×

The chebop system for automatic solution of differential equations. (English) Zbl 1162.65370

Summary: In Matlab, it would be good to be able to solve a linear differential equation by typing u=L\string\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.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
68W30 Symbolic computation and algebraic computation
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35L15 Initial value problems for second-order hyperbolic equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1965.
[2] E. Anderson, et al., LAPACK User’s Guide, SIAM, 1999. · Zbl 0934.65030
[3] Z. Battles, Numerical Linear Algebra for Continuous Functions, DPhil thesis, Oxford University Computing Laboratory, 2006.
[4] Z. Battles and L. N. Trefethen, An extension of Matlab to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004), pp. 1743–1770. · Zbl 1057.65003 · doi:10.1137/S1064827503430126
[5] F. Bornemann, On the numerical evaluation of Fredholm determinants, manuscript, 2008.
[6] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn., Dover, 2001. · Zbl 0994.65128
[7] E. A. Coutsias, T. Hagstrom, and D. Torres, An efficient spectral method for ordinary differential equations with rational function coefficients, Math. Comput., 65 (1996), pp. 611–635. · Zbl 0846.65037 · doi:10.1090/S0025-5718-96-00704-1
[8] M. Dieng, Distribution Functions for Edge Eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé Representations, PhD thesis, University of California, Davis, 2005. · Zbl 1093.60009
[9] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge, 1996. · Zbl 0844.65084
[10] J. R. Gilbert, C. Moler, and R. Schreiber, Sparse matrices in MATLAB: design and implementation, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 333–356. · Zbl 0752.65037 · doi:10.1137/0613024
[11] L. Greengard, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal., 28 (1991), pp. 1071–1080. · Zbl 0731.65064 · doi:10.1137/0728057
[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. · Zbl 0901.65021
[13] N. Mai-Duy and R. I. Tanner, A spectral collocation method based on integrated Chebyshev polynomials for two-dimensional biharmonic boundary-value problems, J. Comput. Appl. Math., 201 (2007), pp. 30–47. · Zbl 1110.65112 · doi:10.1016/j.cam.2006.01.030
[14] B. Muite, A comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems, manuscript, 2007. · Zbl 1188.65140
[15] S. A. Orszag, Accurate solution of the Orr–Sommerfeld equation, J. Fluid Mech., 50 (1971), pp. 689–703. · Zbl 0237.76027 · doi:10.1017/S0022112071002842
[16] R. Pachón, R. Platte, and L. N. Trefethen, Piecewise smooth chebfuns, IMA J. Numer. Anal., submitted. · Zbl 1202.65016
[17] T. W. Tee and L. N. Trefethen, A rational spectral collocation method with adaptively determined grid points, SIAM J. Sci. Comput., 28 (2006), pp. 1798–1811. · Zbl 1123.65105 · doi:10.1137/050641296
[18] C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159 (1994), pp. 151–174. · Zbl 0789.35152 · doi:10.1007/BF02100489
[19] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, 2000. · Zbl 0953.68643
[20] L. N. Trefethen, Computing numerically with functions instead of numbers, Math. Comput. Sci., 1 (2007), pp. 9–19. · Zbl 1145.41302 · doi:10.1007/s11786-007-0001-y
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.