SpringerLink
Log in

An efficient operational matrix approach for the solutions of Burgers' and fractional Burgers' equations using wavelets

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper, we have developed an efficient Chebyshev wavelet-Picard method for solving Burgers’ equation and time-fractional Burgers’ equation. In order to get the solution, we decompose the nonlinear PDE by Picard method and then convert into a system of algebraic equations. Convergence analysis of the proposed wavelet method is also discussed. The proposed wavelet solutions are compared with the solutions obtained by Cole-Hopf transformation and Haar wavelet method. Satisfactory agreement with HAM and analytical solution is observed. Some illustrative examples are given to demonstrate the efficiency and accuracy of the proposed wavelet method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /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.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. J.D. Logan, An Introduction to Nonlinear Partial Differential Equations (Wiley—Inter Science, New York, 1994).

    Google Scholar 

  2. G. Adomian, The diffusion-Brusselator equation. Comput. Math. Appl. 29, 1–3 (1995)

    Article  Google Scholar 

  3. J.M. Burger, A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. I, 171–199 (1948)

    Article  Google Scholar 

  4. J.D. Cole, On a quasilinear parabolic equations occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951)

    Article  Google Scholar 

  5. Ö. Oruç, F. Bulut, A. Esen, A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’equation. J. Math. Chem. 53, 1592–1607 (2015)

    Article  CAS  Google Scholar 

  6. M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burgers’ equations using Adomian-Pade technique. Appl. Math. Comput. 189, 1034–1047 (2007)

    Google Scholar 

  7. M.M. Rashidi, E. Erfani, New analytic method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM. Comput. Phys. Comm. 180, 1539–1544 (2009)

    Article  CAS  Google Scholar 

  8. P.C. Jain, D.N. Holla, Numerical solution of coupled Burgers equations. Int. J. Numer. Methods Eng. 12, 213–222 (1978)

    Google Scholar 

  9. F.W. Wubs, E.D.de.Goede, , An explicit-implicit method for a class of time-dependent partial differential equations. Appl. Numer. Math. 9, 11–19 (1992)

    Article  Google Scholar 

  10. C.A.J. Fletcher, Generating exact solutions of the two-dimensional Burgers’ equation. Int. J. Numer. Methods Fluids 3, 213–216 (1983)

    Article  Google Scholar 

  11. M.M. Rashidi, G. Domairry, S. Dinarvand, Approximate solutions for the Burgers’ and regularized long wave equations by means of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 708–717 (2009)

    Article  Google Scholar 

  12. M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burgers’ and coupled Burgers equations. J. Comput. Appl. Math. 181(2), 245–251 (2005)

    Article  Google Scholar 

  13. E. Varoglu, W.D. Finn, Space-time finite elements incorporating characteristics for the Burgers’ equation. Int. J. Numer. Methods Eng. 16, 171–184 (1980)

    Article  Google Scholar 

  14. M. Gulsu, A finite difference approach for solution of Burgers’ equation. Appl. Math. Comput. 175, 1245–1255 (2006)

    Google Scholar 

  15. C. Basdevant, M. Deville, P. Haldenwang, J.M. Lacroix, J. Ouazzani, R. Peyret, P. Orlandi, A. Patera, Spectral and finite difference solutions of the Burgers’ equation. Comput. Fluids 14, 23–41 (1986)

    Article  Google Scholar 

  16. E.N. Aksan, A. Ozdes, T. Ozis, A numerical solution of Burgers’ equation on least squares approximation. Appl. Math. Comput 176, 270–279 (2006)

    Google Scholar 

  17. K. Mohan, K. Kadalbajoo, A. Awasthi, A numerical method based on Crank–Nicholson scheme for Burgers’ equation. Appl. Math. Comput. 182, 1430–1442 (2006)

    Google Scholar 

  18. T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, Higher-order accurate numerical solution of unsteady Burgers’ equation. Appl. Math. Comput. 250, 701–707 (2015)

    Google Scholar 

  19. A. Gorguis, A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations. Appl. Math. Comput. 173, 126–136 (2006)

    Google Scholar 

  20. U. Lepik, Numerical solution of differential equations using Haar wavelets. Math. Comp. Simul. 68, 127–143 (2005)

    Article  Google Scholar 

  21. M.-Q. Chen, C. Hwang, Y.-P. Shin, The computation of wavelet-Galerkin a approximation on a bounded interval. Int. J. Numer. Methods Eng. 39, 2921–2944 (1996)

