Research On Product Integral Method Of Generalized Jacobi Function For Fractional Ordinary Differential Equations

Fractional differential equations have been widely used in mathematics,economics,physics and other fields,so it is very meaningful to study the solution of fractional differential equations.However,it is often very difficult to find analytical solutions of fractional differential equations,and many equations do not even have analytical solutions,so it is extremely important to solve numerical solutions of fractional differential equations.In this thesis,the initial value problem of nonlinear fractional ordinary differential equations is studied.According to the generalized Jacobi function,a new product integral method is constructed,which is called the product integral method of generalized Jacobi function.solution.First,using the relevant properties of the generalized Jacobi function,the initial value problem of the nonlinear fractional ordinary differential equation is transformed into its equivalent Volterra integral equation.Using the idea of Fourier expansion,the generalized Jacobi function is selected as the basis function to construct a Generalized Jacobi product integral method.Then,a theoretical analysis is made on the product integration method of the generalized Jacobi function.In the study of the properties of the product-integral method of generalized Jacobi function,convergence is introduced under the condition of two smoothness assumptions,and a general discussion is made.The convergence of the Jacobi function product integration method is considered as the condition of iterative convergence.At the same time,the linear stability of the generalized Jacobi function product integration method is studied.The stability region of the function product integration method verifies the stability of the generalized Jacobi function product integration method.Finally,in this thesis,two numerical examples are selected to verify the properties of the product of the generalized Jacobi function under the two smoothness cases.Numerical experiments prove the convergence of various generalized Jacobi function product integration methods in solving initial value problems of nonlinear fractional ordinary differential equations.