Computational methods for integral equations
Computational methods for integral equations
ACM Transactions on Mathematical Software (TOMS)
Applied Numerical Mathematics - Auckl numerical ordinary differential equations (ANODE 98 workshop)
A perspective on the numerical treatment of Volterra equations
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
The double-exponential transformation in numerical analysis
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. V: quadrature and orthogonal polynomials
Some Extended Explicit Bel'tyukov Pairs for Volterra Integral Equations of the Second Kind
SIAM Journal on Numerical Analysis
Numerical Solution of Linear Quasistatic Hereditary Viscoelasticity Problems
SIAM Journal on Numerical Analysis
Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations
SIAM Journal on Scientific Computing
Adaptive, Fast, and Oblivious Convolution in Evolution Equations with Memory
SIAM Journal on Scientific Computing
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
Journal of Computational and Applied Mathematics
Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations
Journal of Computational and Applied Mathematics
Hi-index | 7.31 |
Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.