The chebop system for automatic solution of differential equations
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In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun system in object-oriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.
Key wordschebfun chebop spectral method Chebyshev points object-oriented Matlab differential equations
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- 1.M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1965.Google Scholar
- 2.E. Anderson, et al., LAPACK User’s Guide, SIAM, 1999.Google Scholar
- 3.Z. Battles, Numerical Linear Algebra for Continuous Functions, DPhil thesis, Oxford University Computing Laboratory, 2006.Google Scholar
- 5.F. Bornemann, On the numerical evaluation of Fredholm determinants, manuscript, 2008.Google Scholar
- 6.J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn., Dover, 2001.Google Scholar
- 8.M. Dieng, Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations, PhD thesis, University of California, Davis, 2005.Google Scholar
- 9.B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge, 1996.Google Scholar
- 12.R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, 1998.Google Scholar
- 14.B. Muite, A comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems, manuscript, 2007.Google Scholar
- 16.R. Pachón, R. Platte, and L. N. Trefethen, Piecewise smooth chebfuns, IMA J. Numer. Anal., submitted.Google Scholar