    Article  Google Scholar 

  22. E.R. Benton, G.W. Platzman, A table of solution of the one-dimensional Burgers’ equation. Q. Appl. Math. 9, 195–212 (1972)

    Article  Google Scholar 

  23. T. Ozis, A. Ozodes, A direct variational methods applied to Burgers’ equation. J. Comput. Appl. Math. 71, 163–175 (1996)

    Article  Google Scholar 

  24. E.N. Aksan, A. Ozdes, A numerical solution of Burgers’ equation. Appl. Math. Comput. 156, 395–402 (2004)

    Google Scholar 

  25. J. Caldwell, P. Wanless, A.E. Cook, A finite element approach to Burgers’ equation. Appl. Math. Model. 5, 381–385 (1981)

    Article  Google Scholar 

  26. J. Caldwell, P. Smith, Solution of Burgers’ equation with a large Reynolds number. Appl. Math. Modell. 6(5), 381–385 (1982)

    Article  Google Scholar 

  27. R. Saunders, J. Caldwell, P. Wanless, A variational-Iterative scheme applied to Burgers’ equation. IMA J. Numer. Anal. 4, 349–362 (1984)

    Article  Google Scholar 

  28. S.H. Shu-Sen, Seokchan Kim, Gyungsoo Woo, Sucheol Yi, Numerical solution of one-dimensional Burgers’ equation using reproducing Kernel function. J. Comput. Appl. Math. 214, 417–434 (2008)

    Article  Google Scholar 

  29. G.W. Wei, Y. Gu, Conjugate filter approach for solving Burgers’ equation. J. Comput. Appl. Math. 149(2), 439–456 (2002)

    Article  Google Scholar 

  30. U. Lepik, Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci. Phys. Math. 56(1), 28–46 (2007)

    Article  Google Scholar 

  31. M. Gulsu, T. Ozis, Numerical solution of Burgers’ equation with restrictive Taylor approximation. Appl. Math. Comput. 171(2), 1192–1200 (2005)

    Google Scholar 

  32. Zhi Shi, Li-Yuan Deng, Qing-Jiang Chen, Numerical solution of differential equations by using Haar wavelets, Proc. of the 2007. In: International conference on wavelet analysis and pattern recognistion, Bei**g, China, 2–4 Nov. 2007

  33. K.R. Sharma, Role of turbulence, time dependence on heat transfer in circulating fluidized beds, 90th AIChe Annual Meeting, Miami, FL, USA

  34. P. Jamet, R. Bonnerot, Numerical solution of the Eulerian equations of compressible flow by finite element method which follows the boundary and interfaces. J. Comput. Phys. 18, 21–45 (1975)

    Article  Google Scholar 

  35. T. Ozis, E.N. Aksan, A. Ozdes, A finite element approach for solution of Burgers’ equation. Appl. Math. Comp. 139, 417–428 (2003)

    Article  Google Scholar 

  36. R. Jiwari, A Haar wavelet quasi-linearization approach for numerical simulation of Burgers’ equation. Comput. Phys. Commun. 183, 2413–2423 (2012)

    Article  CAS  Google Scholar 

  37. V. Comincioli, G. Naldi, T. Scapolla, A wavelet-based method for numerical solution of nonlinear evolution equations. Appl. Numer. Math. 33, 291–297 (2000)

    Article  Google Scholar 

  38. A. Haar, Zur theorie der orthogonalen Funktionsysteme. Math. Annal 69, 331–371 (1910)

    Article  Google Scholar 

  39. G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, 1997).

    Google Scholar 

  40. S. Bertoluzza, G. Naldi, J.C. Ravel, Wavelets and the numerical solution of Boundary value problems on the interval, in Wavelets: theory algorithms and applications. ed. by K. Chui, L. Montefusco, L. Puccio (Academic Press, Cambridge, 1994), pp. 425–428

    Google Scholar 

  41. C. Cattani, Haar wavelet spline. J. Interdiscip Math. 4, 35–47 (2001)

    Article  Google Scholar 

  42. C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems. IEE Proc Control Theory Appl 144, 87–94 (1997)

    Article  Google Scholar 

  43. K.R. Catleman, Digital image processing. Englewood Cliffs: Prentice- Hall. (1996)

  44. B.J. Falkowski, Mutual relations between arithmetic and Haar functions. Proc. IEEE Int. Symp. Circ. System (31st ISCAS), Vol. V .Monterey, CA, USA, June, p.138–41 1998

