Chebyshev Spectral-collocation Method And Piecewise Spectral-collocation Method For Volterra Type Equations

Our research covers two major parts, one is the Chebyshev spectral-collocationmethod for a class of weakly singular Volterra integral equations (VIEs) withproportional delay, the other is the piecewise spectral-collocation method for VIEsand Volterra integro-diferential equations (VIDEs), including the case with non-vanishing delay.We investigate Chebyshev spectral-collocation method for a class of singularVIEs with proportional delay. In this method, we choose the Chebyshev Gauss-Lobatto points as the collocation points, approximate the integral term by Gaussquadrature formula. We prove that the numerical errors decay exponentially. Thismethod can be extensively applied to a class of singular VIEs with unsmoothsolution. Chebyshev Gauss-Lobatto points are the ones that can be obtainedeasier than other Jacobi Gauss points. This brings us the convenience to enhancethe degree of the approximate polynomial. We prove the existence and uniquenessof the solution to the objective equation by contraction mapping principle, and thesmoothness by mathematical induction. In order to high accurately approximatethe integral term we divide it into two Jacobi integral terms which can be highaccurately approximated by Gauss quadrature formula.Many methods for the VIEs and VIDEs enhance the accuracy by refninggrids, for example, the fnite element method, fnite diference method, Runge-Kutta method and piecewise polynomial collocation method. The global Legendrespectral-collocation method enhances the accuracy by increasing the degree ofthe approximate polynomial. Combining these methods, we propose a piecewisespectral-collocation method for VIEs and VIDEs. This method can enhance theaccuracy by both refning the grids and increasing the degree of the local approx-imate polynomial. The numerical solution obtained by the proposed method iscontinuous on the global defnition domain. We prove the existence and uniquenessof the corresponding discrete system. We also obtain the numerical approximationto the derivative function of the solution of VIDEs.We extensively apply piecewise spectral-collocation method to VIEs and VIDEs with non-vanishing delay. In this method, we divide the global interval into severalsubinterval according to the primary discontinuous points associated with the non-vanishing delay. This helps the numerical errors converge exponentially withoutthe interference from the primary discontinuous points. Our experiments resultsshow that the proposed method is more efcient than the piecewise polynomial col-location method. The experiments results also show that our method is availablefor non-linear cases including the case that the delay are the functions of solutionto the objective equations.