Volterra-Prabhakar function of distributed order and some applications

The paper studies the exact solution of two kinds of generalized Fokker-Planck equa-tions in which the integral kernels are given either by the distributed order function k1(t) = integral 01 t-mu/Gamma(1 - mu)d mu or the distributed order Prabhakar function k2(alpha, gamma; lambda; t) = integral 01 e-gamma alpha,1-mu(lambda; t) d mu, where the Prabhakar function is denoted as e-gamma alpha,1-mu(lambda; t). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non -negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker-Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of k2(alpha, gamma; lambda; t) and generated by it the probability density function p2(x, t).(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).