorems proved for the new transforms

Application to initial value problem for a third-order PDE involving Laplacian

Abstract

New index transforms, involving the square of Bessel functions of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third order partial differential equation, involving the Laplacian, is obtained.