Skip to main content
SpringerLink
Log in

Numerical Solutions of Riesz Fractional Partial Differential Equations

  • Chapter
  • First Online:
Nonlinear Differential Equations in Physics

Abstract

Nowadays, different applications of fractional differential equations in many areas, such as engineering, physics, chemistry, astrophysics, and many other sciences are observed.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. Wang, X., Liu, F., Chen, X.: Novel second-order accurate implicit numerical methods for the riesz space distributed-order advection-dispersion equations. Adv. Math. Phys. 2015, 14 (2015). (Article ID 590435)

    Google Scholar 

  2. Saha Ray, S.: Exact solutions for time fractional diffusion method by decomposition method. Phys. Scr. 75, 53–61 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Khan, Y., Diblík, J., Faraz, N., Šmarda, Z.: An efficient new perturbative Laplace method for space-time fractional telegraph equations. Adv. Differ. Equ. 2012 (2012). (Article number 204)

    Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461–580 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comp. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu, F., Anh, V., Turner, I.: Numerical solution of the space Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7(4), 753–764 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciesielski, M., Leszczynski, J.: Numerical solutions of a boundary value problem for the anomalous diffusion equation with the Riesz fractional derivative. In: Proceedings of the 16th International Conference on Computer Methods in Mechanics Czestochowa, Poland (2005)

    Google Scholar 

  12. Shen, S., Liu, F., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation. IMA J. Appl. Math. 73(6), 850–872 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Risken, H.: The Fokker-Planck Equation: Methods of solution and Applications. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  14. So, F., Liu, K.L.: A study of the subdiffusive fractional Fokker-Planck equation of bistable systems. Phys. A 331(3–4), 378–390 (2004)

    Article  MathSciNet  Google Scholar 

  15. Saha Ray, S., Gupta, A.K.: A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker-Planck equations in modelling of anomalous diffusion systems. J. Math. Chem. 52(8), 2277–2293 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker-Planck equation. Appl. Math. Model. 33, 256–273 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhuang, P., Liu, F., Turner, I., Anh, V.: Numerical treatment for the fractional Fokker-Plank equation. ANZIAM J. 48, 759–774 (2007)

    Article  Google Scholar 

  18. Odibat, Z., Momani, S.: Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives. Phys. Lett. A 369, 349–358 (2007)

    Article  MATH  Google Scholar 

  19. Vanani, S.K., Aminataei, A.: A numerical algorithm for the space and time fractional Fokker-Planck equation. Int. J. Numer. Meth. Heat Fluid Flow 22(8), 1037–1052 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yildirm, A.: Analytical approach to Fokker-Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method. J. King Saud Univ. (Science) 22, 257–264 (2010)

    Article  Google Scholar 

  21. Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin, Heidelberg (2009)

    Book  MATH  Google Scholar 

  22. Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., Morris, H.C.: Solitons and Nonlinear Wave Solutions. Academic, London (1982)

    MATH  Google Scholar 

  23. Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser, Boston (2005)

    Book  MATH  Google Scholar 

  24. Newell, A.C.: Nonlinear Optics. Addition-Wesley, New York (1992)

    MATH  Google Scholar 

  25. Saha Ray, S.: Numerical solutions and solitary wave solutions of fractional KDV equations using modified fractional reduced differential transform method. Comput. Math. Math. Phys. 53(12), 1870–1881 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Alfimov, G., Pierantozzi, T., Vázquez, L.: Numerical study of a fractional sine-Gordon equation. In: Workshop Preprints/Proceedings of Fractional Differentiation and Its Applications, FDA 2004, pp. 153–162 (2004)

    Google Scholar 

  27. Ivanchenko, YuM, Soboleva, T.K.: Nonlocal interaction in Josephson junctions. Phys. Lett. A 147, 65–69 (1990)

    Article  Google Scholar 

  28. Gurevich, A.: Nonlocal Josephson electrodynamics and pinning in superconductors. Phys. Rev. B. 46, 3187–3190 (1992)

    Article  Google Scholar 

  29. Barone, A., Paterno, G.: Physics and applications of the Josephson effect. Wiley, New York (1982)

    Book  Google Scholar 

  30. Aliev, Y.M., Silin, V.P.: Travelling 4π-kink in nonlocal Josephson electrodynamics. Phys. Lett A, 177(3), 259–262 (1993)

    Article  Google Scholar 

  31. Aliev, Y.M., Ovchinnikov, K.N., Silin V.P., Uryupin, S.A. Perturbations of stationary solutions in a nonlocal model of a josephson junction. J. Exp. Theor. Phys., 80(551) (1995)

