Lebedev's type index transforms with the modified Bessel functions

TL;DR

This paper introduces new Lebedev-type index transforms involving modified Bessel functions, analyzes their properties, and applies them to solve a fourth-order PDE, revealing new solutions and connections to generalized transforms.

Contribution

It develops and studies new Lebedev-type index transforms with modified Bessel functions, including their boundedness, invertibility, and applications to PDEs, extending previous work with novel solutions.

Findings

Transform boundedness and invertibility established

Inversion theorems proved for the new transforms

Application to solve a specific fourth-order PDE

Abstract

New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the modulus of Macdonald's function, recently considered by the author.