Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.