Analytical and semi-analytical solutions for time-fractional Cahn-Allen equation

This paper investigates the analytical and semi-analytical solutions of the time-fractional Cahn-Allen equation, which describes the structure of dynamic for phase separation in Fe-Cr-X (X = Mo, Cu) ternary alloys. We apply a modified auxiliary equation method and the Adomian decomposition method to get distinct solutions to our suggested model. These solutions describe the dynamic of the phase separation in iron alloys and use in solidification and nucleation problems. The applications of this method arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. We apply a conformable fractional derivative to this fractional model to convert it into a nonlinear partial differential equation with integer order. We obtain many analytical wave solutions and also apply a semi-analytical scheme to calculate the absolute value of error. All solutions are verified by using Mathematica software.

Automated summary

This paper investigates the analytical and semi-analytical solutions of the time-fractional Cahn-Allen equation, which describes the dynamic structure of phase separation in Fe-Cr-X ternary alloys. The solutions obtained through modified auxiliary equation method and Adomian decomposition method are applicable in various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The use of conformable fractional derivative converts the fractional model into a nonlinear partial differential equation with integer order, resulting in multiple analytical wave solutions verified using Mathematica software.