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Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

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Abstract

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.

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References

  1. Agarwal, R.P., Benchora, M., Hamani, S.: Boundary value problems for fractional differential equations. Georgian Math. J. 16, 401 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Agarwal, R.P., Benchora, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agarwal, R.P., O’Regan, D., Stanuek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145 (2002)

    Article  MATH  Google Scholar 

  5. Aoki, K., Osteryoung, J.: Square wave voltammetry in a thin-layer cell. J. Electroanal. Chem. 240, 45 (1988)

    Article  Google Scholar 

  6. Caputo, M.: Elasticità e Dissipazione. Zanichelli, Bologna (1969)

    Google Scholar 

  7. Glöcke, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68, 46 (1995)

    Article  Google Scholar 

  8. Gorenflo, R., Luchko, Y., Mainardi, F.: Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2, 383 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time-fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gorenflo, R., Vivoli, A., Mainardi, F.: Discrete and continuous random walk models for space-time fractional diffusion. Nonlinear Dyn. 38, 101 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hahn, M., Umarov, S.: Fractional Fokker–Planck–Kolmogorov type equations and their associated stochastic differential equations. Fract. Calc. Appl. Anal. 14, 56 (2011)

    MathSciNet  Google Scholar 

  12. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011), 51 pages

    Article  MathSciNet  Google Scholar 

  13. Heymans, N.: Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dyn. 38, 221 (2004)

    Article  MATH  Google Scholar 

  14. Hilfer, R.: Classification theory for anequilibrium phase transitions. Phys. Rev. E 48, 2466 (1993)

    Article  MathSciNet  Google Scholar 

  15. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  16. Hilfer, R.: Fractional diffusion based on Riemann–Liouville fractional derivatives. J. Phys. Chem. B 104, 3914 (2000)

    Article  Google Scholar 

  17. Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284, 399 (2002)

    Article  Google Scholar 

  18. Hilfer, R.: On fractional relaxation. Fractals 11, 251 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hilfer, R.: Foundations of fractional dynamics: a short account. In: Klafter, J., Lim, S.C., Metzler, R. (eds.) Fractional Dynamics, Recent Advances. World Scientific, Singapore (2011)

    Google Scholar 

  20. Ishteva, M., Scherer, R., Boyadjiev, L.: On the Caputo operator of fractional calculus and C-Laguerre functions. Math. Sci. Res. J. 9, 161 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  22. Kiryakova, V.S.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liang, J., Chen, Y.Q.: Hybrid symbolic and numerical simulation studies of time-fractional order wavediffusion systems. Int. J. Control 79, 1462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lundberg, H.K., Miller, H.R., Trumper, D.L.: Initial conditions, generalized functions, and the Laplace transform: troubles at the origin. IEEE Control Syst. Mag. 27, 22 (2007)

    Article  Google Scholar 

  25. Lutz, E.: Fractional Langevin equation. Phys. Rev. E 64, 051106 (2001)

    Article  Google Scholar 

  26. Machado, J.A.T.: Discrete-time fractional-order controllers. Fract. Calc. Appl. Anal. 4, 47 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Mainardi, F.: Fractional diffusive waves in viscoelastic solids. In: Wegner, J.L., Norwood, F.R. (eds.) Nonlinear Waves in Solids. Appl. Mech. Rev., Proc. Issue, pp. 93–97 (1994)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mainardi, F., Paradisi, P.: Fractional diffusive waves. J. Comput. Acoust. 9, 1417 (2001)

    MathSciNet  Google Scholar 

  32. Mainardi, F., Pironi, P.: The fractional Langevin equation: Brownian motion revisited. Extr. Math. 11, 140 (1996)

    MathSciNet  Google Scholar 

  33. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mendes, R.V.: A fractional calculus interpretation of the fractional volatility model. Nonlinear Dyn. 55, 395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach. Phys. Rev. Lett. 82, 3563 (1999)

    Article  Google Scholar 

  37. Mirčeski, V.: Charge transfer kinetics in thin-film voltammetry. Theoretical study under conditions of square-wave voltammetry. J. Phys. Chem. B 108, 13719 (2004)

    Article  Google Scholar 

  38. Mirčeski, V., Tomovski, Ž.: Voltammetry based on fractional diffusion. J. Phys. Chem. B 113, 2794 (2009)

    Article  Google Scholar 

  39. Mirčeski, V., Tomovski, Ž.: Modeling of a voltammetric experiment in a limiting diffusion space. J. Solid State Electrochem. 15, 197 (2011)

    Article  Google Scholar 

  40. Nigmatullin, R.R.: Fractional integral and its physical interpretation. Theor. Math. Phys. 90, 242 (1992)

    Article  MathSciNet  Google Scholar 

  41. Ortigueira, M.D.: Introduction to fractional linear systems. Part 1. Continuous-time case. IEEE Proc. Vis. Image Signal Process. 147, 62 (2000)

