New index transforms with the product of Bessel functions

TL;DR

This paper introduces new index transforms involving products of Bessel functions, analyzes their properties, establishes relationships with known transforms, and applies them to solve a fourth-order PDE initial value problem.

Contribution

It develops novel index transforms with Bessel function kernels, studies their invertibility, and applies them to partial differential equations, expanding the mathematical toolkit.

Findings

Established invertibility and mapping properties of the new transforms

Derived relationships with Kontorovich-Lebedev and Fourier cosine transforms

Provided solutions to a specific fourth-order PDE using these transforms

Abstract

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.