A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation

This paper is concerned to present and apply a new generalized fractional derivative, that is the Generalized Hilfer-type (GH) fractional derivative. This derivative unifies various previously defined fractional derivatives of the types Hilfer-Katugampola, Hilfer-Hadamard, Caputo-Hadamard, Hadamard, Hilfer, Riemann-Liouville, Caputo etc into a single form. The mellin transform of this new GH fractional derivative is obtained. As an application, a generalized space-time fractional diffusion equation is constructed using the newly obtained GH fractional derivative in the time-variable and the already existing Riesz fractional derivative in the space-variable. For the solution of this new generalized fractional diffusion equation, the similarity transformations and the mellin transform methods are used where an explicit solution in terms of the Fox’s H-function is derived. The diffusive behavior and the role of various parameters involved in the newly obtained GH fractional derivative are described with the help of 3D plots.