Numerical Methods For Two Classes Of Time Fractional Order Partial Differential Equations

Fractional differential operators are more suitable for describing some of the complex dynamic behavior in real life due to their non-locality.In recent years,the applications of fractional order differential equations pervade many fields of science,but it is not easy to obtain the analytical solution of such kind of equations.Therefore,exploring numerical methods for solving the equation is an important means to qualitatively study such kind of equations,and how to construct a high-precision numerical method need to be further explored.Based on the idea of spline and reproducing kernel theory,this paper studies two classes of time fractional order partial differential equations,i.e.,time variable fractional order Mobile-Immobile advection-dispersion equation and nonlinear time-fractional Schr(?)dinger equation.The main results are listed as follows:Firstly,in this paper,the smoothness of the solution for the time variable fractional order Mobile-Immobile advection-dispersion equation is discussed,and the spline function can be used to approximate the solution of such kind of equations.Using the reproducing kernel function and spline polynomial,the approximate solution in the form of series is presented.At the same time,a simple method for the ε-approximate solution is established.In order to illustrate the validity of our method,the convergence and stability of the algorithm are investigated.Secondly,we study the numerical method for solving a class of nonlinear time-fractional Schr(?)dinger equation.The solution of this kind of equations usually has weak singularity,we apply the fractional integral operator to both sides of the original differential equation and yield the corresponding integral equation.Different from the general fractional differential equation,the solution of this kind of equations belongs to the complex field.In this paper,the solution is divided into real part and imaginary part,then the approximate solution of the two parts with easily computable term is derived in the reproducing kernel space,respectively.At the same time,the existence of any ε-approximate solution is discussed.Numerical results show that the method has good convergence and high precision.