  45. B.J. Falkowski, C.H. Chang, Properties and applications of paired Haar transform Proc. IEEE Int. Conf. Inform, Commun Signal Process (Ist ICICS), Vol. 1. Singapore, September p.48–51 1997

  46. C.H. Hsiao, State analysis of linear time delayed systems via Haar wavelets. Math. Comp. Simul. 44, 457–470 (1997)

    Article  Google Scholar 

  47. C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems. Math. Comput. Simul. 57, 347–353 (2001)

    Article  Google Scholar 

  48. G. Hariharan, K. Kannan, K.R. Sharma, Haarwavelet in estimating the depth profile of soil temperature. Appl. Math. Comput. 210, 119–225 (2009)

    Google Scholar 

  49. G. Hariharan, K. Kannan, K.R. Sharma, Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211, 284–292 (2009)

    Google Scholar 

  50. G. Hariharan, K. Kannan, A comparative study of a Haar wavelet method and a restrictive Taylor’s series method for solving convection–diffusion equations. Int. J. Comput. Methods Eng. Sci. Mech. 11(4), 173–184 (2010)

    Article  Google Scholar 

  51. G. Hariharan, K. Kannan, Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering. Appl. Math. Model. 38(1), 799–813 (2014)

    Article  Google Scholar 

  52. G. Hariharan, K. Kannan, An overview of Haar wavelet method for solving differential and integral equations. World Appl. Sci. J. 23(12), 1–14 (2013)

    Google Scholar 

  53. R. Rajaraman, G. Hariharan, An efficient wavelet based spectral method to singular boundary value problems. J. Math. Chem. (2015). https://doi.org/10.1007/s10910-015-0536-0

    Article  Google Scholar 

  54. R. Rajaraman, G. Hariharan, A new coupled wavelet-based method applied to the nonlinear reactions–diffusion equation arising in mathematical chemistry. J. Math. Chem. 51(9), 2386–2400 (2013). https://doi.org/10.1007/s10910-013-0217-9

    Article  CAS  Google Scholar 

  55. R. Rajaraman, G. Hariharan, An efficient wavelet based approximation method to gene propagation model arising in population biology. J. Membr. Biol. 247, 561–570 (2014). https://doi.org/10.1007/s00232-014-9672-x

    Article  CAS  PubMed  Google Scholar 

  56. G. Hariharan, An efficient wavelet based approximation method to water quality assessment model in a uniform channel. Ain Shams Eng. J. 5, 525–532 (2014)

    Article  Google Scholar 

  57. G. Hariharan, An efficient Legendre wavelet based approximation method for a few-Newell and Allen–Cahn equations. J. Membr. Biol. 247(5), 371–380 (2014)

    Article  CAS  PubMed  Google Scholar 

  58. M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51, 2361–2385 (2013). https://doi.org/10.1007/s10910-013-0217-9

    Article  CAS  Google Scholar 

  59. M.-Q. Chen, C. Hwang, Y.-P. Shih, The computation of Wavelet-Galerkin approximation on a bounded interval. Int. J. Numer. Methods Eng. 39, 2921–2944 (1996)

    Article  Google Scholar 

  60. A. Avudainayagam, C. Vani, Wavelet-Galerkin solutions of quasilinear hyperbolic conservation equations. Commun. Numer. Methods Eng. 15, 589–601 (1999)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the Naval Research Board (Project No.:NRB447/SC/19-20) New Delhi for their financial support. We also acknowledge SASTRA Deemed University, Thanjavur for extending infrastructure support to carry out the study.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur, Tamilnadu, 613 401, India

    G. Hariharan, G. Swaminathan & B. Sripathy

Authors
  1. G. Hariharan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Swaminathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Sripathy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Hariharan.

Additional information

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

Hariharan, G., Swaminathan, G. & Sripathy, B. An efficient operational matrix approach for the solutions of Burgers' and fractional Burgers' equations using wavelets. J Math Chem 59, 554–573 (2021). https://doi.org/10.1007/s10910-020-01206-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-020-01206-2

Keywords

Search

Navigation