On boundary value problem of the nonlinear fractional partial integro-differential equation via inverse operators

This paper is to obtain sufficient conditions for the uniqueness and existence of solutions to a new nonlinear fractional partial integro-differential equation with boundary conditions. Our analysis relies on an equivalent implicit integral equation in series obtained from an inverse operator, the multivariate Mittag-Leffler function, Leray-Schauder’s fixed point theorem as well as Banach’s contractive principle. Several illustrative examples are also presented to show applications of the key results derived. Finally, we consider the generalized fractional wave equation in \({\mathbb {R}}^n\) and deduce the analytic solution for the first time based on the inverse operator method, which leads us a fresh approach to studying some well-known partial differential equations.