New index transforms of the Lebedev- Skalskaya type

TL;DR

This paper introduces new Lebedev-Skalskaya type index transforms using the real part of the modified Bessel function, explores their properties in Lebesgue spaces, and applies them to solve PDE initial value problems.

Contribution

It presents novel index transforms involving the real part of the modified Bessel function and establishes their properties and applications, including solving PDEs.

Findings

Proved boundedness and invertibility of the new transforms in Lebesgue spaces.

Derived inversion theorems for these transforms.

Applied the transforms to solve a second order PDE involving the Laplacian.

Abstract

New index transforms, involving the real part of the modified Bessel function 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 of the initial value problem for the second order partial differential equation, involving the Laplacian, is obtained. It is noted, that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich- Lebedev transform and its inverse.