Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics

Elina Shishkina works in University of Voronezh State University, Voronezh, Russia and Information Technology and Management in Rzeszow, Rzeszow, Poland. Her research focuses on singular differential equations, fractional powers of differential and integral operators, embedding theorems and integral transforms. She earned a specialist in mathematics in 2004 at Voronezh State University, Voronezh, Russia and PhD in mathematics in 2006 from the same university. She started scientific work in 2004 under the guidance of Professor L. N. Lyakhov. In 2019 The International Society for Analysis, its Applications and Computation (ISAAC) presents to E. Shishkina the life membership for achievements in Integral Transforms, Operator Theory, PDE. She is an author of more than 50 articles and 2 monographs. Sergei Sitnik is a Professor at the Chair of Applied Mathematics and Computer Modeling, Institute of Engineering and Digital Technologies, at Belgorod State National Research University (“BSU”), Belgorod, Russia. ?? started scientific work on transmutations and PDE theory in 1980s in Voronezh University under the guidance of Professors I. A. Kipriyanov and V. V. Katrakhov. In his research he combines theoretical and applied problems. His fields of interest are: transmutation operators, integral transforms and special functions, singular and fractional order differential equations, numerical methods and mathematical modeling. S. M. Sitnik is an author of approximately 350 scientific papers and 2 monographs.

IntroductionAcknowledgement and thanks1 Basic definitions and propositions2 Basics of fractional calculus and fractional order differentialequations3 Essential of the transmutations4 Weighted generalized functions generated by quadratic form5 Buschman-Erdelyi integral and transmutation operators6 Integral transforms compositions method for transmutations7 Differential equations with Bessel operator8 Applications of transmutations to the different problems9 Fractional powers of Bessel operators10 B-potentials theory11 Fractional differential equations with singular coefficientsConclusionReferences