Skip to main content
Log in

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this article, we compute numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete \(E_{2}\), \(E_{\infty }\) and \(E_{\text {rms}}\) error norms. Varying the number of nodal points M and time step-size \(\Delta t\), convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Bhatt HP, Khaliq AQM (2016) Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation. Comput Phys Commun 200:117–138

    Article  MathSciNet  Google Scholar 

  • Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent, part II. J R Astral Soc 13:529–539

    Article  Google Scholar 

  • Chen Y, An H-L (2008) Numerical solutions of coupled Burgers equations with time-and space-fractional derivatives. Appl Math Comput 200:87–95

    MathSciNet  MATH  Google Scholar 

  • Chen W, Ye L, Sun H (2010) Fractional diffusion equations by the Kansa method. Comput Math Appl 59:1614–1620

    Article  MathSciNet  Google Scholar 

  • Esipov SE (1995) Coupled Burgers’ equations: a model of polydispersive sedimentation. Phys Rev E 52:3711–3718

    Article  Google Scholar 

  • Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, River Edge

    Book  Google Scholar 

  • Fujita Y (1990) Cauchy problems of fractional order and stable processes. Jpn J Appl Math 7(3):459–476

    Article  MathSciNet  Google Scholar 

  • Golbabai A, Mohebianfar E, Rabiei H (2014) On the new variable shape parameter strategies for radial basis functions. Comput Appl Math 34(2):691–704

    Article  MathSciNet  Google Scholar 

  • Haq S, Hussain M (2018) Selection of shape parameter in radial basis functions for solution of time-fractional Black–Scholes models. Appl Math Comput 335:248–263

    MathSciNet  MATH  Google Scholar 

  • Hayat U, Kamran A, Ambreen B, Yildirim A, Mohyud-din ST (2013) On system of time-fractional partial differential equations. Walailak J Sci Technol 10(5):437–448

    Google Scholar 

  • Hilfer R (1995) Foundations of fractional dynamics. Fractals 3(3):549–556

    Article  Google Scholar 

  • Hilfer R (2000) Fractional diffusion based on Riemann–Liouville fractional derivative. J Phys Chem 104:3914–3917

    Article  Google Scholar 

  • Hussain M, Haq S, Ghafoor A (2019) Meshless spectral method for solution of time-fractional coupled KdV equations. Appl Math Comput 341:321–334

    MathSciNet  MATH  Google Scholar 

  • Kansa EJ (1990) Multiquadrics–a scattered data approximation scheme with application to computation fluid dynamics, II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations. Comput Math Appl 19:149–161

    Article  MathSciNet  Google Scholar 

  • Karkowski J (2013) Numerical experiments with the Bratu equation in one, two and three dimensions. Comput Appl Math 32(2):231–244

    Article  MathSciNet  Google Scholar 

  • Kumar M, Pandit S (2014) A composite numerical scheme for the numerical solution of coupled Burgers’ equation. Comput Phys Commun 185(3):1304–1313

    Article  Google Scholar 

  • Liew KJ, Ramli A, Majid AA (2017) Searching for the optimum value of the smoothing parameter for a radial basis function surface with feature area by using the bootstrap method. Comput Appl Math 36(4):1717–1732

    Article  MathSciNet  Google Scholar 

  • Liu GR, Gu TY (2005) An introduction to meshfree methods and their programming. Springer Press, Berlin

    Google Scholar 

  • Liu J, Huo G (2011) Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method. Appl Math Comput 217:7001–7008

    MathSciNet  MATH  Google Scholar 

  • Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics. Springer, New York, pp 291–348

    Chapter  Google Scholar 

  • Mardani A, Hooshmandasl MR, Heydari MH, Cattani C (2018) A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Comput Math Appl 75(1):122–133

    Article  MathSciNet  Google Scholar 

  • Micchelli CA (1986) Interpolation of scattered data: distance matrix and conditionally positive definite functions. Construct Approx 2:11–22

    Article  Google Scholar 

  • Mittal RC, Jiwari R (2016) Numerical simulation of reaction–diffusion systems by modified cubic B-spline differential quadrature method. Chaos Soliton Fractals 92:9–19

    Article  MathSciNet  Google Scholar 

  • Nee J, Duan J (1998) Limit set of trajectories of the coupled viscous Burgers’ equations. Appl Math Lett 11(1):57–61

    Article  MathSciNet  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, p 198

    MATH  Google Scholar 

  • Shivanian E (2015) A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms. Eng Anal Bound Elem 54:1–12

    Article  MathSciNet  Google Scholar 

  • West BJ, Bologna M, Grigolini P (2003) Physics of fractal operators. Springer, New York

    Book  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable suggestions which improved the quality of the work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sirajul Haq.

Additional information

Communicated by José Tenreiro Machado.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hussain, M., Haq, S., Ghafoor, A. et al. Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method. Comp. Appl. Math. 39, 6 (2020). https://doi.org/10.1007/s40314-019-0985-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-019-0985-3

Keywords

Mathematics Subject Classification

Navigation