Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Abstract

Based on the homogenous balanced principle and subequation method, an improved separation variables function-expansion method is proposed to seek exact solutions of time-fractional nonlinear PDEs. This method is novel and meaningful without using Leibniz rule and chain rule of fractional derivative which have been proved to be incorrect. By using this method, we studied a nonlinear time-fractional PDE with diffusion term. Some general solutions are obtained which contain many arbitrary parameters. Solutions given in related reference are just our especial case. And we also obtained some new type of solutions.

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

“…In [17] and [18], Rui used an invariant subspace and first integral method and a homogenous balance method to solve more fractional dynamical systems. In [19], He and Zhao proposed an expansion method based on the sub-equation and homogenous balance principle and got some interesting exact solutions. For more analysis and applications of the invariant space method, we can read Ma's papers [20] and [21].…”

“…Our given solutions include not only rational functions solutions, but also implicit function solutions in terms of trigonometric function and logarithmic function. Compared with the methods in [16] and [19], our method provides a general variable separation scheme by which more complicated solutions can be found directly without assuming the forms of solutions. For example, for a nonlinear RL action-diffusion equation, we get two solutions with the forms of implicated function in terms of logarithm and arctangent functions (see application 1.1).…”