Study On Finite Point Method For Fractional Convection-diffusion Equation.

Fractional differential equation is an extension of classical differential equation model based on fractional differential.In contrast,the former can better simulate the changing law of natural physical phenomena.Fr actional convection-diffusion equation(FCDE)is included in fractional dynamic equation,which can contain the fractional derivative of time.However,the mathematical theory of the equation is still im mature,especially the numerical solution which needs to further research.This paper mainly studies the numerical solu,tion of time fractional convection-diffusion equation(TFCDE)based on Caputo derivative.The numerical oscillation p henomenon of ten occurs when convection dominates.For the instability of such numerical values,this paper proposes a meshless finite point method(FPM),which can effectively prevent numerical oscillations caused by a small diffusion coefficient.The research work of this paper mainly has the following points:First of all,the research background and the resear ch progress at home and abroad of fractional convection-diffusion equ.ations are given.Next,preliminary knowledge of fractional calculus is introduced,including the definition and properties of fractional integral and derivative.Then the research progress and the related theory of the meshless finite point method are given.Secondly,the finite point algorirthm formats are established for the linear one-dimenional and two-dimensional time fractional convection diffusion equations,respectively.That is,applying a stability term to the equation,construct approximation functions,and discretize the time and space variables respeclively.And then stability of the discrete format is analyzed and demonstrated.Finally,numerical examples are given to compare the finite point method with the finite difference method(FDM),and the errors of the two methods are given at different points and different times.The results show that the finite point method has higher calculation accuracy and can eliminate oscillations.Finally,the finite point method is applied to the solution of one-dimensional and two-dimensional time fractional convection-diffusion equations with nonlinear diffusion term and source term.The time direction of the equation is approximated by L1 interpolation.Then the time semi-discrete format is obtained,and the stability of the format is proved.By applying the stability term,constructing the approximation function,and using the collocation method for the spatial direction,the finite point full discrete format is obtained.At last,the numerical simulation is given.The errors of the finite point method and the finite difference method at different points and different times are compared.The finite point method has higher numerical precision and can avoid numerical oscillations.The feasibility and validity of the proposed method are verified for solving the time fractional convection-diffusion equation.