Skip to main content
SpringerLink
Log in

On the Solution of Time-Fractional Diffusion Models

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2022 (ICCSA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13375))

Included in the following conference series:

  • 721 Accesses

Abstract

In many situations, the analysis of viscoelastic materials, like polymers, takes benefit from the introduction of fractional operators in the mathematical formalization. In addition, fractional differential models have been applied in a wide variety of fields, from biology to thermodynamics, from diffusion of information to dehydration/rehydration of food. Thus, a great interest is paid both in the analytical and in the numerical solution of fractional differential problems. The present paper considers a class of time-fractional diffusion problems with Dirichlet boundary conditions. Using Duhamel’s principle, the analytical solution is found. As usual in this context, the solution is given in series form and depends on the Mittag-Leffler function. We suggest a computational procedure to evaluate the solution with high accuracy, in a computing environment. Some test examples are presented both in the subdiffusion and in the superdiffusion case, to illustrate the behavior of the solution for different values of the fractional index. Test cases have been carried out in Matlab.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abbaszadeh, M., Mohebbi, A.: A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term. Comput. Math. Appl. 66(8), 1345–1359 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bazhlekova, E.: Existence and uniqueness results for a fractional evolution equation in Hilbert space. Fract. Calc. Appl. Anal. 15(2), 232–243 (2012)

    Article  MathSciNet  Google Scholar 

  3. Burrage, K., Cardone, A., D’Ambrosio, R., Paternoster, B.: Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116, 82–94 (2017)

    Article  MathSciNet  Google Scholar 

  4. Cardone, A., Conte, D., Paternoster, B.: A MATLAB implementation of spline collocation methods for fractional differential equations. In: Gervasi, O., et al. (eds.) ICCSA 2021. LNCS, vol. 12949, pp. 387–401. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86653-2_29

    Chapter  Google Scholar 

  5. Cardone, A., Conte, D.: Stability analysis of spline collocation methods for fractional differential equations. Math. Comput. Simulation 178, 501–514 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cardone, A., Conte, D., Paternoster, B.: Stability analysis of two-step spline collocation methods for fractional differential equations, submitted

    Google Scholar 

  7. Cardone, A., Conte, D., Paternoster, B.: Two-step collocation methods for fractional differential equations. Discrete Continuous Dyn. Syst. Ser. B 23(7), 2709–2725 (2018)

    Article  MathSciNet  Google Scholar 

  8. Cardone, A., Conte, D., Paternoster, B.: Numerical treatment of fractional differential models. In: Abdel Wahab, M. (ed.) FFW 2020 2020. LNME, pp. 289–302. Springer, Singapore (2021). https://doi.org/10.1007/978-981-15-9893-7_21

    Chapter  Google Scholar 

  9. Cardone, A., D’Ambrosio, R., Paternoster, B.: A spectral method for stochastic fractional differential equations. Appl. Numer. Math. 139, 115–119 (2019)

    Article  MathSciNet  Google Scholar 

  10. Cardone, A., Frasca-Caccia, G.: Numerical conservation laws of time fractional diffusion PDEs. Fract. Calc. Appl. Anal. (2022). https://doi.org/10.1007/s13540-022-00059-7

  11. Conte, D., Farsimadan, E., Moradi, L., Palmieri, F., Paternoster, B.: Time-delay fractional optimal control problems: a survey based on methodology. Lect. Notes Mech. Eng. pp. 325–337 (2021)

    Google Scholar 

  12. Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005)

    Article  MathSciNet  Google Scholar 

  13. Di Paola, M., Fiore, V., Pinnola, F., Valenza, A.: On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials. Mech. Mater. 69(1), 63–70 (2014)

    Article  Google Scholar 

  14. Di Paola, M., Pirrotta, A., Valenza, A.: Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results. Mech. Mater. 43(12), 799–806 (2011)

    Article  Google Scholar 

  15. Diethelm, K., Ford, N., Freed, A.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)

    Article  MathSciNet  Google Scholar 

  16. Diethelm, K., Freed, A.: The fracPECE subroutine for the numerical solution of differential equations of fractional order. In: Forschung und wissenschaftliches Rechnen, 1999. pp. 57–71. Göttingen (1998)

    Google Scholar 

  17. Ezzat, M., El-Bary, A.: Unified GN model of electro-thermoelasticity theories with fractional order of heat transfer. Microsyst. Technol. 24(12), 4965–4979 (2018)

    Article  Google Scholar 

  18. Failla, G., Zingales, M.: Advanced materials modelling via fractional calculus: challenges and perspectives. Philo. Trans. R. Soc. A 378(2172), article no. 20200050 (2020). https://doi.org/10.1098/rsta.2020.0050

  19. Farsimadan, E., Moradi, L., Conte, D., Paternoster, B., Palmieri, F.: Comparison between protein-protein interaction networks CD4\(^+\)T and CD8\(^+\)T and a numerical approach for fractional hiv infection of CD4\(^+\)T cells. Lect. Notes Comput. Sci. 12949, 78–94 (2021)

    Article  Google Scholar 

  20. Foroozani, A., Ebrahimi, M.: Nonlinear anomalous information diffusion model in social networks. Commun. Nonlinear Sci. Numer. Simul. 103, Paper No. 106019, 18 (2021)

    Google Scholar 

  21. Garrappa, R.: The Mittag-Leffler function, March 2015. (http://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function), MATLAB Central File Exchange. Accessed February 24, 2022

  22. Garrappa, R.: On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87(10), 2281–2290 (2010)

    Article  MathSciNet  Google Scholar 

  23. Garrappa, R.: Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53(3), 1350–1369 (2015). https://doi.org/10.1137/140971191

    Article  MathSciNet  MATH  Google Scholar 

  24. Lemus-Mondaca, R., Nuñez, H., Jaques, A., Ramírez, C., Simpson, R.: The anomalous diffusion model based on a fractional calculus approach applied to describe the rehydration process of dried vegetal food matrices. J. Food Process Eng. 44(9), e13773 (2021)

    Google Scholar 

  25. Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15(1), 141–160 (2012)

    Article  MathSciNet  Google Scholar 

  26. Luchko, Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59(5), 1766–1772 (2010)

    Article  MathSciNet  Google Scholar 

  27. Mainardi, F.: Fractional calculus. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. ICMS, vol. 378, pp. 291–348. Springer, Vienna (1997). https://doi.org/10.1007/978-3-7091-2664-6_7

    Chapter  MATH  Google Scholar 

  28. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific (2010)

    Google Scholar 

  29. Moradi, L., Conte, D., Farsimadan, E., Palmieri, F., Paternoster, B.: Optimal control of system governed by nonlinear Volterra integral and fractional derivative equations. Comput. Appl. Math. 40(4), Paper No. 157, 15 (2021)

    Google Scholar 

  30. Nigmatullin, R., Nelson, S.: Recognition of the “fractional” kinetics in complex systems: Dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz. Signal Process. 86(10), 2744–2759 (2006)

    Google Scholar 

  31. Podlubny, I.: Fractional differential equations, Mathematics in Science and Engineering, vol. 198. Academic Press Inc, San Diego (1999)

    MATH  Google Scholar 

  32. Povstenko, Y.: Fractional Thermoelasticity. SMIA, vol. 219. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15335-3

    Book  MATH  Google Scholar 

  33. Povstenko, Y.: Time-fractional thermoelasticity problem for a sphere subjected to the heat flux. Appl. Math. Comput. 257, 327–334 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Seemab, A., ur Rehman, M.: A note on fractional Duhamel’s principle and its application to a class of fractional partial differential equations. Appl. Math. Lett. 64, 8–14 (2017)

    Google Scholar 

  35. Shen, J., Li, C., Wu, H., Kalantari, M.: Fractional order viscoelasticity in characterization for atrial tissue. Korea Aust. Rheol. J. 25(2), 87–93 (2013)

    Article  Google Scholar 

  36. Sherief, H.H., El-Hagary, M.A.: Fractional order theory of thermo-viscoelasticity and application. Mech. Time Depend. Mater. 24(2), 179–195 (2019). https://doi.org/10.1007/s11043-019-09415-2

    Article  Google Scholar 

  37. Simpson, R., Ramírez, C., Nuñez, H., Jaques, A., Almonacid, S.: Understanding the success of page’s model and related empirical equations in fitting experimental data of diffusion phenomena in food matrices. Trends Food Sci. Technol. 62, 194–201 (2017)

    Article  Google Scholar 

  38. Umarov, S.: On fractional Duhamel’s principle and its applications. J. Differ. Eq. 252(10), 5217–5234 (2012)

    Article  MathSciNet  Google Scholar 

  39. Umarov, S., Saydamatov, E.: A fractional analog of the Duhamel principle. Fract. Calc. Appl. Anal. 9(1), 57–70 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Yang, W., Chen, Z.: Fractional single-phase lag heat conduction and transient thermal fracture in cracked viscoelastic materials. Acta Mech. 230(10), 3723–3740 (2019). https://doi.org/10.1007/s00707-019-02474-z

    Article  MathSciNet  MATH  Google Scholar 

  41. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are members of the GNCS group. This work is supported by GNCS-INDAM project and by the Italian Ministry of University and Research, through the PRIN 2017 project (No. 2017JYCLSF) “Structure preserving approximation of evolutionary problems”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Salerno, Via Giovanni Paolo II n. 132, 84084, Fisciano, SA, Italy

    Angelamaria Cardone & Gianluca Frasca-Caccia

Authors
  1. Angelamaria Cardone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluca Frasca-Caccia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Angelamaria Cardone .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Universidad de Málaga, Malaga, Spain

    Eligius M. T. Hendrix

  4. Monash University, Clayton, VIC, Australia

    David Taniar

  5. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cardone, A., Frasca-Caccia, G. (2022). On the Solution of Time-Fractional Diffusion Models. In: Gervasi, O., Murgante, B., Hendrix, E.M.T., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2022. ICCSA 2022. Lecture Notes in Computer Science, vol 13375. Springer, Cham. https://doi.org/10.1007/978-3-031-10522-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-10522-7_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-10521-0

  • Online ISBN: 978-3-031-10522-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation