Advertisement

BIT Numerical Mathematics

, Volume 35, Issue 4, pp 488–503 | Cite as

Stable approximation of fractional derivatives of rough functions

  • Dinh Nho Hào
  • H. -J. Reinhardt
  • A. Schneider
Article

Abstract

The process of numerical fractional differentiation is well known to be an ill-posed problem, and it has been discussed by many authors, and a large number of different solution methods has been proposed. However, available approaches require a knowledge of a bound of the second or third derivatives of the function under consideration which are not always available. In this paper the following mollification method is suggested for this problem: if the data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely, by elements of an appropriater-regular multiresolution approximation {V j }j ∈ ℤ ofL2(ℝ). WithinV j the problem of fractional differentiation is well-posed, and we can find a mollification parameterJ depending on the noise level ɛ in the data, such that the error estimation is of Hölder type. The method can be numerically implemented by fundamental results by Beylkin, Coifman and Rokhlin ([2]) on representing differential operators in wavelet bases. It is worthwhile to note that there are very few papers concerning the question of stable numerical fractional differentiation of very rough functions. It is interesting that by our method, in a certain sense, we can approximate the derivatives of very rough functions (functions fromH s (ℝ),s ε ℝ) which have no derivative in the classical sense, like the hat functions, step functions..., in a stable way. Our method is of interest, since it is local. This means that to approximate the fractional derivative of a function at a point with improperly given data, we need only local information about this function in an appropriate neighbourhood of this point.

Key words

Fractional derivative rough function ill-posed problems mollification wavelets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Baumeister,Stable Solution of Inverse Problems, Vieweg & Sohn, Braunschweig/Wiesbaden, 1987.Google Scholar
  2. 2.
    G. Beylkin, R. Coifman and V. Rokhlin,Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 44 (1991), pp. 141–183.Google Scholar
  3. 3.
    I. Daubechies,Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41 (1988), pp. 909–996.Google Scholar
  4. 4.
    I. Daubechies,Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992.Google Scholar
  5. 5.
    Dinh Nho Hào,A mollification method for ill-posed problems, Numer. Math., 68 (1994), pp. 469–506.Google Scholar
  6. 6.
    Dinh Nho Hào, H.-J. Reinhardt and F. Seiffarth,Stable numerical fractional differentiation by mollification, Numer. Funct. Anal. Optimiz., 15 (1994), pp. 635–659.Google Scholar
  7. 7.
    R. Gorenflo and S. Vessella,Abel Integral Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1991Google Scholar
  8. 8.
    G. H. Hardy and J. E. Littlewood,Some properties of fractional integrals, I. Math. Zeit., 27 (1928), pp. 565–606.Google Scholar
  9. 9.
    J. L. Lions and E. Magenes,Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin, Heidelberg, New York, 1972.Google Scholar
  10. 10.
    A. K. Louis,Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989.Google Scholar
  11. 11.
    Y. Meyer,Wavelets and Operators, Cambridge University Press, 1992.Google Scholar
  12. 12.
    V. Perrier,Towards a method for solving partial differential equations using wavelet bases, in Wavelets, J. M. Combes et al., eds., Springer-Verlag, Berlin, pp. 269–283, 1990.Google Scholar
  13. 13.
    Vu Kim Tuan and R. Gorenflo,The Grünwald Letnikov difference operator and regularization of the Weyl fractional differentiation, Zeitschrift für Analysis und ihre Anwendungen (ZAA), 14(1994), pp. 537–545.Google Scholar
  14. 14.
    H. Weyl,Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung, Vierteljahrsschrift der naturforschenden Gesellschaft in Zürich, 62(1917), pp. 296–302.Google Scholar

Copyright information

© BIT Foundation 1995

Authors and Affiliations

  • Dinh Nho Hào
    • 1
  • H. -J. Reinhardt
    • 1
  • A. Schneider
    • 1
  1. 1.Universität-GH-SiegenFB MathematikSiegenGermany

Personalised recommendations