Applications of Integrable Nonlinear Diffusion Equations in Industrial Modelling

Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)


There are useful integrable nonlinear diffusion equations that can be transformed directly to linear partial differential equations. The possibility of linearisation allows us to incorporate a much broader class of boundary conditions than would be available under reduction by a one-parameter Lie symmetry. By this means we can solve nonlinear boundary value problems of practical significance. Examples are given in the solidification of multi-phase materials with nonlinear thermal transport coefficients, infiltration of water in unsaturated soil and evolution of a metal surface by fourth-order curvature-driven diffusion. From this approach, there arise some open mathematical problems.


Nonlinear diffusion Stefan problems Steel solidification Unsaturated flow Infiltration Surface diffusion Grain boundaries Integrable models 


  1. 1.
    Bluman, G.W., Kumei, S., Reid, G.J.: New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Broadbridge, P.: Nonintegrability of nonlinear diffusion-convection equations in two spatial dimensions. J. Phys. A Math. Gen. 19, 1245–1257 (1986)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Broadbridge, P., Goard, J.M.: Grain boundary evolution with time dependent material properties. J. Eng. Math. 66(1–3), 87–102 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Broadbridge, P., Rogers, C.: On a nonlinear reaction-diffusion boundary-value problem: application of a Lie-BŁcklund symmetry. J. Austral. Math. Soc. (b) 34, 318–332 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Broadbridge, P., Vassiliou, P.J.: The role of symmetry and separation in surface evolution and curve shortening. Symmetry, integrability and geometry: methods and applications (SIGMA), 7, 052, 19 pages (2011).
  6. 6.
    Broadbridge, P., White, I.: Constant rate rainfall infiltration: a versatile non-linear model. 1. Analytic solution. Water Resour. Res. 24, 145–154 (1988)CrossRefGoogle Scholar
  7. 7.
    Cannon, J.R., Douglas, J., Hill, C.D.: A multi-boundary Stefan problem and the disappearance of phases. J. Math. Mech. 17, 21–33 (1967)MATHMathSciNetGoogle Scholar
  8. 8.
    Clarkson, P.A., Fokas, A.S., Ablowitz, M.J.: Hodograph transformations on linearizable partial differential equations. SIAM J. Appl. Math. 49, 1188–1209 (1989)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Edwards, M.P.: Classical symmetry reductions of nonlinear diffusion convection equations. Phys. Lett. A. 190, 149–154 (1994)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Fujita, H.: The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part I. Textile Res. J. 22(11), 757–760 (1952)CrossRefGoogle Scholar
  11. 11.
    Fujita, H.: The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part II. Textile Res. J. 22(12), 823–827 (1952)CrossRefGoogle Scholar
  12. 12.
    Fujita, H.: The exact pattern of a concentration-dependent diffusion in a semi-infinite medium. Part III. Textile Res. J. 24, 234–240 (1954)CrossRefGoogle Scholar
  13. 13.
    Fulford, G.R., Broadbridge, P.: Industrial Mathematics: Case Studies in the Diffusion of Heat and Matter. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  14. 14.
    Ibragimov, N.H.: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1. CRC Press, Boca Raton (1993)Google Scholar
  15. 15.
    Kenmochi, N.: A new proof of the uniqueness of solutions to two-phase Stefan problems for nonlinear parabolic equations. In: Hoffmann, K.-H., Sprekels, J. (eds.) Free Boundary Value Problems, pp. 101–126. Birkhauser, Basel (1990)CrossRefGoogle Scholar
  16. 16.
    Kirchhoff. :Theorie der Wärme. Leipzig (1891)Google Scholar
  17. 17.
    Mikhailov, A.V., Shabat, A.B., Sokolov, V.V.: The symmetry approach to classification of integrable equations. In: Zakharov, V.E. (ed.) What is Integrability ?, pp. 115–184. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  18. 18.
    Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957)CrossRefGoogle Scholar
  19. 19.
    Philip, J.R.: Theory of infiltration. Adv. Hydrosci. 5, 215–296 (1969)CrossRefGoogle Scholar
  20. 20.
    Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. Chapman-Hall/CRC Press, Boca Raton (2003)MATHGoogle Scholar
  21. 21.
    Sander, G.C., Parlange, J.-Y., Kuhnel, V., Hogarth, W.L., Lockington, D., O’Kane, J.P.J.: Exact nonlinear solution for constant flux infiltration. J. Hydrol. 97(34), 341–346 (1988)CrossRefGoogle Scholar
  22. 22.
    Storm, M.L.: Heat conduction in simple metals. J. Appl. Phys. 22, 940–951 (1951)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Touloukian, Y.S. (ed.): Thermophysical Properties of Matter, vol. 1. Plenum, New York (1970)Google Scholar
  24. 24.
    Triadis, D., Broadbridge, P.: Analytical model of infiltration under constant-concentration boundary conditions. Water Resour. Res. 46(3), W03526 (2010). doi: 10.1029/2009WR008181 CrossRefGoogle Scholar
  25. 25.
    Tritscher, P., Broadbridge, P.: A similarity solution of a multiphase Stefan problem incorporating general nonlinear heat conduction. Int. J. Heat Mass Transfer 37, 2113–2121 (1994)CrossRefMATHGoogle Scholar
  26. 26.
    Tritscher, P., Broadbridge, P.: Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove. Proc. Roy. Soc. A 450, 569–587 (1995)CrossRefMATHGoogle Scholar
  27. 27.
    White, I., Broadbridge, P.: Constant rate rainfall infiltration: a versatile non-linear model. 2. Applications of solutions. Water Resour. Res. 24, 155–162 (1988)CrossRefGoogle Scholar

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© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

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