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


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.


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  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

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