Time-fractional diffusion equation with ψ-Hilfer derivative N. Vieira ‡ , M.M. Rodrigues ‡ , and M. Ferreira §,‡ ‡ CIDMA - Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro Campus Universit´ ario de Santiago, 3810-193 Aveiro, Portugal. Emails: nloureirovieira@gmail.com; mrodrigues@ua.pt; mferreira@ua.pt § School of Technology and Management Polytechnic of Leiria P-2411-901, Leiria, Portugal. E-mail: milton.ferreira@ipleiria.pt July 6, 2022 Abstract In this work, we consider the multidimensional time-fractional diffusion equation with the ψ-Hilfer deriva- tive. This fractional derivative enables the interpolation between Riemann-Liouville and Caputo fractional derivatives and its kernel depends on an arbitrary positive monotone increasing function ψ thus encompassing several fractional derivatives in the literature. This allows us to obtain general results for different families of problems that depend on the function ψ selected. By employing techniques of Fourier, ψ-Laplace, and Mellin transforms, we obtain a solution representation in terms of convolutions involving Fox H-functions for the Cauchy problem associated with our equation. Series representations of the first fundamental solution are explicitly obtained for any dimension as well as the fractional moments of arbitrary positive order. For the one-dimensional case, we show that the series representation reduces to a Wright function and we prove that it corresponds to a probability density function for any admissible ψ. Finally, some plots of the fundamental solution are presented for particular choices of the function ψ and the order of differentiation. Keywords: Time-fractional diffusion equation; ψ-Hilfer fractional derivative; ψ-Laplace transform; Fun- damental solution; Fractional moments. MSC 2010: 35R11; 26A33; 35A08; 35A22; 35C15. 1 Introduction Since the beginning of fractional calculus, several definitions of fractional integrals and fractional derivatives have been introduced in the literature. The main difference between them lies in their kernel and this makes the number of definitions wide. This diversity allows certain problems to be treated with specific fractional operators. In [16,26] was proposed a fractional integral operator with respect to another function ψ, obtaining a general operator, in the sense that it is enough to choose a function ψ with certain properties to obtain most of the existing fractional integral operators. Attempting to incorporate a large number of definitions of fractional derivatives into one formulation, the concept of the fractional derivative of a function with respect to another function was recently introduced. In 2017, Almeida [2] proposed a new fractional derivative called ψ-Caputo with respect to a function ψ that generalizes a class of fractional derivatives in the Caputo sense. The same idea can be adapted to define the ψ-Riemmann-Liouville fractional derivative. In 2018, Sousa and Oliveira [29] unified The final version is published in Computational and Applied Mathematics, 41-No.6 (2022), Article ID: 230 (26 pp). It as available via the website https://link.springer.com/article/10.1007/s40314-022-01911-5 1