Research And Application Of Difference Methods For Generalized Time Fractional Allen-Cahn Equations

The properties of the material depend on the distribution of the microstructure of the material.As a method at the mesoscopic scale,phase field simulation is widely used in practical problems related to the evolution of material microstructure.In this paper,the application background of phase field equations is firstly introduced,and the development status of integer-order and fractional-order phase field equations and their solutions are reviewed.The physical properties of the energy decrement in the phase field equation are expounded,and the prediction-correction numerical algorithm is introduced in detail.Secondly,a differential approximation scheme of the generalized time fractional Caputo derivative is given,and the convergence order of this scheme is proved to be 2-α order,and the convergence order is verified by numerical examples.Based on the Newton interpolation method,an approximation scheme of the generalized time fractional Caputo derivative is constructed,and its convergence order is proved to be 3-α by means of the integral by parts and the Newton interpolation method.Thirdly,the time derivative term of the one-dimensional generalized time fractional Allen-Cahn equation is discretized by the finite difference method,and the spatial derivative term is discretized by the second-order central difference scheme.The fully discrete scheme of the order of 1;the generalized time fractional-order Allen-Cahn equations under three different boundary conditions are numerically solved with the constructed discrete scheme,and the physical properties of the energy decreasing of the phase field equation under different boundary conditions are verified;numerical results It shows that different boundary conditions have no obvious effect on the phase field evolution results.The same method is used to construct the full discrete scheme of the two-dimensional generalized time fractional Allen-Cahn equation;and the equations in the case of complex initial values and random initial values are numerically solved;the numerical results show that the constructed discrete scheme has different initial values.The energy decrement can be guaranteed under all conditions,and the result of the phase field evolution process is significantly dependent on the initial value.Finally,the integer-order phase field evolution equation of the phase fraction in the "Fe-Cu-Mn-Ni co-precipitation and evolution process" is extended to a generalized fractional-order phase field evolution equation.Numerical simulation of fractional evolution;and the effects of different orders and initial conditions of the generalized Allen-Cahn equation on the evolution of the phase field are analyzed.