Based On Legendre Functions To Research Three Kinds Of Problems Of Fractional Order Calculus

In recent years,the rapid development of fractional order calculus has been widely applied in various fields.The fractional calculus can be used to describe complex physical and engineering problems accurately and concisely.Therefore,the research on this field has great value.In general,to obtain the analytic solution of fractional order differential equation is fairly difficult.Thus,it is necessary to propose a numerical method with high accuracy.In addition,fractional order controllers with better transient response and robustness can be obtained through establish appropriate fractional order models.Therefore,the study of numerical solution and parameter identification of fractional order differential equation is an important research topic.One of the powerful tools of numerical calculation is function approximation.The numerical solution obtained by approximating unknown function has high accuracy with less computation time.In this paper,we study the problems by combining orthogonal Legendre functions with approximation theory.Firstly,the development and research status of fractional order calculus and the research status of numerical calculation are introduced respectively.The research background and the significance of approximation based on Legendre orthogonal function are given.Then the definition and properties of Legendre polynomial are listed.In the following,related basic knowledge of fractional calculus and variable order fractional calculus are also introduced.Secondly,in this paper,fractional order differential operator matrix of the shifted Legendre polynomials and generalized Legendre polynomials are derived.Based on the obtained operator matrix,numerical solutions for variable order fractional functional differential equation and fractional order diffusion equation are presented respectively.The functional terms and boundary conditions are dealt with of function approximation.The integral term is approximated by Gauss-Legendre integral formula.Finally,the operator matrix is applied to rewrite the original equation into an algebraic equation.Numerical simulations illustrate the feasibility and effectiveness of the proposed methods.Finally,based on the differential operator matrix of Legendre wavelet function,coefficients and fractional orders of a class of fractional order systems with noise are identified.Optimization process can be executed to identify unknown parameters by minimizing the error between the output of the real system and the one of the algebraic approximate system.Numerical simulations confirm the efficiency and robustness of the proposed method.