Numerical Methods For Three Classes Of Variable Order Fractional Differential-integal Equations

Variable-order fractional calculus is the development of fractional calculus. As the development of fractional calculus, it is only theoretical studies after being presented. In recent years, with the development of science and technology, variable-order fractional calculus is applied to some fields of engineering, because one of the characteristics of variable-order fractional calculus is it can describe the genetic and memory characteristics over a wide frequency range of materials. However, there are short of the literatures about variable-order fractional calculus, which are not enough to solve all the practical problems about variable-order fractional calculus. This article presents a numerical method for three classes of variable-order fractional differential-integral equations. And I hope the study is more or less helpful in other fields referring to variable-order fractional calculus. The paper mainly includes the following aspects:First of all, based on the shift Chebyshev polynomials, solutions of the one dimensional function are approximated, and then the paper gives approximation function about the error estimation and convergence analysis of the shift Chebyshev polynomials. Based on the characteristics of the definition of the variable-order fractional calculus, the operational matrices of variable-order fractional differential operators are derived. With the operational matrices, the one-dimensional linear variable-order fractional differential-integral equations can be transformed into algebraic equation, and put the discrete points into algebraic equation, we get algebraic equations.Secondly, for the one-dimensional nonlinear variable-order fractional calculus problem, we deduce the first-order matrix integral operator and differential operator matrix of shifted Chebyshev polynomials. The one-dimensional nonlinear variable-order fractional differential equations are transformed into algebraic equation based on the integral and differential operational matrices. The original problem by discrete variables will be converted into algebraic equations, thus numerical solution of the original equations obtained. In the end, a comparison between the proposed method and the finite difference method is also proposed through examples to demonstrate the effectiveness and efficiency of the method.Finally, the paper generalizes the method to two-dimensional variable-order fractional differential equations. By comparing the method with finite difference method, it can prove the effectiveness and efficiency of the method.