Journal of Engineering Mathematics

, Volume 67, Issue 3, pp 233–240 | Cite as

Some results for an \({\mathcal{N}}\)-dimensional nonlinear diffusion equation with radial symmetry

  • E. K. LenziEmail author
  • M. K. Lenzi
  • T. M. Gimenez
  • L. R. da Silva


The solutions of a nonlinear diffusion equation by considering the radially symmetric \({\mathcal{N}}\)-dimensional case are investigated. This equation has the nonlinearity present in the diffusive term and external force. The solutions are obtained by using a similarity method and connected to the q-exponential and q-logarithmic functions which emerge from the Tsallis formalism. In addition, the results obtained here may be useful to investigate a rich class of situations related to anomalous diffusion.


Anomalous diffusion Nonlinear diffusion equation Power-law diffusion coefficient Similarity Tsallis formalism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Frank TD (2005) Nonlinear Fokker–Planck equations. Springer-Verlag, HeidelbergzbMATHGoogle Scholar
  2. 2.
    Zola RS, Lenzi MK, Evangelista LR, Lenzi EK, Lucena LS, da Silva LR (2008) Exact solutions for a diffusion equation with a nonlinear external force. Phys Lett A 372: 2359–2363CrossRefADSGoogle Scholar
  3. 3.
    Assis PC Jr, da Silva PC, da Silva LR, Lenzi EK, Lenzi MK (2006) Nonlinear diffusion equation and nonlinear external force: exact solution. J Math Phys 47: 103302CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Assis PC Jr, da Silva LR, Lenzi EK, Malacarne LC, Mendes RS (2005) Nonlinear diffusion equation. Tsallis formalism and exact solutions. J Math Phys 46: 123303CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Grarilov VP, Klepikova NV, Rodean HC (1995) Trial of a nonlinear diffusion equation as a model of turbulent-diffusion. Atmos Environ 29: 2317–2322CrossRefGoogle Scholar
  6. 6.
    Muskat M (1937) The flow of homogeneous fluids through porous media. McGraw-Hill, MichiganzbMATHGoogle Scholar
  7. 7.
    Chavanis PH (2008) Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations. Eur Phys J B 62: 179–208CrossRefADSGoogle Scholar
  8. 8.
    Daly E, Porporato A (2004) Similarity solutions of nonlinear diffusion problems related to mathematical hydraulics and the Fokker-Planck equation. Phys Rev E 70: 056303CrossRefADSGoogle Scholar
  9. 9.
    De Wiest JM (eds). (1969) Flow through porous media. Academic Press, New YorkGoogle Scholar
  10. 10.
    Crank J (1956) The mathematics of diffusion. Oxford University Press, LondonzbMATHGoogle Scholar
  11. 11.
    Havlin S, Ben-Avraham D (1987) Diffusion in disordered media. Adv Phys 36: 695–798CrossRefADSGoogle Scholar
  12. 12.
    Lee BP (1994) Renormalization-group calculation for the reaction kA → 0. J Phys A 27: 2633–2652CrossRefADSGoogle Scholar
  13. 13.
    Alemany PA, Zanette DH, Wio HS (1994) Time-dependent reactivity for diffusion-controlled annihilation and coagulation in 2 dimensions. Phys Rev E 50: 3646–3655CrossRefADSGoogle Scholar
  14. 14.
    Gilchrist J, Van der Beek CJ (1994) Nonlinear diffusion in hard and soft superconductors. Physica C 231: 147–156CrossRefADSGoogle Scholar
  15. 15.
    Vinokur VM, Feigel’man MV, Geshkenbein VB (1991) Exact solution for flux creep with logarithmic u(j) dependence—self-organized critical state in high-tc superconductors. Phys Rev Lett 67: 915–918CrossRefADSGoogle Scholar
  16. 16.
    Mayergoyz I (1998) Nonlinear diffusion of electromagnetic fields. Academic Press, New YorkGoogle Scholar
  17. 17.
