Transmutations and Applications: a survey
arXiv:1012.3741
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.
141 pages, 393 references, english contents and russian text
References in corpus (5)
- Fractional Calculus: Integral and Differential Equations of Fractional Order
- Inverse spectral problems for Sturm-Liouville operators with singular potentials
- Inequalities and monotonicity of ratios for generalized hypergeometric function
- Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity
- Series expansions for the third incomplete elliptic integral via partial fraction decompositions
Cited by in corpus (17)
- Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
- Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane
- Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems
- Spectral parameter power series for perturbed Bessel equations
- Construction of transmutation operators and hyperbolic pseudoanalytic functions
- Transmutations and spectral parameter power series in eigenvalue problems
- A short survey of recent results on Buschman-Erdelyi transmutations
- The general form of the Euler--Poisson--Darboux equation and application of transmutation method
- Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems
- Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators
- Complete families of solutions for the Dirac equation: an application of bicomplex pseudoanalytic function theory and transmutation operators
- On Evaluation of Riesz Constants for Systems of Shifted Gaussians
- Analytic approximation of transmutation operators and related systems of functions
- Applications of integral transforms composition method (ITCM) to wave-type singular differential equations and index shift transmutations
- Buschman-Erdelyi transmutations, classification and applications
- Transmutation operators: construction and applications
- On two classes of generalized fractional operators (with short historical survey of fractional calculus)