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Analytical and semi-analytical solutions for time-fractional Cahn-Allen equation. (English) Zbl 1470.35114

Summary: 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.

MSC:

35C07 Traveling wave solutions
35C08 Soliton solutions
35K57 Reaction-diffusion equations
65J15 Numerical solutions to equations with nonlinear operators
83C15 Exact solutions to problems in general relativity and gravitational theory

Software:

Mathematica
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