Transmutations and Applications: a survey

TL;DR

This survey comprehensively reviews transmutation theory and its diverse applications, including new results, covering various operators, fractional integrals, and open problems in the field.

Contribution

It provides an extensive overview of transmutation theory with original results and discusses numerous applications and open problems in the area.

Findings

Analysis of transmutations for Sturm--Liouville operators

Discussion of transmutations for singular differential operators

Identification of open problems in transmutation theory

Abstract

This is an extended version of originally published survey in the book: "Advances in Modern Analysis and Mathematical Modeling". Editors: Yu.F.Korobeinik, A.G.Kusraev. Vladikavkaz: Vladikavkaz Scientific Center, of the Russian Academy of Sciences and Republic of North Ossetia--Alania, 2008. P. 226-293. (In Russian). In this survey we consider main transmutation theory topics with many applications, including author's own results. The topics covered are: transmutations for Sturm--Liouville operators, Vekua-Erdelyi-Lowndes transmutations, transmutations for general differential operators with variable coefficients, Sonine and Poisson transmutations, transmutations and fractional integrals, Buschman--Erdelyi transmutations, in the search for Volterra unitary transmutations, transmutations for singular differential operators with variable coefficients, composition method for transmutations, some applications and open problems.