Advertisement

Acta Mechanica

, Volume 230, Issue 3, pp 839–850 | Cite as

On Stefan-type moving boundary problems with heterogeneity: canonical reduction via conjugation of reciprocal transformations

  • Colin RogersEmail author
Original Paper
  • 40 Downloads

Abstract

Here, two distinct kinds of reciprocal transformation are employed in conjunction to reduce a wide class of nonlinear moving boundary problems with heterogeneity to analytically solvable canonical forms. These are associated, in turn, with a classical Stefan problem and with one with variable latent heat relevant to the analysis of the evolution of seepage fronts in soil mechanics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Rubenstein, L.I.: The Stefan Problem, American Mathematical Society Translations, vol. 27. American Mathematical Society, Providence (1971)Google Scholar
  2. 2.
    Friedman, A.: Variational Principles and Free Boundary Problems. Wiley, New York (1982)zbMATHGoogle Scholar
  3. 3.
    Elliot, C.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, vol. 59. Pitman, New York (1982)Google Scholar
  4. 4.
    Crank, J.: Free and Moving Boundary Value Problems. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar
  5. 5.
    Alexides, V., Solomon, A.D.: Mathematical Modelling of Melting and Freezing Processes. Taylor and Francis, Washington (1996)Google Scholar
  6. 6.
    Tarzia, D.A.: A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and Related Problems. MAT Ser A 2, 1–297 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Storm, M.L.: Heat conduction in simple metals. J. Appl. Phys. 22, 940–951 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rogers, C.: Application of a reciprocal transformation to a two-phase Stefan problem. J. Phys. A Math. Gen. 18, L105–L109 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rogers, C.: On a class of moving boundary problems in nonlinear heat conduction. Application of a Bäcklund transformation. Int. J. Nonlinear Mech. 21, 249–256 (1986)CrossRefzbMATHGoogle Scholar
  10. 10.
    Natale, M.F., Tarzia, D.A.: Explicit solutions to the two-phase Stefan problem for Storm-type materials. J. Phys. A Math. Gen. 33, 395–404 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Briozzo, A.C., Natale, M.F.: Nonlinear Stefan problem with convective boundary condition in Storm’s materials. Zeit ang. Math. Phys. 67, 19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rogers, C.: On a class of reciprocal Stefan moving boundary problems. Zeit. ang. Math. Phys. 66, 2069–2079 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Calogero, F., De Lillo, S.: The Burgers equation on the semi-infinite and finite intervals. Nonlinearity 2, 37–43 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rogers, C.: Moving boundary problems for the Harry Dym equation and its reciprocal associates. Zeit. ang. Math. Phys. 66, 3205–3220 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rogers, C.: Moving boundary problems for an extended Dym equation. Reciprocal connections. Meccanica 52, 3531–3540 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rogers, C.: On a class of moving boundary problems for the potential mkdV equation. Special issue, waves and stability. Ricerche di Matematica 65, 563–577 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rogers, C., Wong, P.: On reciprocal Bäcklund transformations of inverse scattering schemes. Physica Scr. 30, 10–14 (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Oevel, W., Rogers, C.: Gauge transformations and reciprocal links in 2+1-dimensions. Rev. Math. Phys. 5, 299–330 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hone, A.N.W.: Reciprocal transformations. Painlevé property and solutions of energy-dependent Schrödinger hierarchies. Phys. Lett. A 249, 46–54 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  21. 21.
    Rogers, C., Schief, W.K.: Ermakov-type systems in nonlinear physics and continuum mechanics. In: Euler, N. (ed.) Nonlinear Systems and Their Remarkable Mathematical Structures. CRC Press, Cambridge (2018)Google Scholar
  22. 22.
    Voller, V.R., Swenson, J.B., Paola, C.: An analytical solution for a Stefan problem with variable latent heat. Int. J. Heat Mass Transf. 47, 5387–5390 (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Salva, N.N., Tarzia, D.A.: Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions. J. Math. Anal. Appl. 379, 240–244 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Broadbridge, P.: Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media. J. Math. Phys. 29, 622–626 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rogers, C., Stallybrass, M.P., Clements, D.L.: On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation. Nonlinear Anal. Theory Methods Appl. 7, 785–799 (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Keller, J.B.: Melting and freezing at constant speed. Phys. Fluids 29, 2013 (1986)CrossRefGoogle Scholar
  27. 27.
    Karal, F.C., Keller, J.B.: Elastic wave propagation in homogeneous and inhomogeneous media. J. Acoust. Soc. Am. 31, 694–705 (1959)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Barclay, D.W., Moodie, T.B., Rogers, C.: Cylindrical impact waves in inhomogeneous Maxwellian visco-elastic media. Acta Mech. 29, 93–117 (1978)CrossRefzbMATHGoogle Scholar
  29. 29.
    Clements, D.L., Atkinson, C., Rogers, C.: Antiplane crack problems for an inhomogeneous elastic material. Acta Mech. 29, 199–211 (1978)CrossRefzbMATHGoogle Scholar
  30. 30.
    Rogers, C.: Reciprocal relations in non-steady one-dimensional gasdynamics. Zeit. ang. Math. Phys. 19, 58–63 (1968)CrossRefzbMATHGoogle Scholar
  31. 31.
    Rogers, C.: Invariant transformations in non-steady gasdynamics and magnetogasdynamics. Zeit. ang. Math. Phys. 20, 370–382 (1969)CrossRefzbMATHGoogle Scholar
  32. 32.
    Donato, A., Ramgulam, U., Rogers, C.: The 3+1-dimensional Monge-Ampère equation in discontinuity wave theory: application of a reciprocal transformation. Meccanica 27, 257–262 (1992)CrossRefzbMATHGoogle Scholar
  33. 33.
    Rogers, C., Ruggeri, T.: A reciprocal Bäcklund transformation: application to a nonlinear hyperbolic model in heat conduction. Lett. Il. Nuovo Cimento 44, 298–296 (1985)Google Scholar
  34. 34.
    Fokas, A.S., Rogers, C., Schief, W.K.: Evolution of methocrylate distribution during wood saturation. A nonlinear moving boundary problem. Appl. Math. Lett. 18, 321–328 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rogers, C., Schief, W.K.: The classical Korteweg capillarity system: geometry and invariant transformations. J. Phys. A Math. Theor. 47, 345201 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rogers, C., Malomed, B.: On Madelung systems in nonlinear optics: a reciprocal invariance. J. Math. Phys. 59, 051506 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Bollati, J., Tarzia, D.A.: Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face. Zeit. ang. Math. Phys. 69, 38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rogers, C., Broadbridge, P.: On a nonlinear moving boundary problem with heterogeneity: application of a reciprocal transformation. Zeit. ang. Math. Phys. 39, 122–128 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

Personalised recommendations