Numerical Methods For Three Classes Of Delay Differential-integal Equations

Fractional order calculus equation and delay calculus equation are widely used in natural science and social science,etc.The researchers put the delay phenomenon join the fractional order calculus equation and get the fractional order delay differential equations.The related problems of such equations had attracted more and more scholars.This article mainly studies the following three classes of delay calculus equations: the nonlinear variable order fractional calculus equation with constant delay,the proportion of fractional order delay differential equation and the linear fractional order delay partial differential equation.The thesis uses the shifted Bernstein polynomial approximation unknown function to solve the problems of numerical solution,the main steps are combined with operator matrix and the derivation process of the numerical algorithm is given.Then through the collocation method to discrete variables,the matrix multiplication is expressed as the form of the linear or nonlinear algebraic equations form and using computer to program to realize the numerical solution.Then we give numerical solution and exact solution of the data and graph at several points in time.Firstly,Bernstein polynomial approximation function and equation of solvability for proof are given.Based on shifted Bernstein polynomials and their basic properties we seek the differential operator matrix and delay of the first order differential operator matrix.Using Matlab software combined with the least square method,we will get coefficient matrix of the polynomial,then we can obtain numerical solution of the original equation.Numerical results show that the proposed method is feasible.Secondly,we solve the proportion of fractional order delay differential equation defined on the extended interval.The proportion of delays calculus operator matrix is deduced,the derivation process of numerical algorithm and convergence analysis are given.Using a lower order shifted Bernstein polynomial solving numerical example,and give the absolute error under different constraints.Finally,the thesis generalizes the method to two-dimensional fractional order delay partial differential equation.The numerical example verifies the feasibility and efficiency of this method by comparing the method with finite difference method.