Skip to main content

Wavelets and differential equations

  • Conference paper
  • First Online:
Applied Parallel Computing Industrial Computation and Optimization (PARA 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1184))

Included in the following conference series:

  • 201 Accesses

Abstract

Wavelet applications to date have been dominated by signal and image processing. While perhaps not immediately appealing as a means of solving differential equations, the growing body of literature in this area indicates that wavelets have a role to play here, too. We give here some of the basic background and an example illustrating how wavelets can be used to solve differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. Amaratunga, J. R. Williams, S. Qian and J. Weiss, Wavelet-Galerkin solutions for one-dimensional partial differential equations, Int. J. Numer. Methods Eng. 37, 2703–2716, 1994.

    Google Scholar 

  2. P. Auscher, Wavelets with boundary conditions on the interval, in Wavelets: A Tutorial in Theory and Applications, C. K. Chui (Ed.), Academic Press, San Diego, CA, 217–236, 1992.

    Google Scholar 

  3. V. A. Barker, Computing connection coefficients, Technical Report 1996-5, Department of Mathematical; Modelling, Technical University of Denmark, Lyngby, 1996.

    Google Scholar 

  4. S. Bertoluzza, G. Naldi and J.-C. Ravel, Wavelet methods for the numerical solution of boundary value problems on the interval, in Wavelets: Theory, Algorithms and Applications, C. K. Chui, L. B. Montefusco and L. Puccio (eds.), Academic Press, San Diego, CA, 425–448, 1994.

    Google Scholar 

  5. G. Beylkin, On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal. 29, 1716–1740, 1992.

    Google Scholar 

  6. P. Charton and V. Perrier, A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equations, Matemática Aplicada e Computacional (submitted).

    Google Scholar 

  7. W. Dahmen, A. Kunoth, K. Urban, A wavelet Galerkin method for the Stokes equations, Computing 56, 259–301, 1996.

    Google Scholar 

  8. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.

    Google Scholar 

  9. M. Dorobantu, Wavelet-based Algorithms for Fast PDE Solvers, Ph.D. Thesis, Technical Report TRITA-NA-9507, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm, 1995.

    Google Scholar 

  10. B. Engquist, S. Osher and S. Zhong, Fast wavelet algorithms for linear evolution equations, SIAM J. Sci. Comput. 15, 755–775, 1994.

    Google Scholar 

  11. R. Glowinski, J. Periaux, M. Ravachol, T.-W. Pan, R. O. Wells, Jr. and X. Zhou, Wavelet methods in computational fluid dynamics, in Algorithmic Trends in Computational Fluid Dynamics, M. Y. Hussaini, A. Kumar and M. D. Salas (eds.), Springer, New York, 259–276, 1993.

    Google Scholar 

  12. L. Jameson, On the Daubechies-based wavelet differentiation matrix, ICASE Report No. 93-95 NASA Langley Research Center Hampton, VA, 1993.

    Google Scholar 

  13. A. Kunoth, On the fast evaluation of integrals of refinable functions, in Wavelets, Images, and Surface Fitting, P.J. Laurent, A. Le Méhauté, L.L. Schumaker (eds.), AKPeters, Boston, 327–334, 1994.

    Google Scholar 

  14. A. Latto, K. L. Resnikoff and E. Tenenbaum, The evaluation of connection coefficients of compactly supported wavelets, in Proceedings, French-USA Workshop on Wavelets and Turbulence, Princeton Univ., June 1991, Y. Maday (Ed.), Springer, Berlin 1992.

    Google Scholar 

  15. P. Monasse and V. Perrier, Orthonormal wavelet bases adapted for partial differential equations with boundary conditions, Preprint 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Jerzy Waśniewski Jack Dongarra Kaj Madsen Dorte Olesen

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Barker, V.A. (1996). Wavelets and differential equations. In: Waśniewski, J., Dongarra, J., Madsen, K., Olesen, D. (eds) Applied Parallel Computing Industrial Computation and Optimization. PARA 1996. Lecture Notes in Computer Science, vol 1184. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62095-8_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-62095-8_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62095-2

  • Online ISBN: 978-3-540-49643-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics