BIT Numerical Mathematics

, Volume 48, Issue 4, pp 701–723 | Cite as

The chebop system for automatic solution of differential equations

  • Tobin A. Driscoll
  • Folkmar Bornemann
  • Lloyd N. Trefethen
Article

Abstract

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 words

chebfun chebop spectral method Chebyshev points object-oriented Matlab differential equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1965.Google Scholar
  2. 2.
    E. Anderson, et al., LAPACK User’s Guide, SIAM, 1999.Google Scholar
  3. 3.
    Z. Battles, Numerical Linear Algebra for Continuous Functions, DPhil thesis, Oxford University Computing Laboratory, 2006.Google Scholar
  4. 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.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    F. Bornemann, On the numerical evaluation of Fredholm determinants, manuscript, 2008.Google Scholar
  6. 6.
    J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn., Dover, 2001.Google Scholar
  7. 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.MATHCrossRefMathSciNetGoogle Scholar
  8. 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. 9.
    B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge, 1996.Google Scholar
  10. 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.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    L. Greengard, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal., 28 (1991), pp. 1071–1080.MATHCrossRefMathSciNetGoogle Scholar
  12. 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
  13. 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.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    B. Muite, A comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems, manuscript, 2007.Google Scholar
  15. 15.
    S. A. Orszag, Accurate solution of the Orr–Sommerfeld equation, J. Fluid Mech., 50 (1971), pp. 689–703.MATHCrossRefGoogle Scholar
  16. 16.
    R. Pachón, R. Platte, and L. N. Trefethen, Piecewise smooth chebfuns, IMA J. Numer. Anal., submitted.Google Scholar
  17. 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.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159 (1994), pp. 151–174.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, 2000.MATHGoogle Scholar
  20. 20.
    L. N. Trefethen, Computing numerically with functions instead of numbers, Math. Comput. Sci., 1 (2007), pp. 9–19.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Tobin A. Driscoll
    • 1
  • Folkmar Bornemann
    • 2
  • Lloyd N. Trefethen
    • 3
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Zentrum Mathematik – M3Technical University of MunichGarching bei MünchenGermany
  3. 3.Computing LaboratoryUniversity of OxfordOxfordUK

Personalised recommendations