On the existence and uniqueness of solutions of fractional order partial integro-differential equations

The main objective of this paper is to investigate and prove the existence and uniqueness theorems of the solution of nonlinear partial integro-differential equations of fractional order. The theorems of two types for the nonlinear fractional order partial integro-differential equations such as one- and two-dimensional are proved by applying the fixed point theorem of Banach space couple with contraction mapping principle, in which the sufficient conditions are presented in order to ensure the existence and uniqueness of a unique fixed point related to the integro-differential equation in operator form.