Skip to main content
SpringerLink
Log in

Statistical solutions of PDEs by nonstandard densities

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Statistical solutions of partial differential equations are discussed in the framework of nonstandard analysis. They are described by means of a nonstandard differential equation for their suitably defined densities. As a necessary tool we present a hyperfinite version of weak solutions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albeverio, S., Fenstad, J. E., Hoegh-Krohn, R., Lindstrom, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics. London: Academic Press. 1986.

    Google Scholar 

  2. Anderson, R. M.: Star-finite representation of measure spaces. Trans. Amer. Math. Soc.271, 667–687 (1982).

    Google Scholar 

  3. Capiński, M.: A statistical approach to the heat equation. Zeszyty Nauk. Uniw. Jagielloń. Prace Mat.20, 111–133 (1979).

    Google Scholar 

  4. Capiński, M., Cutland, N. J.: A simple example of intrinsic turbulence. To appear in J. Diff. Equations.

  5. Cutland, N. J.: Infinitesimals in action. J. London Math. Soc.35, 202–216 (1987).

    Google Scholar 

  6. Cutland, N. J. (ed.): Nonstandard Analysis and Its Applications. Cambridge: Univ. Press. 1988.

    Google Scholar 

  7. Foia§, C.: Statistical study of Navier-Stokes equations I. Rend. Sem. Math. Univ. Padova48, 219–348 (1973).

    Google Scholar 

  8. Feng Hanqiao, Mary, D. F. St., Wattenberg, F.: Applications of nonstandard analysis to partial differential equations I. The diffusion equation. Mathematical Modelling7, 507–523 (1986).

    Google Scholar 

Download references

Author information

Author notes

  1. M. Capiński

    Present address: Institute of Mathematics, Jagiellonian University, Krakow, Poland

Authors and Affiliations

  1. Department of Pure Mathematics, University of Hull, Hull, England

    M. Capiński & N. J. Cutland

Authors
  1. M. Capiński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. J. Cutland
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research was supported by an SERC Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Capiński, M., Cutland, N.J. Statistical solutions of PDEs by nonstandard densities. Monatshefte für Mathematik 111, 99–117 (1991). https://doi.org/10.1007/BF01332349

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01332349

Keywords

Search

Navigation