  In the paper we study applications of integral transforms composition method
(ITCM) for obtaining transmutations via integral transforms. It is possible to
derive wide range of transmutation operators by this method. Classical integral
transforms are involved in the integral transforms composition method (ITCM) as
basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin,
Laplace and some generalized transforms. The ITCM and transmutations obtaining
by it are applied to deriving connection formulas for solutions of singular
differential equations and more simple non-singular ones. We consider
well-known classes of singular differential equations with Bessel operators,
such as classical and generalized Euler-Poisson-Darboux equation and the
generalized radiation problem of A.Weinstein. Methods of this paper are applied
to more general linear partial differential equations with Bessel operators,
such as multivariate Bessel-type equations, GASPT (Generalized Axially
Symmetric Potential Theory) equations of A.Weinstein, Bessel-type generalized
wave equations with variable coefficients,ultra B-hyperbolic equations and
others. So with many results and examples the main conclusion of this paper is
illustrated: the integral transforms composition method (ITCM) of constructing
transmutations is very important and effective tool also for obtaining
connection formulas and explicit representations of solutions to a wide class
of singular differential equations, including ones with Bessel operators.

Classical Analysis and ODEs