    Article  Google Scholar 

  42. Ortigueira, M.D.: On the initial conditions in continuous-time fractional linear systems. Signal Process. 83, 2301 (2003)

    Article  MATH  Google Scholar 

  43. Ortigueira, M.D.: An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst. Mag. 8, 19 (2008)

    Article  Google Scholar 

  44. Ortigueira, M.B., Batista, A.G.: A fractional linear system view of the fractional Brownian motion. Nonlinear Dyn. 38, 295 (2004)

    Article  MATH  Google Scholar 

  45. Ortigueira, M.D., Coito, F.J.: System initial conditions vs derivative initial conditions. Comput. Math. Appl. 59, 1782 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ortigueira, M.D., Coito, F.J.: On the usefulness of Riemann–Liouville and Caputo derivatives in describing fractional shift-invariant linear systems. J. Appl. Nonlinear Dyn. 1, 113 (2012)

    Google Scholar 

  47. Ortigueira, M.D., Magin, R.L., Trujillo, J.J., Velasco, M.P.: A real regularised fractional derivative. SIViP 6, 351 (2012)

    Article  Google Scholar 

  48. Paneva-Konovska, J.: Convergence of series in Mittag-Leffler functions. C. R. Acad. Bulgare Sci. 63, 815 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Podlubny, I.: Fractional Differential Equations. Acad. Press, San Diego (1999)

    MATH  Google Scholar 

  50. Sandev, T., Tomovski, Ž.: The general time fractional Fokker–Planck equation with a constant external force. In: Proc. Symp. Fract. Sig. Syst., Coimbra, 4–5 November 2009, pp. 27–39 (2009)

    Google Scholar 

  51. Sandev, T., Tomovski, Ž.: Wave equation for a vibrating string in presence of a fractional friction. In: Proc. Symp. Fract. Sig. Syst., Lisbon, 4–6 November 2009 (2009)

    Google Scholar 

  52. Sandev, T., Tomovski, Ž.: The general time fractional wave equation for a vibrating string. J. Phys. A, Math. Theor. 43, 055204 (2010)

    Article  MathSciNet  Google Scholar 

  53. Sandev, T., Tomovski, Ž.: Asymptotic behavior of a harmonic oscillator driven by a generalized Mittag-Leffler noise. Phys. Scr. 82, 065001 (2010)

    Article  Google Scholar 

  54. Sandev, T., Metzler, R., Tomovski, Ž.: Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative. J. Phys. A, Math. Theor. 44, 255203 (2011)

    Article  MathSciNet  Google Scholar 

  55. Sandev, T., Tomovski, Ž., Dubbeldam, J.L.A.: Generalized Langevin equation with a three parameter Mittag-Leffler noise. Physica A 390, 3627 (2011)

    Article  Google Scholar 

  56. Sandev, T., Metzler, R., Tomovski, Ž.: Velocity and displacement correlation functions for fractional generalized Langevin equations. Fract. Calc. Appl. Anal. 15, 426 (2012)

    MathSciNet  Google Scholar 

  57. Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 209, 299 (2004)

    Article  Google Scholar 

  58. Srivastava, H.M., Tomovski, Ž.: Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel. Appl. Math. Comput. 211, 198 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tihonov, A.N., Samarskii, A.A.: Mathematical Physics Equations, 7th edn. Nauka, Moscow (2004) (in Russian)

    Google Scholar 

  60. Tomovski, Ž.: Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator. Nonlinear Anal. 75, 3364 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Tomovski, Ž., Sandev, T.: Effects of a fractional friction with power-law memory kernel on string vibrations. Comput. Math. Appl. 62, 1554 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tomovski, Ž., Sandev, T.: Fractional wave equation with a frictional memory kernel of Mittag-Leffler type. Appl. Math. Comput. 218, 10022 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Tomovski, Ž., Hilfer, R., Srivastava, H.M.: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 21, 797 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  64. Tomovski, Ž., Sandev, T., Metzler, R., Dubbeldam, J.: Generalized space-time fractional diffusion equation with composite fractional time derivative. Physica A 391, 2527 (2012)

    Article  MathSciNet  Google Scholar 

  65. Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3, 231 (2000)

    MathSciNet  MATH  Google Scholar 

  66. Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, vol. 4. Cambridge University Press, Cambridge (1927)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Saints Cyril and Methodius University, 1000, Skopje, Macedonia

    Živorad Tomovski

  2. Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020, Skopje, Macedonia

    Trifce Sandev

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  1. Živorad Tomovski
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  2. Trifce Sandev
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Tomovski, Ž., Sandev, T. Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions. Nonlinear Dyn 71, 671–683 (2013). https://doi.org/10.1007/s11071-012-0710-x

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