ARTICLE | doi:10.20944/preprints202301.0290.v1
Subject: Computer Science And Mathematics, Analysis Keywords: generalized Sonin condition; general fractional integral; general fractional derivative of arbitrary order; regularized general fractional derivative of arbitrary order; 1st level general fractional derivative; 1st fundamental theorem of fractional calculus; 2nd fundamental theorem of fractional calculus
Online: 17 January 2023 (01:34:12 CET)
In this paper, the 1st level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and posses an integrable singularity of power law type at the origin. The 1st level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann-Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st level general fractional derivatives of arbitrary order, some important properties including the 1st and the 2nd fundamental theorems of Fractional Calculus are formulated and proved.
ARTICLE | doi:10.20944/preprints201711.0072.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: multi-dimensional diffusion-wave equation; neutral-fractional diffusion-wave equation; fundamental solution; Mellin-Barnes integral; integral representation; Wright function; generalized Wright function
Online: 12 November 2017 (08:30:34 CET)
In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.