Reproducing Kernel Method For Solving Allen-Cahn Equation

Establishing model equations has become a way of scientific research by using mathematical logic to simplify practical problems.Allen-Cahn equation is a mathematical model established by abstracting practical problems in materials science,biology,environmental science and other disciplines,which can reveal microscopic phenomena intuitively.Hence,it has caused many researchers to grope it.Due to the variability of practical problems,the exact solution of Allen-Cahn equation is generally difficult to obtain.Therefore,finding a suitable numerical method to solve the equation has become a hot spot.In this paper,the reproducing kernel method is adopted to numerically solve the Allen-Cahn equation.Firstly,based on the polynomial space,a reproducing kernel space in the form of Legendre polynomial is defined and the corresponding reproducing kernel function is obtained.Moreover,the first-order semi-implicit scheme is used to process the time variables and the Quasi-Newton method is used to deal with the space variables,for the sake of dispersing the equation in turns.Finally,the equation is transformed into an operator equation,so that we can solve the equation approximately through the reproducing kernel method.At the same time,the proposed semi-discrete scheme is analyzed,and it is verified that the scheme maintains the energy dissipation characteristics of Allen-Cahn equation.Numerical examples show that the proposed scheme is practicable and effective.