# 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
88 Citations

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## 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

• Mathematics
• 1983
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