Reasearch On High Accuracy Numerical Method For Three Class Of Fractional And Variable Fractional Differential Equations

With the deepening of fractional calculus research,fractional differential has been widely used in various fields such as physics,engineering science,and biology.In the process of rapid development of fractional derivatives,the variable fractional derivative is continuously developed.And the problem can be effectively analyzed according to the flexibility of the variable fractional derivative.Therefore,it is worthwhile to study the numerical solutions of fractional and variable fractional order equations by using the function approximation algorithm.In this paper,the shift Chebyshev polynomials approximation algorithm is used to study the fractional-order nonlinear Sine-Gorden equation,the variable fractional-order variable-coefficient nonlinear differential equation and the variable fractional-order partial differential equations.Firstly,the paper introduces the physical background of the fractional-order nonlinear Sine-Gorden equation.Based on the basic definitions of fractional Caputo-type differential and shifted Chebyshev polynomials,combined with the idea of ?function approximation.And the integer order and fractional differential operator matrices are derived.The form of the algebraic equation is obtained by discrete variables.Then,the numerical solution of the fractional-order nonlinear Sine-Gorden equation is obtained.Secondly,the matrices of nonlinear product differential operator and variable coefficient product differential operator are obtained by the knowledge of variable fractional order theory.And applied to the variable fractional-order variable coefficient nonlinear differential equation.Then,the numerical solution of differential equation can be obtained by choosing suitable collocation points.After that,the numerical solution is corrected base on the error correction theory.Finally,the approximation formulas of binary functions are obtained by using the format of function approximation,which lays a foundation for the study of variable fractional partial differential equations.According to the variation form of the variable fractional differential operator,the variable fractional differential operator for time and space are obtained.Then the numerical solution for the system of partial differential equations of variable fractional order is solved and the error correction is carried out.