Fractional Reproducing Kernel Method For Fractional Diffusion Equation

Fractional differential equations can simulate natural phenomena more accu-rately than integer-order differential equations in many applied sciences,especially fractional order diffusion equations can more accurately describe some anomalous diffusion phenomena,such as simulation of osmotic structure,turbulence,the mo-tion process of groundwater pollution and chaotic dynamical systems in physics,so fractional diffusion equations are one of the research topics that people are very con-cerned about.In this paper,the fractional reproducing kernel method for solving the fractional diffusion equation is proposed.In the third chapter of this article introduces how to solve the initial value prob-lem of space fractional diffusion equation by combining the implicit Euler method and the fractional reproducing kernel method.For the time direction,the implicit Euler method is used to discretize,and then the original problems are transformed into a semi-discrete problem.Next,construct a new fractional reproducing kernel s-pace(2₂[0,1],in which the equation is discussed,and its reproducing kernel function is taken as part of the base,and finally the collocation method is used to numerically approximate the solution of the semi-discrete problem,and the numerical analysis of the time direction and space direction is comprehensively and systematically.Fi-nally given the corresponding numerical experiments and the experimental results were analyzed to verify the high efficiency of the method.Chapter 4 of this paper combines the1 method,which is a classical Caputo fractional derivative discretization method,and the fractional reproducing kernel collocation method to solve the initial value problem of the fractional space-time diffusion equation.The1 method is used to discretize the time fractional deriva-tive which can obtain a semi-discrete problem semi-discrete problems.This paper continues to discuss this problem in fractional reproducing kernel space(2₂[0,1],and organically combines its reproducing kernel function with polynomial function to generate the basis.The collocation method is used to numerically approximate the solution of the semi-discrete problem,finally the convergence analysis of the semi-discrete scheme is given,and the stability of the semi-discrete scheme is dis-cussed.By numerical example,the algorithm in this paper is compared with other algorithms in terms of CPU time and error,and from the perspective of data and images,this method has obvious advantages over other methods.