Solving partial fractional differential equations using the FAFA-transform

In this article, we introduce the generalized Fourier transform (FAFA-transform) and derive an inversion formula and convolution product for this transform. Furthermore, the fundamental solutions of the single-order and distributed-order Cauchy type fractional diffusion equations are given by means of the appropriate FAFA-transform in terms of the Wright functions. Also, applicability of this transform for the explicit solution of the generalized Hilbert type singular integral equation is discussed.