Skip to main content
Springer Nature Link
Log in

Applications of two numerical methods for solving inverse Benjamin–Bona–Mahony–Burgers equation

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this paper, two numerical techniques are presented to solve the nonlinear inverse generalized Benjamin–Bona–Mahony–Burgers equation using noisy data. These two methods are the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We show that the convergence rate of the proposed methods is \(\textit{O}(k^2+h^3)\) and \(\textit{O}\left( \frac{1}{M}\right) \) for the quartic B-spline and Haar wavelet method, respectively. In fact, this work considers a comparative study between quartic B-spline and Haar wavelet methods to solve some nonlinear inverse problems. Results show that an excellent estimation of the unknown functions of the nonlinear inverse problem has been obtained.

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
$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.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Ganji Z, Ganji D, Bararnia H (2009) Approximate general and explicit solutions of nonlinear bbmb equations by exp-function method. Appl Math Model 33(4):1836–1841

    MathSciNet  MATH  Google Scholar 

  2. Noor MA, Noor KI, Waheed A, Al-Said EA (2011) Some new solitonary solutions of the modified Benjamin–Bona–Mahony equation. Comput Math Appl 62(4):2126–2131

    MathSciNet  MATH  Google Scholar 

  3. Eilenberger G (2012) Solitons: mathematical methods for physicists, vol 19. Springer, Berlin

    MATH  Google Scholar 

  4. Whitham G (1974) Linear and nonlinear waves. Wiley, New York

    MATH  Google Scholar 

  5. Gray P, Scott S (1996) Chemical oscillations and instabilities. J Fluid Mech 314(1):406–406

    Google Scholar 

  6. Hasegawa A (2012) Plasma instabilities and nonlinear effects, vol 8. Springer, Berlin

    Google Scholar 

  7. Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform, vol 4. Siam, New Delhi

    MATH  Google Scholar 

  8. Wu X-HB, He J-H (2008) Exp-function method and its application to nonlinear equations. Chaos Solitons Fractals 38(3):903–910

    MathSciNet  MATH  Google Scholar 

  9. Ganji D, Afrouzi GA, Talarposhti R (2007) Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations. Phys Lett A 368(6):450–457

    MATH  Google Scholar 

  10. Abdollahzadeh M, Ghanbarpour M, Hosseini A, Kashani S (2010) Exact travelling solutions for Benjamin–Bona–Mahony–Burgers equations by (g’/g)-expansion method. Int J Appl Math Comput 3(1):70–76

    Google Scholar 

  11. Tari H, Ganji D (2007) Approximate explicit solutions of nonlinear bbmb equations by he’s methods and comparison with the exact solution. Phys Lett A 367(1–2):95–101

    MATH  Google Scholar 

  12. Benjamin TB, Bona JL, Mahony JJ (1972) Model equations for long waves in nonlinear dispersive systems. Phil Trans R Soc Lond A 272(1220):47–78

    MathSciNet  MATH  Google Scholar 

  13. Korteweg DJ, De Vries G (1895) Xli. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. London Edinb Dublin Philos Mag J Sci 39(240):422–443

    MATH  Google Scholar 

  14. Abbasbandy S, Shirzadi A (2010) The first integral method for modified Benjamin–Bona–Mahony equation. Commun Nonlinear Sci Numer Simul 15(7):1759–1764

    MathSciNet  MATH  Google Scholar 

  15. Omrani K (2006) The convergence of fully discrete galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation. Appl Math Comput 180(2):614–621

    MathSciNet  MATH  Google Scholar 

  16. Kannan R, Chung S (2002) Finite difference approximate solutions for the two-dimensional burgers’ system. Comput Math Appl 44(1–2):193–200

    MathSciNet  MATH  Google Scholar 

  17. Al-Khaled K, Momani S, Alawneh A (2005) Approximate wave solutions for generalized Benjamin–Bona–Mahony–Burgers equations. Appl Math Comput 171(1):281–292

    MathSciNet  MATH  Google Scholar 

  18. Zarebnia M, Parvaz R (2013) Cubic b-spline collocation method for numerical solution of the Benjamin–Bona–Mahony–Burgers equation. WASET Int J Math Computat Phys Electr Comput Eng 7(3):540–543

    Google Scholar 

  19. Aksan E (2006) Quadratic b-spline finite element method for numerical solution of the burgers’ equation. Appl Math Comput 174(2):884–896

    MathSciNet  MATH  Google Scholar 

  20. Samarskii AA, Vabishchevich PN (2008) Numerical methods for solving inverse problems of mathematical physics, vol 52. Walter de Gruyter, Berlin

    MATH  Google Scholar 

  21. Beck J, Blackwell B, StClair C (1985) Inverse heat conduction: Ill-posed problems. A Wiley-Interscience, New York

    MATH  Google Scholar 

  22. Pourgholi R, Rostamian M (2010) A numerical technique for solving ihcps using tikhonov regularization method. Appl Math Model 34(8):2102–2110

    MathSciNet  MATH  Google Scholar 

  23. Foadian S, Pourgholi R, Hashem Tabasi S (2018) Cubic b-spline method for the solution of an inverse parabolic system. Appl Anal 97(3):438–465

    MathSciNet  Google Scholar 

  24. Mazraeh HD, Pourgholi R, Houlari T (2017) Combining genetic algorithm and sinc-galerkin method for solving an inverse diffusion problem. TWMS J Appl Eng Math 7(1):33

    MathSciNet  MATH  Google Scholar 

  25. Pourgholi R, Saeedi A (2017) Applications of cubic b-splines collocation method for solving nonlinear inverse parabolic partial differential equations. Numer Meth Partial Differ Equ 33(1):88–104

