Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in Rn with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a generalized Mittag-Leffler function, as well as a few popular fixed-point theorems. These studies have good applications in general since uniqueness, existence and stability are key and important topics in many fields. Several examples are presented to demonstrate applications of results obtained by computing approximate values of the generalized Mittag-Leffler functions. (ii) We use the inverse operator method and newly established spaces to find analytic solutions to a number of notable partial differential equations, such as a multi-term time-fractional convection problem and a generalized time-fractional diffusion-wave equation in Rn with initial conditions only, which have never been previously considered according to the best of our knowledge. In particular, we deduce the uniform solution to the non-homogeneous wave equation in n dimensions for all n >= 1, which coincides with classical results such as d'Alembert and Kirchoff's formulas but is much easier in the computation of finding solutions without any complicated integrals on balls or spheres.