Cauchy Type Problems

Abstract

We now analyze Cauchy type problems of differential equations of fractional order with Hilfer and Hilfer-Prabhakar derivative operators. The existence and uniqueness theorems for n-term nonlinear fractional differential equations with Hilfer fractional derivatives of arbitrary orders and types will be proved. Cauchy type problems for integro-differential equations of Volterra type with generalized Mittag-Leffler function in the kernel will be considered as well. Using the operational method of Mikusinski, the solution of a Cauchy type problem for a linear n-term fractional differential equations with Hilfer fractional derivatives will be obtained. We will show utility of operational method to solve Cauchy type problems of a wide class of integro-differential equations with variable coefficients, involving Prabhakar integral operator and Laguerre derivatives. For this purpose, following some recent works, we choose the examples which, by means of fractional derivatives, generalize the well-known ordinary differential equations and partial differential equations, related to time fractional heat equations, free electronic laser equation, some evolution and boundary value problems, and finally some Cauchy type problems for the generalized fractional Poisson process.