General convolutions of integral transforms and their application to ode and pde problems

Abstract

The present research is devoted to some polyconvolutions generated by various integral transforms. For example, we study convolutions of the Hankel transform with the following factorization properties: Hv[h](x) = Hmu[f](x)Hmu+v[g](x), Hv+ m[h](x) = x-vHv+m[f](x)H v[g](x), H[h](x) = x-vHmu[f](x)H mu[g](x), Hmu[h](x) = x-vHv[f](x) Hmu[g](x), where Hv[f](x) is the Hankel transform. Conditions for the existence of the constructed polyconvolutions are found. The results of this research are applied for solvability of ODEs and PDEs by the method of integral transforms. The derived constructions allow us to solve various nonuniform equations. © 2006 Technika ISSN 1392-6292.