    Google Scholar 

  32. Alfimov, G.L., Silin, V.P.: On small perturbations of stationary states in a nonlinear nonlocal model of a Josephson junction. Phys. Lett. A 198(2), 105–112 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Alfimov, G.L., Popkov, A.F.: Magnetic vortices in a distributed Josephson junction with electrodes of finite thickness. Phys. Rev. B: Condens. Matter 52(6), 4503–4510 (1995)

    Article  Google Scholar 

  34. Mintz, R.G., Snapiro, I.B.: Dynamics of Josephson pancakes in layered superconductors. Phys. Rev. B: Condens. Matter 49(9), 6188–6192 (1994)

    Article  Google Scholar 

  35. Cunha, M.D., Konotop, V.V., Vázquez, L.: Small-amplitude solitons in a nonlocal sine-Gordon model. Phys. Lett. A 221(5), 317–322 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vázquez, L., Evans, W.A., Rickayzen, G.: Numerical investigation of a non-local sine-Gordon model. Phys. Lett. A 189(6), 454–459 (1994)

    Article  Google Scholar 

  37. Wu, G., Baleanu, D., Deng, Z., Zeng, S.: Lattice fractional diffusion equation in terms of a Riesz-Caputo difference. Physica A-Stat. Mech. Appl. 438, 335–339 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rabei, E.M., Rawashdeh, I.M., Muslih, S., Baleanu, D.: Hamilton-Jacobi formulation for systems in terms of Riesz’s fractional derivatives. Int. J. Theor. Phys. 50(5), 1569–1576 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Fahd, J., Thabet, A., Baleanu, D.: On Riesz-caputo formulation for sequential fractional variational principles. Abstr. Appl, Anal. 2012, 1–15 (2012). (Article Number: 890396)

    MathSciNet  MATH  Google Scholar 

  40. Magin, R.L., Abdullah, O., Baleanu, D., Zhou, X.J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. Magn. Reson. 190(2), 255–270 (2008)

    Article  Google Scholar 

  41. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, Q., Liu, F., Turner, I.: Computationally efficient numerical methods for time- and space-fractional Fokker-Planck equations. Phys. Scr. 2009, T136 (2009)

    Google Scholar 

  43. Çelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Debnath, L.: Integral Transforms and Their Applications. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  45. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Taylor and Francis, London (1993)

    MATH  Google Scholar 

  46. Saha Ray, S.: Fractional Calculus with Applications for Nuclear Reactor Dynamics. CRC Press, Taylor and Francis Group, Boca Raton, New York (2015)

    Google Scholar 

  47. Tarasov, V.E.: Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer, Berlin, Heidelberg, New York (2011)

    Google Scholar 

  48. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)

    Book  MATH  Google Scholar 

  49. Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fractional Calculus Appl. Anal. 1(2), 167–191 (1998)

    MathSciNet  MATH  Google Scholar 

  50. Saha Ray, S.: Numerical Analysis with Algorithms and Programming. CRC Press, Taylor and Francis Group, Boca Raton, New York (2016)

    MATH  Google Scholar 

  51. Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press Inc., New York, USA (1985)

    Google Scholar 

  52. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston (1994)

    Book  MATH  Google Scholar 

  53. Khan, Y., Taghipour, R., Falahian, M., Nikkar, A.: A new approach to modified regularized long wave equation. Neural Comput. Appl. 23, 1335–1341

    Article  Google Scholar 

  54. Jafari, H., Sayevand, K., Khan, Y., Nazari, M.: Davey-Stewartson equation with fractional coordinate derivatives. Sci World J 2013, 8 pp (2013). (Article ID 941645)

    Article  Google Scholar 

  55. Kaya, D.: A numerical solution of the Sine-Gordon equation using the modified decomposition method. Appl. Math. Comp. 143, 309–317 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  56. Wei, G.W.: Discrete singular convolution for the sine-Gordon equation. Physica D 137, 247–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  57. Batiha, B., Noorani, M.S.M., Hashim, I.: Numerical solution of sine-Gordon equation by variational iteration method. Phys. Lett. A 370, 437–440 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Liao, S.: The proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai (1992) (in English)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Odisha, India

    Santanu Saha Ray

Authors
  1. Santanu Saha Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Santanu Saha Ray .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Saha Ray, S. (2020). Numerical Solutions of Riesz Fractional Partial Differential Equations. In: Nonlinear Differential Equations in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1656-6_4

Download citation

Publish with us

Policies and ethics

Search

Navigation