Approximate Solution of Fractional Allen-Cahn Equation by the Mittag-Leffler Type Kernels

Abstract

This study presents the approximate analytic solution of the fractional Allen-–Cahn equation involving fractional-order derivatives with the Mittag-Leffler type kernels. The fractional derivative contains three parameters that can adjust the model. We utilize the homotopy analysis method (HAM) to generate the solution of the fractional differential equations. The effect of the fractional parameters on
the solution behaviors is studied, and the approximate analytical solution of the fractional Allen-–Cahn equation has been acquired successfully. Numerical results are given through graphs and tables. Since the exact solution of the obtained differential equation is unknown, we calculate the residual error to demonstrate the algorithm’s efficiency.

Key words and phrases. analytic solution , homotopy analysis method, fractional Allen-Cahn equation, fractional calculus.

2010 Mathematics Subject Classification. 65D15, 26A33, 35R11