Fractional differential equations, compatibility, and exact solutions

In this study, we propose the concept of compatibility between a fractional-order differential equation (FDE) and an integer-order one. For a given FDE, we obtain integer-order quasilinear partial differential equations (PDEs) that are compatible with it by imposing compatibility criteria. The criteria are constructed by considering the FDE and the PDEs. Then, using the general solutions of integer-order equations, we derive the exact solutions to the FDE. To illustrate the novelty and advantage of the method, we consider the time/space-fractional diffusion equations to show that the compatibility with the first-order PDE leads quickly and easily to the solutions obtained by the nonclassical Lie symmetry analysis. More importantly, compatibility with the second-order PDEs gives us new exact solutions to the above-mentioned fractional equations.