Numerical Solution For Four Classes Of Fractional Differential Equations Based On Shifted Chebyshev Polynomials

Nowadays,with the rapid development of science and technology,the mathematical model of integer linear or nonlinear system can not meet the needs of practical problems in the field of physics and engineering science.Fractional derivatives have been applied to the viscoelastic materials and the dynamics of the oscillator and so on,which often estab-lish mathematical models using fractional differential equations.Therefore,combining the definition of Caputo types of fractional derivatives and the thought of operator matrix,using shifted Chebyshev polynomials to approach the unknown function in the studied equation-s,the paper explores the numerical solutions of four classes of fractional order differential equations.The thesis includes the following contents:Firstly,in the paper,the definition of Caputo types of fractional derivatives and shift-ed Chebyshev polynomials are introduced,the numerical solutions of a class of multi-order fractional differential equation by using the shifted Chebyshev polynomials are calculated,the fractional differential operator matrix is derived.The multi-order fractional error differ-ential equation is constructed to obtain the approximation error function,which can make the numerical solution be corrected and the precision of numerical solution be improved.Secondly,in the paper,the numerical problems of fractional linear and nonlinear delay differential equation of pantograph type are discussed,the convergence and related theorems of the shifted Chebyshev polynomials are described and the linear and nonlinear proportional time-delay coefficient matrix based on shifted Chebyshev polynomials are derived.Using the derived matrix,the two equations can be transformed into the form of matrix product.The numerical solutions of two equations by MATLAB programming can be studied.Finally,in the paper,the numerical solutions of a class of fractional delay partial dif-ferential equation of pantograph type are researched,shifted Chebyshev polynomials can be used to approximate the binary functions.And the original equation is expressed as the form of operator matrix product,the distribution points are rationally selected to discrete variables,the coefficient matrix of polynomials is obtained,then the numerical problem of a class of fractional delay partial differential equation of pantograph type can be studied.