The Hp-version Jacobi Spectral Collocation Method For Two Kinds Of Volterra Integral Equations

Numerical simulation of Volterra integral equation is of great significance in physics,engineering and other fields.This type of equation is models of evolutionary problems with memory and requires long-term simulation,which brings difficulties to the solution of Volterra integral equations.How to design an efficient numerical algorithm to solve the Volterra integral equation is a hot issue in current research,and it is also a difficult problem.The first part of this work proposes the hp-version Jacobi spectral collocation method for the second type of nonlinear Volterra integral equations.First,we introduce Jacobi polynomial and the shifted Jacobi polynomial,and derive the spectral collocation scheme of the Volterra integral equation.Second,we select the shifted Jacobi polynomial as the basis function on each subinterval,and establish the hp-version error on the entire interval under reasonable nonlinear assumptions.Finally,we perform numerical simulations on linear/nonlinear Volterra integral equations,and the results reveal the high efficiency of the hp-version Jacobi spectral collocation method,which is consistent with the theoretical analysis.The second part of the work proposes the hp-version Jacobi spectral collocation method for the nonlinear Volterra integral equation with vanishing variable delays.First,the interval is divided into coarse/fine grid,and the shifted Jacobi polynomial is introduced.Secondly,the hp-version spectral collocation scheme of the nonlinear Volterra integral equation with vanishing variable delays is designed,and the optimal estimation over the entire interval is established under reasonable nonlinear assumptions.Finally,we perform numerical simulations on linear models,nonlinear models,oscillating models,and discontinuous models.The results show that our method is efficient.