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Mathematical Methods in Engineering pp 23–55Cite as

Sub-diffusion equations of fractional order and their fundamental solutions

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Abstract

The time-fractional diffusion equation is obtained by generalizing the standard diffusion equation by using a proper time-fractional derivative of order 1 — β in the Riemann-Liouville (R-L) sense or of order β in the Caputo (C) sense, with β ∈ (0, 1). The two forms are equivalent and the fundamental solution of the associated Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process, related to a phenomenon of sub- diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time-derivatives of order less than one. Then the two forms are no longer equivalent. However, the fundamental solution still is a probability density of a non-Markovian process but one exhibiting a distribution of time-scales instead of being self-similar: it is expressed in terms of an integral of Laplace type suitable for numerical computation. We consider with some detail two cases of diffusion of distributed order: the double order and the uniformly distributed order discussing the differences between the R-L and C approaches. For these cases we analyze in detail the behaviour of the fundamental solutions (numerically computed) and of the corresponding variance (analytically computed) through the exhibition of several plots. While for the R-L and for the C cases the fundamental solutions seem not to differ too much for moderate times, the behaviour of the corresponding variance for small and large times differs in a remarkable way.

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References

  1. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports, 339, 1–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. Math. Gen., 37, R161–R208 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Piryatinska, A., Saichev, A.I., Woyczynski, W.A.: Models of anomalous diffusion: the subdiffusive case. Physica A, 349, 375–420 (2005)

    Article  Google Scholar 

  4. Zaslavsky, G.M.: Chaos, fractional kinetics and anomalous transport. Phys. Reports, 371, 461–580 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Klafter, J., Sokolov, I.M.: Anomalous diffusion spreads its wings. Physics World, 18, 29–32 (2005)

    Google Scholar 

  6. Sokolov, I.M., Klafter, J.: From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos, 15, 026103–026109 (2005)

    Article  MathSciNet  Google Scholar 

  7. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations, J. Math. Phys., 30, 134–144 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mainardi, F.: On the initial value problem for the fractional diffusion-wave equation. In: Rionero, S., Ruggeri, T. (ed) Waves and Stability in Continuous Media. World Scientific, Singapore (1994)

    Google Scholar 

  9. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals, 7, 1461–1477 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (ed) Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag, Wien New-York (1997) [Reprinted in http://www.fracalmo.org]

    Google Scholar 

  11. Gorenflo, R., Rutman, R.: On ultraslow and intermediate processes. In: Rusev, P., Dimovski, I., Kiryakova, V. (ed) Proc. Workshop on Transform Methods and Special Functions (Sofia 1994). Science Culture Technology, Singapore (1995)

    Google Scholar 

  12. Gorenflo, R., Luchko, Yu., Mainardi, F.: Analytical properties and applications of the Wright function. Fractional Calculus and Applied Analysis, 2, 383–414 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Gorenflo, R., Luchko, Yu., Mainardi, F.: Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Computational and Applied Mathematics, 118, 175–191 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mainardi, F., Pagnini, G.: The Wright functions as solutions of the time-fractional diffusion equations. Appl. Math. and Comp., 141, 51–62 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B, 133, 425–430 (1986)

    Article  Google Scholar 

  16. Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in random media. Physica A, 185, 87–97 (1992)

    Article  Google Scholar 

  17. Metzler, R., Glöckle, W.G., Nonnenmacher, T.F.: Fractional model equation for anomalous diffusion. Physica A, 211, 13–24 (1994)

    Article  Google Scholar 

  18. Saichev, A., Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos, 7, 753–764 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gel’fand, I.M., Shilov, G.E.: Generalized Functions. Vol. I, Academic Press, New York London (1964)

    MATH  Google Scholar 

  20. Gorenflo, R., Mainardi, F.: Fractional diffusion processes: probability distributions and continuous time random walk. In: Rangarajan, G., Ding, M. (ed) Processes with Long Range Correlations. Springer Verlag, Berlin (2003) [Lecture Notes in Physics, No. 621]

    Chapter  Google Scholar 

  21. Gorenflo, R., Mainardi, F.: Simply and multiply scaled diffusion limits for continuous time random walks. In: Benkadda, S., Leoncini, X., Zaslavsky, G. (ed) Proceedings of the International Workshop on Chaotic Transport and Complexity in Fluids and Plasmas, Carry Le Rouet (France) 20–25 June 2004. IOP (Institute of Physics) Journal of Physics: Conference Series 7 (2005)

    Google Scholar 

  22. Mainardi, F., Vivoli, A., Gorenflo, R.: Continuous time random walk and time fractional diffusion: a numerical comparison between the fundamental solutions. Fluctuation and Noise Letters, 5, L291–L297 (2005)

    Article  Google Scholar 

  23. Scalas, E., Gorenflo, R., Mainardi, F.: Uncoupled continuous-time random walks: solution and limiting behaviour of the master equation. Physical Review E, 69, 011107-1/8 (2004)

    Article  MathSciNet  Google Scholar 

  24. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  25. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  26. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Miscellaneous Functions. In: Higher Transcendental Functions. Bateman Project, Vols. 1–3, McGraw-Hill, New York (1955)

    Google Scholar 

  27. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (ed) Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag, Wien New York (1997) [Reprinted in http://www.fracalmo.org]

    Google Scholar 

  28. Gorenflo, R., Iskenderov, A., Luchko, Yu.: Mapping between solutions of fractional diffusion-wave equations. Fractional Calculus and Applied Analysis, 3, 75–86 (2000)

