Advertisement

On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study

  • Nataliya Vasylyeva
  • Lyudmyla Vynnytska
Article

Abstract

We study the anomalous diffusion version of the quasistationary Stefan problem (the fractional quasistationary Stefan problem) in the multidimensional case \({\Omega(t) \subset R^{n},\, n \geq 2}\). This free boundary problem is a mathematical model of a solute drug released from a polymer matrix (\({n = \overline{1,3}}\)). We prove the existence and uniqueness of the classical solution for this moving boundary problem locally in time. A numerical solution is constructed in the two-dimensional case.

Mathematics Subject Classification

Primary 35R35 35C15 Secondary 35B65 35R11 

Keywords

Quasistationary Stefan problem Anomalous diffusion Caputo derivative Coercive estimates 

References

  1. 1.
    Atkinson C.: Moving boundary problems for time fractional and composition dependent diffusion. Frac. Calc. Appl. Anal. 15(2), 207–221 (2012)zbMATHGoogle Scholar
  2. 2.
    Bazaliy B.V.: On a Stefan problem. Dokl. AN USSR Ser. A 11, 3–7 (1986)Google Scholar
  3. 3.
    Bazaliy B.V.: On a proof of the classical solvability of the Hele-Shaw problem with a free boundary. Ukr. Math. J. 50, 1452–1462 (1998)Google Scholar
  4. 4.
    Bazaliy B.V., Friedman A.: The Hele-Shaw problem with surface tension in a half-plane: a model problem. J. Differ. Equ. 216, 387–438 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bazaliy B.V., Vasylyeva N.: The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension. J. Math. Phys. Anal. Geom. 10(1), 3–43 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Di Benedetto E., Friedman A.: The ill-posed Hele-Shaw and Stefan problems for supercoold water. Trans. Am. Math. Soc. 282, 183–203 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bizhanova G., Solonnikov V.: On problems with free boundaries for second-order parabolic equations. Algebra i Analiz 12(6), 98–139 (2000)MathSciNetGoogle Scholar
  8. 8.
    Bouchaud J.-P., Georges A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen S., Merriman B., Osher S., Smereka P.: A simple level set method for solving Stefan problems. J. Comput. Phys. 135(1), 8–29 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diethelm K., Ford N.J., Freed A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1-4), 3–22 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diethelm K., Ford N.J., Freed A.D., Luchko Y.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Method. Appl. M 194(6), 743–773 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Elliott, C., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problem. Pitman, London (1982)Google Scholar
  13. 13.
    Erdélyi, A.: Higher Transcendental Functions, vol. 3. Mc Graw-Hill, New York (1955)Google Scholar
  14. 14.
    Escher J., Simonett G.: Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28, 1028–1047 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gilbarg, D, Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)Google Scholar
  16. 16.
    Hohlov Y.E., Howison S.: The classification of solutions in the free boundary Hele-Shaw problem. Dokl. Acad. Nauk USSR 325, 1161–1166 (1992)Google Scholar
  17. 17.
    Howison, S.D.: Bibliography of free and moving boundary problems in Hele-Shaw and Stokec flow. http://www.maths.ox.ac.uk/howison/Hele-Shaw (2006)
  18. 18.
    Hanzawa E.I.: Classical solution of the Stefan problem. Tohoku Math. J. 33, 297–335 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kilbas A.: Fractional calculus of the generalized Wright functions. Fract. Calc. Appl. Anal. 8, 113–126 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006)Google Scholar
  21. 21.
    Kochubei A.N.: Fractional-parabolic systems. Potential Anal. 37 1, 1–30 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Krasnoschok, M., Vasylyeva, N.: Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition. Int. J. Part. Differ. Equ. 2013 (ID 796430, 20 p) (2013). doi: 10.1155/2013/7964300
  23. 23.
    Krasnoschok M., Vasylyeva N.: On a nonclassical fractional boundary-value problem for the Laplace operator. J. Differ. Equ. 257(6), 1814–1839 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Krasnoschok, M., Vasylyeva, N.: Local solvability of the two-dimensional Hele-Shaw problem with a fractional derivative in time. Math. Trudy. 17(2), 1–30 (2014)Google Scholar
  25. 25.
    Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Parabolic Equations. Academic Press, New York (1968)Google Scholar
  26. 26.
    Ladyzhenskaia, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)Google Scholar
  27. 27.
    Li X., Xu M., Wang S.: Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices. J. Phys. A: Math. Theor. 40, 12131–12141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu B.-T., Hsu J.-P.: Theoretical analysis on diffusional release from ellipsoidal drug delivery devices. Chem. Eng. Sci. 61, 1748–1752 (2006)CrossRefGoogle Scholar
  29. 29.
    Liu J., Xu M: An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices. Z. Angew. Math. Mech. 84(1), 22–28 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu J., Xu M.: Some exact solutions to Stefan problems with fractional differential equations. J. Math. Anal. Appl. 351, 536–542 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Logg, A., Mardal, K.A., Wells, G. (eds.): Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book. 84, Springer, New York (2012)Google Scholar
  32. 32.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications 16. Birkhäuser Verlag, Basel (1995)Google Scholar
  33. 33.
    Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Garpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, New York (1997)Google Scholar
  34. 34.
    Mainardi, F., Tomirotti, M.: On a special function arising in the time fractional diffusion-wave equation. In: Rusev, P., Dimovski, L., Kiryakova, V. (eds.) Transform Methods and Special Functions, pp. 171–183. Science Culture Technology, Sofia (1995)Google Scholar
  35. 35.
    Metzler R., Klafter J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ochoa-Tapia J.A., Valdes-Parada F.J., Alvarez-Ramirez J.: A fractional-order Darcy’s law. J. Phys. A 374, 1–14 (2007)MathSciNetGoogle Scholar
  37. 37.
    Pskhu, A.V.: Partial Differential Equations of the Fractional Order, (in Russian). Nauka, Moscow (2005)Google Scholar
  38. 38.
    Pskhu A.V.: The fundamental solution of a diffusion-wave equation of fractional order (in Russian). Izvestia RAN 73, 141–182 (2009)MathSciNetGoogle Scholar
  39. 39.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Philadelphia (1993)Google Scholar
  40. 40.
    Solonnikov V.A.: Estimates for the solution of the second initial-boundary value problem for the Stokes system in spaces of functions with Hölder-continuous derivatives with respect to the space variables. J. Math. Sci. 109(5), 1997–2017 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Vasylyeva N.: On the solvability of the Hele-Shaw problem in the case of nonsmooth initial data in weighted Hölder classes. Ukr. Math. Bull. 2(3), 323–349 (2005)MathSciNetGoogle Scholar
  42. 42.
    Voller, V.R.: An overview of numerical methods for solving phase change problems. In: Minkowycz, W.J., Sparrow, E.M. (eds.) Advances in Numerical Heat Transfer, vol. 1, pp. 341–375. Taylor & Francis, Washington, DC (1996)Google Scholar
  43. 43.
    Voller V.R.: An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Int. J. Heat Mass Transf. 53, 5622–5625 (2010)CrossRefzbMATHGoogle Scholar
  44. 44.
    Voller V.R., Falcini F., Garra R.: Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects. Phys. Rew. E 87, 042401 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics of NAS of UkraineDonetskUkraine
  2. 2.Simula Research LaboratoryLysakerNorway

Personalised recommendations