High-order methods for the numerical solution of Volterra integro-differential equations
Journal of Computational and Applied Mathematics
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
An efficient spectral method for ordinary differential equations with rational function coefficients
Mathematics of Computation
Spectral methods in MatLab
A note on the numerical solution of high-order differential equations
Journal of Computational and Applied Mathematics
An Extension of MATLAB to Continuous Functions and Operators
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Inexact Inverse Iteration with Variable Shift for Nonsymmetric Generalized Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
GMRES for the Differentiation Operator
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
Optimal Gegenbauer quadrature over arbitrary integration nodes
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Computation of frequency responses for linear time-invariant PDEs on a compact interval
Journal of Computational Physics
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Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization. The same objects can be used in matrix-free iterative methods in linear algebra, in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients. As a further application of the ability to implement operator equations, a method of Greengard [1] for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems. In the integral form, large condition numbers associated with differentiation matrices in high-order problems are avoided. The ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun. The integral method also can be extended to first-order systems, although chebfun syntax does not currently allow easy implementation in this case.