Acta Mechanica

, Volume 229, Issue 10, pp 4215–4228 | Cite as

Nonlinear diffusion in arterial tissues: a free boundary problem

  • Diletta BuriniEmail author
  • Silvana De Lillo
  • Gioia Fioriti
Original Paper


A free boundary problem on a finite interval is formulated and solved for a nonlinear diffusion–convection equation. The model is suitable to describe drug diffusion in arterial tissues after the drug is released by an arterial stent. The problem is reduced to a system of nonlinear integral equations, admitting a unique solution for small time. The existence of an exact solution corresponding to a moving front is also shown, which is in agreement with numerical results existing in the literature.


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  1. 1.
    McGinty, S.: A decade of modelling drug release from arterial stents. Math. Biosci. 257, 80–90 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hossainy, S., Prabhu, S.: A mathematical model for predicting drug release from a biodurable drug-eluting stent coating. J. Biomed. Mater. Res. A 87, 487–493 (2008)CrossRefGoogle Scholar
  3. 3.
    McGinty, S., Pontrelli, G.: A general model of coupled drug release and tissue absorption for drug delivery devices. J. Control. Release 217, 327–336 (2015)CrossRefGoogle Scholar
  4. 4.
    Pontrelli, G., de Monte, F.: Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transf. 50, 3658–3669 (2007)CrossRefGoogle Scholar
  5. 5.
    Hwang, C.W., Wu, D., Edelman, E.R.: Physiological transport forces govern drug distribution for stent-based delivery. Circulation 104, 600–605 (2001)CrossRefGoogle Scholar
  6. 6.
    Friedman, A.: Variational Principles and Free-Boundary Problems. Wiley, New York (1982)zbMATHGoogle Scholar
  7. 7.
    Elliot, C.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems, vol. 59. Pitman Publishing, New York (1982)Google Scholar
  8. 8.
    Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar
  9. 9.
    Rosen, G.: Method for the exact solution of a nonlinear diffusion–convection equation. Phys. Rev. Lett. 49, 1844 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fokas, A.S., Yortsos, Y.C.: On the exactly solvable equation \(S_{t}=\left[\left(\beta S+\gamma \right)^{-2}S_{x}\right]_{x}+\alpha \left(\beta S+\gamma \right)^{-2}S_{x}\) occurring in two-phase flow in porous media. SIAM J. Appl. Math. 42, 318 (1982)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fokas, A.S., Pelloni, B.: Generalized Dirichlet to Neumann map for moving initial-boundary value problems. J. Math. Phys. 48, 013502 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    De Lillo, S., Fokas, A.S.: The Dirichlet-to-Neumann map for the heat equation on a moving boundary. Inverse Probl. 23, 1699–1710 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fokas, A.S., De Lillo, S.: The unified transform for linear, linearizable and integrable nonlinear partial differential equations. Phys. Scr. 89, 1–10 (2014)CrossRefGoogle Scholar
  14. 14.
    Rogers, C.: Application of a reciprocal transformation to a two-phase Stefan problem. J. Phys. A Math. Gen. 18, L 105 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rogers, C.: On a class of moving boundary problems in non-linear heat conduction: application of a Bäcklund transformation. Int. J. Nonlinear Mech. 21, 249–256 (1986)CrossRefGoogle Scholar
  16. 16.
    Friedman, A.: Free boundary problems in science and technology. Not. AMS 47, 854–861 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ablowitz, M.J., De Lillo, S.: On a Burgers–Stefan problem. Nonlinearity 13, 471–478 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kingston, J.G., Rogers, C.: Reciprocal Bäcklund transformations of conservation laws. Phys. Lett. A 92, 261–264 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fokas, A.S., Rogers, C., Schief, W.K.: Evolution of methacrylate distribution during wood saturation. Appl. Math. Lett. 18, 321–328 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    De Lillo, S., Salvatori, M.C., Sanchini, G.: On a free boundary problem in a nonlinear diffusive–convective system. Phys. Lett. A 310, 25–29 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    De Lillo, S., Lupo, G.: A two-phase free boundary problem for a nonlinear diffusion–convection equation. J. Phys. A Math. Theor. 41, 145207 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Burini, D., De Lillo, S.: An inverse problem for a nonlinear diffusion–convection equation. Acta Appl. Math. 122, 69–74 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Calogero, F., De Lillo, S.: The Burgers equation on the semi-infinite and finite intervals. Nonlinearity 2, 37–43 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Burini, D., De Lillo, S.: Nonlinear heat diffusion under impulsive forcing. Math. Comput. Model. 55, 269–277 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Friedman, A.: Partial Differential Equations of Parabolic Type, Chap. 8. Prentice Hall, Englewood Cliffs (1964)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of PerugiaPerugiaItaly
  2. 2.Department of Mathematics and Computer Science “Ulisse Dini”University of FlorenceFlorenceItaly

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