Solving Time-space Fractional Partial Differential Equations In Fractional Reproducing Kernel Space

As a generalization of integer-order calculus,fractional calculus has attracted the attention of scholars because it has unique advantages over integer-order calculus because it is more in line with the actual phenomenon.Fractional differential equations have important applications in fluid mechanics,economics,control theory,and other fields.Although fractional differential equations can describe actual phenomena more accurately,analytical solutions to fractional differential equations are difficult to obtain due to the non-local nature of fractional derivatives.Therefore,it is necessary to seek an effective numerical method to solve fractional differential equations.In this paper,the numerical solution of two types of fractional order differential equations is studied based on the fractional-order reproducing kernel method.Firstly,considering the relatively simple fractional ordinary differential equation model,a suitable univariate fractional order reproducing kernel space is constructed according to the characteristics of the equation.The definition of fractional reproducing kernel space and the expression of reproducing kernel are given,and the fractionalorder differential equation is solved by the reproducing kernel method.Then,the convergence of the algorithm is proved and the effectiveness of the fractional order reproducing kernel method is illustrated by example.Secondly,the space-time fractional partial differential equation containing two fractional derivatives is numerically solved.On the basis of the univariate fractional regenerative kernel space,a binary fractional order space is constructed.The reproducing kernel method is used to convert the problem into a system of algebraic equations,and then the numerical solution of fractional partial differential equations is obtained by solving the system of equations.Finally,the convergence of the algorithm is proved and the numerical examples show that the method is good and effective.