Fractional order solutions to fractional order partial differential equations

In this paper, we formulate a class of fractional order partial differential equations by concentrating on fractional order partial derivatives. We consider an initiative towards the formulation of fractional order Schrödinger equations, heat equation, and wave equation in the realm of fractional calculus. For a given fractional number, we obtain their respective fractional order solutions. In the light of fractional order quantum mechanics, we extend our analysis further for various potential functions for the case of the time-independent and time-dependent Schrödinger equations. Hereby, our proposal gives a refined solution to discrete dynamical systems. Namely, instead of taking the integral-valued steps in the evolution of a chosen dynamical system, it takes fractional-valued steps while performing an integral or solving a system of differential equations. Finally, concerning the extended analysis of our fractional order solutions, we anticipate interesting applications of the above formulation in other domains of physical sciences and natural systems. © 2019, Sociedad Española de Matemática Aplicada.