    MathSciNet  MATH  Google Scholar 

  26. Pourgholi R, Saeedi A (2016) Solving a nonlinear inverse problem of identifying an unknown source term in a reaction-diffusion equation by adomian decomposition method. TWMS J Appl Eng Math 6(1):150

    MathSciNet  MATH  Google Scholar 

  27. Saeedi A, Pourgholi R (2017) Application of quintic b-splines collocation method for solving inverse rosenau equation with dirichlet’s boundary conditions. Eng Comput 33(3):335–348

    Google Scholar 

  28. Pourgholi R, Esfahani A, Houlari T, Foadian S (2017) An application of sinc-galerkin method for solving the tzou equation. Appl Comput Math 16(3):240–256

    MathSciNet  MATH  Google Scholar 

  29. Pourgholi R, Tabasi SH, Zeidabadi H (2018) Numerical techniques for solving system of nonlinear inverse problem. Eng Comput 34(3):487–502

    MATH  Google Scholar 

  30. Dehghan M, Yousefi SA, Rashedi K (2013) Ritz-galerkin method for solving an inverse heat conduction problem with a nonlinear source term via bernstein multi-scaling functions and cubic b-spline functions. Inverse Probl Sci Eng 21(3):500–523

    MathSciNet  MATH  Google Scholar 

  31. Isakov V (1990) Inverse source problems, vol 34. American Mathematical Soc, Providence

    MATH  Google Scholar 

  32. Pourgholi R, Dana H, Tabasi SH (2014) Solving an inverse heat conduction problem using genetic algorithm: sequential and multi-core parallelization approach. Appl Math Model 38(7–8):1948–1958

    MathSciNet  MATH  Google Scholar 

  33. Esfahani A, Pourgholi R (2014) Dynamics of solitary waves of the rosenau-rlw equation. Differ Equ Dyn Syst 22(1):93–111

    MathSciNet  MATH  Google Scholar 

  34. Aziz I, Khan F et al (2014) A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations. J Comput Appl Math 272:70–80

    MathSciNet  MATH  Google Scholar 

  35. Kumar M, Pandit S (2014) A composite numerical scheme for the numerical simulation of coupled burgers’ equation. Comput Phys Commun 185(3):809–817

    MathSciNet  MATH  Google Scholar 

  36. Patra A, Ray SS (2014) Two-dimensional haar wavelet collocation method for the solution of stationary neutron transport equation in a homogeneous isotropic medium. Ann Nucl Energy 70:30–35

    Google Scholar 

  37. Ray SS, Gupta A (2014) Comparative analysis of variational iteration method and haar wavelet method for the numerical solutions of burgers-huxley and huxley equations. J Math Chem 52(4):1066–1080

    MathSciNet  MATH  Google Scholar 

  38. Çelik İ (2013) Haar wavelet approximation for magnetohydrodynamic flow equations. Appl Math Model 37(6):3894–3902

    MathSciNet  MATH  Google Scholar 

  39. Ray SS (2012) On haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation. Appl Math Model 218(9):5239–5248

    MathSciNet  MATH  Google Scholar 

  40. Ray SS, Patra A (2013) Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system. Appl Math Comput 220:659–667

    MathSciNet  MATH  Google Scholar 

  41. Pourgholi R, Tavallaie N, Foadian S (2012) Applications of haar basis method for solving some ill-posed inverse problems. J Math Chem 50(8):2317–2337

    MathSciNet  MATH  Google Scholar 

  42. Pourgholi R, Foadian S, Esfahani A (2013) Haar basis method to solve some inverse problems for two-dimensional parabolic and hyperbolic equations. TWMS J Appl Eng Math 3(1):10–32

    MathSciNet  MATH  Google Scholar 

  43. Pourgholi R, Esfahani A, Foadian S, Parehkar S (2013) Resolution of an inverse problem by haar basis and legendre wavelet methods. Int J Wavelets Multiresolut Inf Process 11(05):1350034

    MathSciNet  MATH  Google Scholar 

  44. Foadian S, Pourgholi R, Tabasi SH, Damirchi J (2017) The inverse solution of the coupled nonlinear reaction-diffusion equations by the haar wavelets. Int J Comput Math 96:1–21

    MathSciNet  Google Scholar 

  45. Prenter PM et al (1975) Splines and variational methods. John Wiley, NewYork

    MATH  Google Scholar 

  46. Rubin SG, Graves RA (1975) A cubic spline approximation for problems in fluid mechanics

  47. Hall C (1968) On error bounds for spline interpolation. J Approx Theory 1(2):209–218

    MathSciNet  MATH  Google Scholar 

  48. Rudin W et al (1976) Principles of mathematical analysis, vol 3. McGraw-hill, New York

    MATH  Google Scholar 

  49. Rashidinia J, Ghasemi M, Jalilian R (2010) A collocation method for the solution of nonlinear one-dimensional parabolic equations. Math Sci 4:87–104

    MathSciNet  MATH  Google Scholar 

  50. Cabeza JMG, García JAM, Rodríguez AC (2005) A sequential algorithm of inverse heat conduction problems using singular value decomposition. Int J Therm Sci 44(3):235–244

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran

    Akram Saeedi, Saedeh Foadian & Reza Pourgholi

Authors
  1. Akram Saeedi
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Saedeh Foadian
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Reza Pourgholi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Reza Pourgholi.

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

Saeedi, A., Foadian, S. & Pourgholi, R. Applications of two numerical methods for solving inverse Benjamin–Bona–Mahony–Burgers equation. Engineering with Computers 36, 1453–1466 (2020). https://doi.org/10.1007/s00366-019-00775-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-019-00775-4

Keywords

Mathematics subject classification