6 - Integral transforms composition method for transmutations

Abstract

In the present chapter we study applications of the integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive a wide range of transmutation operators by this method. Classical integral transforms are involved in the ITCM as basic blocks; among them are Fourier, sine- and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. The ITCM and transmutations obtained by it are applied to derive connection formulas for solutions of singular differential equations and more simple nonsingular ones. We consider well-known classes of singular differential equations with Bessel operators, such as classical and the generalized Euler–Poisson–Darboux equation and the generalized radiation problem of A. Weinstein. Methods of this chapter are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel type equations, generalized axially symmetric potential theory (GASPT) equations of A. Weinstein, Bessel type generalized wave equations with variable coefficients, B-ultrahyperbolic equations, and others.