A note on collocation methods for Volterra integro-differential equations with weakly singular kernels

@article{Tang1993ANO,
  title={A note on collocation methods for Volterra integro-differential equations with weakly singular kernels},
  author={Tao Tang},
  journal={Ima Journal of Numerical Analysis},
  year={1993},
  volume={13},
  pages={93-99}
}
  • T. Tang
  • Published 1993
  • Mathematics
  • Ima Journal of Numerical Analysis
Spline collocation methods can be used to solve Volterra integro-differential equations with weakly singular kernels. In order to obtain optimal convergence behavior, collocation on suitably graded meshes was considered by H. Brunner [1]. This work extends his results to more practical values of the grading exponent 

Tables from this paper

A Spline Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

The piecewise polynomial collocation method is discussed to solve linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids, global

Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.

Spline collocation methods for weakly singular Volterra integro-differential equations

Two piecewise polynomial collocation methods are constructed to solve linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. The attainable orders of

Piecewise Polynomial Approximations for Linear Volterra Integro-Differential Equations with Nonsmooth Kernels

The piecewise polynomial collocation method is discussed to solve linear Volterra-Basset integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids,

Collocation approximations for weakly singular Volterra integro‐differential equations 1

Abstract A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of

Superconvergence of Numerical Solutions to Volterra Integral Equations with Singularities

In this paper we discuss the $\beta$-polynomial discrete collocation method (based on practical meshes) for Volterra integral equations with weakly singular kernels. It will be shown that

An hp-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels

The hp-version error bounds of the collocation method under the $$H^1$$H1-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived.

On the approximate solution of weakly singular integro-differential equations of Volterra type

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1–7]. The present paper
...

References

SHOWING 1-8 OF 8 REFERENCES

Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels

Etude de l'ordre de convergence (globale) des approximations de collocation dans certains espaces de splines-polynomiales appliquees a la solution des equations integrodifferentiales de Volterra avec

On the order of the error in discretization methods for weakly singular second kind volterra integral equations with non-smooth solutions

In general, second kind Volterra integral equations with weakly singular kernels of the form k(t, s)(t-s) -~ posses solutions which have discontinuous derivatives at t = 0. A discrete Gronwall

Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line

We consider Galerkin methods for monotone Abel-Volterra integral equations of the second kind on the half-line. The L2 theory follows from Kolodner's theory of monotone Hammerstein equations. We

Product Integration Methods for the Nonlinear Basset Equation

Product integration methods are derived through backward difference interpolation for the solution of a nonlinear generalisation of the Basset equation; this equation models the motion of a particle