Time-fractional diffusion equation with $$\psi $$-Hilfer derivative

In this work, we consider the multidimensional time-fractional diffusion equation with $$\psi $$ -Hilfer derivative. This fractional derivative enables interpolation between Riemann–Liouville and Caputo derivatives its kernel depends on an arbitrary positive monotone increasing function ,$$ thus encompassing several in literature. allows us to obtain general results for different families of problems that depend selected. By employing techniques Fourier, -Laplace, Mellin transforms, a solution representation terms convolutions involving Fox H-functions Cauchy problem associated our equation. Series representations first fundamental are explicitly obtained any dimension as well moments order. For one-dimensional case, show series reduces Wright prove it corresponds probability density admissible . Finally, some plots presented particular choices order differentiation.