Solving Three Kinds Of Nonliear Variable Order Fractional Differential-integal Equations By Shifted Jacobi Polynomials

In recent years, with the technology advances, some mathematics and other scientific fields of physics and engineering model is no longer a fractional order linear or nonlinear systems, but variable order fractional linear or nonlinear systems. The variable order fractional differential has been successfully applied to the study of viscoelastic materials and oscillator dynamics and control problems, so how to solve the variable order fractional differential integral equation is the key to dealing with these systems, which becomes a hot subject. In numerical analysis, the theory of polynomial approximation function is widely used, which is using polynomials to approximate analytic formula or unknown analytic functions. Thus based on the shifted Jacobi polynomial and combined with variable order fractional calculus concepts and operator matrix, the numerical solutions are acquired. It is not only that the computer procedure is relatively simple, but also only a small number o f shifted Jacobi polynomials are needed to obtain a satisfactory result. The paper is organized as follows:First, the paper introduces the historical background and research status of fractional calculus and variable order fractional calculus. The definitions and properties of the fractional differential, integral, variable order fractional differential and Jacobi polynomials are given. Besides, the shifted Jacobi polynomials are required.Secondly, in chapter 3 and 4, Jacobi polynomial approximation function and convergence analysis are given. Based on shift Jacobi polynomials and their basic properties and the caputo definition of variable order fractional differential, we seek the first order differential operational matrix and variable order fractional differential operational matrix of shifted Jacobi polynomials. Then using the MATLAB process of computation, the numerical solutions of the variable order fractional Riccati differential equation and one kind of nonlinear differential equations are obtain ed. Some numerical examples prove that the method is effective.Finally, in chapter 5, we seek the first order integral operator matrix of shift of Jacobi polynomials. Combined with chapter 3 and 4, the numerical solution of variable order fractional nonlinear differential and integral equations is obtained.