    Tsallis C, Bukman DJ (1996) Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys Rev E 54: R2197–R2200CrossRefADSGoogle Scholar
  18. 18.
    Borland L, Pennini F, Plastino AR, Plastino A (1999) The nonlinear Fokker-Planck equation with state-dependent diffusion—a nonextensive maximum entropy approach. Eur Phys J B 12: 285–297CrossRefADSGoogle Scholar
  19. 19.
    Malacarne LC, Mendes RS, Pedron IT, Lenzi EK (2001) Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution. Phys Rev E 63: 030101CrossRefADSGoogle Scholar
  20. 20.
    Malacarne LC, Mendes RS, Pedron IT, Lenzi EK (2002) N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients. Phys Rev E 65: 052101CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Giordano C, Plastino AR, Casas M, Plastino A (2001) Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions. Eur Phys J B 22: 361–368CrossRefADSGoogle Scholar
  22. 22.
    Drazer G, Wio HS, Tsallis C (2000) Anomalous diffusion with absorption: exact time-dependent solutions. Phys Rev E 61: 1417–1422CrossRefADSGoogle Scholar
  23. 23.
    Rigo A, Plastino AR, Casas M, Plastino A (2000) Anomalous diffusion coupled with Verhulst-like growth dynamics: exact time-dependent solutions. Phys Lett A 276: 97–102zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Plastino AR, Casas M, Plastino A (2000) A nonextensive maximum entropy approach to a family of nonlinear reaction-diffusion equations. Physica A 280: 289–303CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Lenzi EK, Anteneodo C, Borland L (2001) Escape time in anomalous diffusive media. Phys Rev E 63: 051109CrossRefADSGoogle Scholar
  26. 26.
    Tsallis C (1988) Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys 52: 479–487zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Plastino AR, Plastino A (1995) Non-extensive statistical mechanics and generalized Fokker-Planck equation. Physica A 222: 347–354CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Frank TD (2002) On a general link between anomalous diffusion and nonextensivity. J Math Phys 43: 344–350zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Borland L (1998) Microscopic dynamics of the nonlinear Fokker-Planck equation: a phenomenological model. Phys Rev E 57: 6634–6642CrossRefADSGoogle Scholar
  30. 30.
    Woyczynski WA (1999) Burgers-KPZ turbulence: Gottingen lectures. In: Lecture notes in mathematics. Springer-Verlag, TelosGoogle Scholar
  31. 31.
    Frish U, Bec J (2000) Burgulence. In: Lesieur M, David F, Yaglom AM (eds) Proceedings of the Les Houches Summer School, New trends in turbulence. Springer EDP-Sciences, BerlinGoogle Scholar
  32. 32.
    Olesen P (2003) Integrable version of Burgers equation in magnetohydrodynamics. Phys Rev E 68: 016307CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Witelski TP (2003) Intermediate asymptotics for Richards’ equation in a finite layer. J Eng Math 45: 379–399zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Grundy RE (1983) Asymptotic solution of a model nonlinear convective diffusion equation. IMA J Appl Math 31: 121–137CrossRefMathSciNetGoogle Scholar
  35. 35.
    Dawson CN, van Duijn CJ, Grundy RE (1996) Large time asymptotics in contaminant transport in porous media. SIAM J Appl Math 56: 965–993zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    van Duyn CJ, de Graaf JM (1987) Limiting profiles in contaminant transport through porous media. SIAM J Math Anal 18: 728–743zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Shaughnessy B, Procaccia I (1985) Analytical solutions for diffusion on fractal objects. Phys Rev Lett 54: 455–458CrossRefADSGoogle Scholar
