Study On Numerical Solutions Of Three Kinds Of Differential Equations With Bernstein And Fractional Bernstein Polynomials

Based on the outstanding contribution of functional approach theory in numerical computation and the rapid development of differential equations in practical applications,the approach of selecting functional approximation theory to study the numerical solution of various differential equations has become popular.However,there is a high order and a fractional order in high-order differential equations and fractional-order differential equations,which make it more difficult to solve such equations than traditional low-order integer-order differential equations.So solving higher order and fractional differentials The numerical solution of the equation becomes the main research content of this paper.The paper takes Bernstein polynomials and fractional Bernstein polynomials as the basis functions,and discusses the application of function approximation theory in higher order differential equations and fractional differential equations.Firstly,in chapter 1 introduces the development history and research status of the function approximation theory and the Bernstein polynomial,as well as the researching significance of the error correction.Secondly,in chapter 2,based on the recursion formula and ascending formula in the definitions and properties of the Riemann derivatives of Bernstein polynomials,two types of higher order differential operator matrices are given respectively:LM and DS.Combined with the theory of function approximation,the higher order differential equations are transformed into the form of algebraic equations which can be easily solved.The application of Bernstein polynomials in the numerical solution of high-order differential equations is discussed.Two numerical examples are given.The computational results prove the effectiveness of the algorithm.Finally,the application of the function approximation theory in fractional order ordinary differential equations and partial differential equations is discussed in chapter 3 and 4.In chapter 3,we specially give the form of the fractional Bernstein polynomial differential operator matrices and the approach of nonlinear terms,and do a certain theoretical analysis on the error correction.The three examples of fractional linear and non-linear fully illustrate the applicability of the algorithm.In chapter 4,the idea of ?function approximation of binary functions and the coping method of fractional operator matrices in time and space are given emphatically.In two cases,we apply the algorithm in this chapter to calculate the results before and after correction,the contrast strongly illustrates the feasibility of the method and the advantages of correction.