Transmutations and Applications: a survey

2010/12/16 by С. М. Ситник, Sitnik, Sergei M.
Mathematics · #44A02 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1012.3741

openalex publication_date 2010/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Inverse spectral problems for Sturm Liouville operators with singular potentials
• Differential-difference operators associated to reflection groups

Related

• Toward a general theory of transmutation
• Standard transmutation operators for the one dimensional Schr "odinger\n operator with a locally integrable potential
• The Transmutation Operators and Corresponding Hyperbolic Equations
• On construction of transmutation operators for perturbed Bessel\n equations
• On two classes of generalized fractional operators (with short historical survey of fractional calculus)