Skip to main content
SpringerLink
Log in

A Linear B-Spline Approximation for a Class of Nonlinear Time and Space Fractional Partial Differential Equations

  • Chapter
  • First Online:
Numerical Solutions of Realistic Nonlinear Phenomena

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 31))

Abstract

This chapter is devoted to the numerical solution of partial differential equations with fractional derivatives. A linear B-spline approximates the fractional derivatives along with an upwind finite difference method and the Du Fort–Frankel algorithm for the temporal and spatial discretizations. The proposed techniques include an explicit Adams–Bashforth method to improve their numerical stability and accuracy. Several numerical problems, such as the fractional Kolmogorov–Petrovskii–Piskunov, Newell–Whitehead–Segel, and FitzHugh–Nagumo equations, are presented to illustrate the efficiency and reliability of the algorithms.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Similar content being viewed by others

References

  1. Carella, A.R., Dorao, C.A.: Least-Squares spectral method for the solution of a fractional advection–dispersion equation. J. Comput. Phys. 232(1), 33–45 (2013). Doi: https://doi.org/10.1016/j.jcp.2012.04.050

    Article  MathSciNet  Google Scholar 

  2. Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32(4), 1740–1760 (2010). Doi: https://doi.org/10.1137/090771715

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, W., Zhang, J., Zhang, J.: A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures. Fractional Calc. Appl. Anal. 16(1), 76–92 (2013). Doi: https://doi.org/10.2478/s13540-013-0006-y

  4. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics (2010). Doi: https://doi.org/10.1007/978-3-642-14574-2

    Book  MATH  Google Scholar 

  5. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29(1/4), 3–22 (2002). Doi: https://doi.org/10.1023/a:1016592219341

    Article  MathSciNet  Google Scholar 

  6. Du Fort, E.C., Frankel, S.P.: Conditions in the Numerical Treatment of Parabolic Differential Equations. Mathematical Tables and Other Aids to Computation, Vol. 7(43), pp. 135–152 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006). Doi: https://doi.org/10.1002/num.20112

    Article  MathSciNet  MATH  Google Scholar 

  8. Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45(2), 572–591 (2007). Doi: https://doi.org/10.1137/050642757

    Article  MathSciNet  MATH  Google Scholar 

  9. Ferr\(\acute {a}\)s, L.L., Ford, N.J., Morgado, M.L., Rebelo, M.: A numerical method for the solution of the time-fractional diffusion equation. In: International Conference on Computational Science and Its Applications. Springer, Cham (2014). Doi: https://doi.org/10.1007/978-3-319-09144-0-9

  10. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17(4), 257–278 (1955). Doi: https://doi.org/10.1007/bf02477753

    Article  Google Scholar 

  11. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961). Doi: https://doi.org/10.1016/s0006-3495(61)86902-6

    Article  Google Scholar 

  12. Fu, Z.J., Chen, W., Ling, L.: Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng. Anal. Boundary Elem. 57, 37–46 (2015). Doi: https://doi.org/10.1016/j.enganabound.2014.09.003

    Article  MathSciNet  MATH  Google Scholar 

  13. Jassim, H.K.: Homotopy perturbation algorithm using Laplace transform for Newell–Whitehead–Segel equation. Int. J. Adv. Appl. Math. Mech. 2(4), 8–12 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC (2015)

    Google Scholar 

  15. Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316, 614–631 (2016). Doi: https://doi.org/10.1016/j.jcp.2016.04.039

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212(2), 435–445 (2009). Doi: https://doi.org/10.1016/j.amc.2009.02.047

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015). Doi: https://doi.org/10.1016/j.jcp.2014.06.001

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. Moghaddam, B.P., Tenreiro Machado, J.A., Morgado, M.L.: Numerical approach for a class of distributed order time fractional partial differential equations. Appl. Numer. Math. 136, 152–162 (2019). Doi: https://doi.org/10.1016/j.apnum.2018.09.019

    Article  MathSciNet  MATH  Google Scholar 

  20. Moghaddam, B.P., Tenreiro Machado, J.A.: A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput. Math. Appl. 73(6), 1262–1269 (2017). Doi: https://doi.org/10.1016/j.camwa.2016.07.010

    Article  MathSciNet  MATH  Google Scholar 

  21. Moghaddam, B.P., Yaghoobi, S., Tenreiro Machado, J.A.: An extended predictor–corrector algorithm for variable-order fractional delay differential equations. J. Comput. Nonlinear Dyn. 11(6), 061001 (2016). Doi: https://doi.org/10.1115/1.4032574

  22. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962). Doi: https://doi.org/10.1109/jrproc.1962.288235

    Article  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, Elsevier, San Diego (1998)

    MATH  Google Scholar 

  24. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \(\mathbb {R}^2\). J. Comput. Appl. Math. 193(1), 243–268 (2006). Doi: https://doi.org/10.1016/j.cam.2005.06.005

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  26. Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A Stat. Mech. Appl. 388(21), 4586–4592 (2009). Doi: https://doi.org/10.1016/j.physa.2009.07.024

    Article  Google Scholar 

  27. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Institute for Nonlinear Science. Springer, New York (2003). Doi: https://doi.org/10.1007/978-0-387-21746-8

    Book  Google Scholar 

  28. Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014). Doi: https://doi.org/10.1137/130918174

    Article  MathSciNet  MATH  Google Scholar 

  29. Yong, Z.H.O.U., Jinrong, W., Lu, Z.: Basic Theory of Fractional Differential Equations. World Scientific (2016)

    Google Scholar 

  30. Zhao, Z., Li, C.: Fractional difference/finite element approximations for the time–space fractional telegraph equation. Appl. Math. Comput. 219(6), 2975–2988 (2012). Doi: https://doi.org/10.1016/j.amc.2012.09.022

    Article  MathSciNet  MATH  Google Scholar 

  31. Zheng, Y., Li, C., Zhao, Z.: A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 59(5), 1718–1726 (2010). Doi: https://doi.org/10.1016/j.camwa.2009.08.071

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46(2), 1079–1095 (2008). Doi: https://doi.org/10.1137/060673114

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760–1781 (2009). Doi: https://doi.org/10.1137/080730597

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

    Behrouz Parsa Moghaddam

  2. Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    José António Tenreiro Machado

  3. School of Engineering Technology, Eastern Michigan University, Ypsilanti, MI, USA

    Arman Dabiri

Authors
  1. Behrouz Parsa Moghaddam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José António Tenreiro Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arman Dabiri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Behrouz Parsa Moghaddam .

Editor information

Editors and Affiliations

  1. Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

  2. Department of Mathematics, Balıkesir University, Balıkesir, Turkey

    Necati Özdemir

  3. Department of Mathematics & Computer Science, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Moghaddam, B.P., Machado, J.A.T., Dabiri, A. (2020). A Linear B-Spline Approximation for a Class of Nonlinear Time and Space Fractional Partial Differential Equations. In: Machado, J., Özdemir, N., Baleanu, D. (eds) Numerical Solutions of Realistic Nonlinear Phenomena. Nonlinear Systems and Complexity, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-37141-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation