Error Estimate Of Legendre Collocation Methods For Volterra Proportional Delay Integro-differential Equations

Delay integro-differential equations are used in natural science and engineering technology,such as mathematical biology,population dynamics,numerical control.Volterra delay integro-differential equation which is usually used to describe certain biological problems and physical phenomena is one type of delay integro-differential equations.This paper mainly studies the Volterra integro-differential equations with proportional delays,the numerical schemes are respectively established by single-step collocation method and multiple-domain collocation method,and the error of the two methods will be given.In the last two chapters,the numerical results of several concrete examples are used to verify the conclusions of the paper.When the minimal delay is small enough,the single-step collocation method is used to solve Volterra integro-differential equation with proportional delays.Firstly,the integral interval of the equation is divided in line with the primary discontinuity points.Then,the numerical scheme of single-step collocation method is established by using the shifted Legendre polynomial as the basis function,and the error precision of the single-step collocation method is derived.Finally,the numerical results of two examples are used to verify the error precision of the theoretical analysis.When the minimal delay is not small enough,the multiple-domain collocation method is used to solve Volterra integro-differential equation with proportional delays.Firstly,the results of the integral interval division of the single-step collocation method are redivided to ensure that the primary discontinuity points are inside the new partition point set.Then,the numerical scheme of multiple-domain collocation method is established,and the error precision of the method is derived.Finally,the numerical results of two examples are used to verify the error precision of the theoretical analysis.Theoretical analysis and numerical results show that single-step collocation method can recieve exponential convergence rate,moreover,at the same order of the shifted Legendre polynomial,the smaller the minimal delay,the greater the rate of convergence.At the same time,multiple-domain collocation method can also obtain the desired exponential convergence rate,but compared with the single-step collocation method,the convergence rate is smaller when the order of the shifted Legendre polynomial is the same.Moreover,the convergence rate of multiple-domain collocation does not increase indefinitely as the order of the shifted Legendre polynomial increases,but gradually tends to be stable.