Skip to main content
Log in

The role of the Crank-Gupta model in the theory of free and moving boundary problems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A very brief review is given of the striking way in which the Crank-Gupta model has enhanced our understanding of the well-posedness of free and moving boundary problems.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. D. R. Atthey, A finite difference scheme for melting problems, J. Inst. Math. Appl. 13 (1974) 353–366.

    Google Scholar 

  2. A. E. Berger, M. Ciment and J. C. W. Rogers, SIAM J. Numer Anal. 12 (1975) 646–672.

    Article  Google Scholar 

  3. J. Crank,Free and Moving Boundary Problems (Oxford University Press, Oxford, 1984).

    Google Scholar 

  4. J. Crank and R. S. Gupta, A moving boundary problem arising in the diffusion of oxygen in absorbing tissue, J. Inst. Math. Appl. 10 (1972) 19–44.

    Google Scholar 

  5. J. Crank and R. S. Gupta, A method for solving moving boundary problems in heat-flow using cubic splines or polynomials, J. Inst. Math. Appl. 10 (1972) 296–304.

    Google Scholar 

  6. C. M. Elliott and V. Janovsky, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 93–107.

    Google Scholar 

  7. C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, Pitman Res. Notes Math. 59 (1982).

  8. A. Fasano, S. D. Howison, M. Primicerio and J. R. Ockendon, Some remarks on the regularisation of supercooled one-phase Stefan problems in one-dimension, Quart. Appl. Math. 48 (1990) 153–168.

    Google Scholar 

  9. A. Fasano and M. Primicerio, A critical case for the solvability of Stefan-like problems, Math. Methods Appl. Sci. 5 (1983) 84–96.

    Google Scholar 

  10. A. Fasano, M. Primicerio and A. A. Lacey, New results on some classical parabolic free-boundary problems, Quart. Appl. Math. 38 (1981) 439–460.

    Google Scholar 

  11. M. Herrero and J. J.-L. Velazquez, Singularity formation in the one-dimensional supercooled Stefan problem, European J. Appl. Math. 7, to appear.

  12. S. D. Howison and J. R. King, Explicit solutions to six free-boundary problems in fluid flow and diffusion, IMA J. Appl. Math. 42 (1989) 155–175.

    Google Scholar 

  13. J. R. King, Development of singularities in some moving boundary problems, European J. Appl. Math. 6 (1995) 491–508.

    Google Scholar 

  14. J. R. King, private communication.

  15. J. R. King, A. A. Lacey and J. J.-L. Velazquez, Persistence of corners in free-boundaries in Hele-Shaw flow, European J. Appl. Math. 6 (1995) 455–490.

    Google Scholar 

  16. A. A. Lacey, Bounds on solutions of one-phase Stefan problems, European J. Appl. Math. 6 (1995) 509–516.

    Google Scholar 

  17. A. A. Lacey, S. D. Howison and J. R. Ockendon, Irregular morphologies in unstable Hele-Shaw free boundary problems, Quart J. Mech. Appl. Math. 43 (1990) 387–405.

    Google Scholar 

  18. A. A. Lacey and J. R. Ockendon, Ill-posed free boundary problems, Control and Cybernet. 14 (1985) 275–296.

    Google Scholar 

  19. P. Ya. Polubarinova-Kochina,Theory of Groundwater Movement (Princeton University Press, 1962).

  20. S. Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972) 609–618.

    Google Scholar 

  21. L. I. Rubinstein,The Stefan Problem, Transl. Math. Monographs 27 (Amer. Math. Soc., Providence, RI, 1971).

  22. P. G. Saffman and G. I. Taylor, The penetration of fluid into a porous medium or Hele-Shaw cell, Proc. Roy. Soc. London Ser. A 254 (1958) 312–329.

    Google Scholar 

  23. D. G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977) 133–144.

    Google Scholar 

  24. A. J. Schatz, J. Math. Anal. Appl. 28 (1969) 569–580.

    Article  Google Scholar 

  25. B. Sherman, A general one-phase Stefan problem, Quart. Appl. Math. 28 (1970) 377–382.

    Google Scholar 

  26. J. J.-L. Velazquez, Cusp formation for the undercooled Stefan problem in two and three dimensions, European J. Appl. Math. 7, to appear.

  27. P. Wilmott, S. D. Howison and J. N. Dewynne,The Mathematics of Financial Derivatives (Cambridge University Press, Cambridge, 1995).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to Professor J. Crank on the occasion of his 80th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ockendon, J.R. The role of the Crank-Gupta model in the theory of free and moving boundary problems. Adv Comput Math 6, 281–293 (1996). https://doi.org/10.1007/BF02127708

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02127708

Keywords

Navigation