Reproducing Kernel Method For Solving The Fractional Partial Differential Equations With The Delay

Today fractonal differential equations and fractional partial differential equations are applied in various fields,so many schalors begin to study problems of delay fractional differential equations,such as the existence and uniqueness of solutions,stability.Generally,it's difficult to give the exact solution.Under the sense of Caputo fractional derivative,we present a new method for solving a kind of the fractional partial differential equation with delay in reproducing kernel space and obtain the numerical solution.In this paper,the development process and research status of the theory of reproducing kernel and fractional calculus are briefly introduced and the basic definitions of fractional calculus and some practical examples are given.Then the basic theory of reproducing kernel space is introduced and the expression of the reproducing kernel function in the reproducing kernel space is obtained.When the fractional partial differential equation with delay is solved,the initial boundary value conditions are transformed via the homogeneous functions,the reproducing kernel spaces are constructed according to the operator equation,and the expression of the reproducing kernel function is obtained.In reproducing kernel space,the original equation is transformed into an equivalent operator equation,and using the reproducing kernel function one can construct a system of functions which is a complete system of reproducing kernel space.The series expression forms of the solution of the equation are given.Finally,numerical algorithm is given and numerical examples show that the method presented in this paper is valid.