  38. 38.
    Sokolov IM, Klafter J, Blumen A (2000) Ballistic versus diffusive pair dispersion in the Richardson regime. Phys Rev E 61: 2717–2722CrossRefADSGoogle Scholar
  39. 39.
    Vedenov AA (1967) Theory of a weakly turbulent plasma. Rev Plasma Phys 3: 229ADSGoogle Scholar
  40. 40.
    Vlad MO (1994) Fractional diffusion equation on fractals: self-similar stationary solutions in a force field derived from a logarithmic potential. Chaos Solitons Fractals 4: 191–199zbMATHCrossRefGoogle Scholar
  41. 41.
    Frank TD (2002) Generalized Fokker-Planck equations derived from generalized linear nonequilibrium thermodynamics. Physica A 310: 397–412zbMATHCrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Risken H (1984) The Fokker–Planck equation. Springer-Verlag, New YorkzbMATHGoogle Scholar
  43. 43.
    Gardiner CW (1996) Handbook of stochastic methods: for physics, chemistry and the natural sciences, springer series in synergetics. Springer-Verlag, New YorkGoogle Scholar
  44. 44.
    Debnath L (1997) Nonlinear partial differential equations for scientists and engineers. Birkhäuser, BostonzbMATHGoogle Scholar
  45. 45.
    Logan JD (1994) An introduction to nonlinear partial differential equations. Wiley, New YorkzbMATHGoogle Scholar
  46. 46.
    Bluman GW, Cole JD (1974) Similarity methods for differential equations. Springer, BerlinzbMATHGoogle Scholar
  47. 47.
    Ovsiannikov LV (1980) The group analysis of differential equations. Academic Press, New YorkGoogle Scholar
  48. 48.
    Olver P (1986) Applications of Lie groups to differential equations. Springer, BerlinzbMATHGoogle Scholar
  49. 49.
    Cherniha R (1998) New non-Lie ansätze and exact solutions of nonlinear reaction-diffusion-convection equations. J Phys A 31: 8179–8198zbMATHCrossRefMathSciNetADSGoogle Scholar
  50. 50.
    Cherniha R, King JR (2005) Non-linear reaction–diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. J Math Anal Appl 308: 11–35zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Silva AT, Lenzi EK, Evangelista LR, Lenzi MK, da Silva LR (2007) Fractional nonlinear diffusion equation, solutions and anomalous diffusion. Physica A 375: 65–71CrossRefMathSciNetADSGoogle Scholar
  52. 52.
    Wang J, Zhang W-J, Liang J-R, Zhang P, Ren F-Y (2008) Fractional nonlinear diffusion equation and first passage time. Physica A 387: 764–772CrossRefADSGoogle Scholar
  53. 53.
    Liang J-R, Ren F-Y, Qiu W-Y, Xiao J-B (2007) Exact solutions for nonlinear fractional anomalous diffusion equations. Physica A 385: 80–94CrossRefMathSciNetADSGoogle Scholar
  54. 54.
    Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkzbMATHGoogle Scholar
  55. 55.
    Tsallis C, Lenzi EK (2002) Anomalous diffusion: nonlinear fractional Fokker-Planck equation. Chem Phys 284:341–347 (Erratum, Chem Phys 287:295, 2003)Google Scholar
  56. 56.
    Schwämmle V, Curado EMF, Nobre FD (2007) A general nonlinear Fokker-Planck equation and its associated entropy. Eur Phys J B 58: 159–165CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • E. K. Lenzi
    • 1
    Email author
  • M. K. Lenzi
    • 2
  • T. M. Gimenez
    • 3
  • L. R. da Silva
    • 4
  1. 1.Departamento de FísicaUniversidade Estadual de MaringáMaringáBrazil
  2. 2.Departamento de Engenharia QuímicaUniversidade Federal do ParanáCuritibaBrazil
  3. 3.Departamento de Engenharia QuímicaUniversidade Estadual de MaringáMaringáBrazil
  4. 4.Departamento de FísicaUniversidade Federal do Rio Grande do NorteNatalBrazil

Personalised recommendations