    MATH  MathSciNet  Google Scholar 

  29. Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus and Applied Analysis, 4, 153–192 (2001) [Reprinted in http://www.fracalmo.org]

    MATH  MathSciNet  Google Scholar 

  30. Caputo, M.: Elasticità e Dissipazione. Zanichelli, Bologna (1969) [in Italian]

    Google Scholar 

  31. Caputo, M.: Mean fractional-order derivatives differential equations and filters. Ann. Univ. Ferrara, Sez VII, Sc. Mat., 41, 73–84 (1995)

    MATH  MathSciNet  Google Scholar 

  32. Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fractional Calculus and Applied Analysis, 4, 421–442 (2001)

    MATH  MathSciNet  Google Scholar 

  33. Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations. Int. J. Appl. Math., 2, 865–882, 965–987 (2000)

    MATH  MathSciNet  Google Scholar 

  34. Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E, 66, 046129/1–6 (2002)

    Article  Google Scholar 

  35. Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar, V.Yu.: Distributed order time fractional diffusion equation. Fractional Calculus and Applied Analysis, 6, 259–279 (2003)

    MATH  MathSciNet  Google Scholar 

  36. Chechkin, A.V., Klafter, J., Sokolov, I.M.: Fractional Fokker-Planck equation for ultraslow kinetics. Europhysics Lett., 63, 326–332 (2003)

    Article  Google Scholar 

  37. Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Physica Polonica, 35, 1323–1341 (2004)

    Google Scholar 

  38. Naber, M.: Distributed order fractional subdiffusion. Fractals, 12, 23–32 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. Langlands, T.A.M.: Solution of a modified fractional diffusion equation. Physica A, 367, 136–144 (2006)

    Article  MathSciNet  Google Scholar 

  40. Umarov, S., Gorenflo, R.: Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations: Part one. Journal for Analysis and its Applications (ZAA), 24, 449–466 (2005)

    MATH  MathSciNet  Google Scholar 

  41. Mainardi, F., Pagnini, G.: The role of the Fox-Wright functions in fractional subdiffusion of distributed order. J. Computational and Appl. Mathematics. (2006), in press.

    Google Scholar 

  42. Paris, R.B., Kaminski, D.: Asymptotic and Mellin-Barnes Integrals. Cambridge Univ. Press, Cambridge (2001)

    Google Scholar 

  43. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc., 13, 529–539 (1967)

    Google Scholar 

  44. Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure and Applied Geophysics (Pageoph), 91, 134–147 (1971)

    Article  Google Scholar 

  45. Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento (Ser. II), 1, 161–198 (1971)

    Google Scholar 

  46. Gorenflo, R., Vessella, S.: Abel Integral Equations: Analysis and Applications. Springer Verlag, Berlin (1991)

    MATH  Google Scholar 

  47. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  48. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  49. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  50. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer Verlag, New York (2003)

    Google Scholar 

  51. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  52. Schneider, W.R.: Completely monotone generalized Mittag-Leffler functions. Expositiones Mathematicae, 14, 3–16 (1996)

    MATH  MathSciNet  Google Scholar 

  53. Miller, K.S., Samko, S.G.: A note on the complete monotonicity of the generalized Mittag-Leffler function. Real Anal. Exchange, 23, 753–755 (1997)

    MATH  MathSciNet  Google Scholar 

  54. Miller, K.S., Samko, S.G.: Completely monotonic functions. Integral Transforms and Special Functions, 12, 389–402 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  55. Ghizzetti, A., Ossicini, A.: Trasformate di Laplace e Calcolo Simbolico. UTET, Torino (1971)

    MATH  Google Scholar 

  56. Feller, W.: An Introduction to Probability Theory and its Applications. Vol. 2, Wiley, New York (1971)

    MATH  Google Scholar 

  57. Djrbashian, M.M.: Integral Transforms and Representations of Functions in the Complex Plane. Nauka, Moscow (1966) [in Russian] [There is also the transliteration as Dzherbashian]

    Google Scholar 

  58. Mainardi, F., Gorenflo, R.: On Mittag-Leffler type functions in fractional evolution processes. J. Comput. and Appl. Mathematics, 118, 283–299 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  59. Marichev, O.I.: Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables. Chichester, Ellis Horwood (1983)

    MATH  Google Scholar 

  60. Temme, N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126, Bologna, Italy

    Francesco Mainardi & Antonio Mura

  2. ENEA: National Agency for New Technologies, Energy and the Environment, Centre “E. Clementel”, Via Martiri di Monte Sole 4, I-40129, Bologna, Italy

    Gianni Pagnini

  3. Department of Mathematics and Informatics, Free University of Berlin, Arnimallee 3, D-14195, Berlin, Germany

    Rudolf Gorenflo

Authors
  1. Francesco Mainardi
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  2. Antonio Mura
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  3. Gianni Pagnini
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  4. Rudolf Gorenflo
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Editor information

Editors and Affiliations

  1. Çankaya University, Balgat-Ankara, Turkey

    K. Taş  & D. Baleanu  & 

  2. Institute of Engineering of Porto, Porto, Portugal

    J. A. Tenreiro Machado

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Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R. (2007). Sub-diffusion equations of fractional order and their fundamental solutions. In: Taş, K., Tenreiro Machado, J.A., Baleanu, D. (eds) Mathematical Methods in Engineering. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5678-9_3

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  • DOI: https://doi.org/10.1007/978-1-4020-5678-9_3

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  • Online ISBN: 978-1-4020-5678-9

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