Skip to main content
SpringerLink
Log in

An efficient Chebyshev wavelet based analytical algorithm to steady state reaction–diffusion models arising in mathematical chemistry

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

Abstract

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equation containing a nonlinear term related to non-Michaelis–Menten kinetics of the reaction. In this paper, an efficient Chebyshev wavelet based approximation method is introduced for solving the steady state reaction–diffusion equations. Illustrative examples are given to demonstrate the validity and applicability of the method. The power of the manageable method is confirmed. Moreover the use of Chebyshev wavelet method is found to be simple, flexible, efficient, small computation costs and computationally attractive.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. R. Jebari, I. Ganmi, A. Boukricha, Adomian decomposition method for solving nonlinear diffusion equation with convection term. Int. J. Pure Appl. Sci. Technol. 12(1), 49–58 (2012)

    Google Scholar 

  2. A.S. Arife, The modified variational iteration transform method (MVITM) for solve non linear partial differential equation (NLPDE). World Appl. Sci. J. 12(12), 2274–2278 (2011)

    Google Scholar 

  3. O. Kiymaz, Variational iteration method for a class of nonlinear differential equations. Int. J. Contemp. Math. Sci. 5(37), 1819–1826 (2010)

    Google Scholar 

  4. V.G. Gupta, S. Gupta, Application of homotopy analysis method for solving nonlinear Cauchy problem. Surv. Math. Appl. 7, 105–116 (2012)

    Google Scholar 

  5. F. Mabood, Comparison of optimal homotopy asymptotic method and homotopy perturbation method for strongly nonlinear equation. J. Assoc. Arab. Univ. Basic Appl. Sci. 16, 21–26 (2014)

    Google Scholar 

  6. F. Scheller, F. Schubeert, Biosensor, vol. 7 (Elsevier, Amsterdam, 1988)

    Google Scholar 

  7. R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. The Theory of the Steady State (Clarendon Press, Oxford, 1975)

    Google Scholar 

  8. L.K. Bieniasz, D. Britz, Recent developments in digital simulation of electroanalytical experiments. Pol. J. Chem. 78, 1195–1219 (2004)

    CAS  Google Scholar 

  9. G. Hariharan, K. Kannan, Review of wavelet methods for the solution of reaction diffusion problems in science and engineering. Appl. Math. Model. 38, 799–813 (2013)

    Article  Google Scholar 

  10. 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 

  11. 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  Google Scholar 

  12. G. Hariharan, K. Kannan, Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211(2), 284–292 (2009)

    Article  Google Scholar 

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

    Google Scholar 

  14. G. Hariharan, R. Rajaraman, 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)

    Article  CAS  Google Scholar 

  15. 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). doi:10.1007/s10910-013-0217-9

    Article  CAS  Google Scholar 

  16. P. Pirabaharan, R.D. Chandrakumar, G. Hariharan, An efficient wavelet based approximation method for estimating the concentration of species and effectiveness factors in porous catalysts. MATCH 73(3), 705–727 (2015)

    Google Scholar 

  17. M. Mahalakshmi, G. Hariharan, An efficient wavelet based approximation method to steady state reaction–diffusion model arising in mathematical chemistry. J. Membr. Biol. 247(3), 263–271 (2014)

    Article  CAS  Google Scholar 

  18. P. Manimozhi, A. Subbiah, L. Rajendran, Solution of steady-state substrate concentration in the action of biosensor response at mixed enzyme kinetics. Sensors Actuators B 147, 290–297 (2010)

    Article  CAS  Google Scholar 

  19. C.V. Pao, Mathematical analysis of enzyme-substrate reaction diffusion in some biochemical systems. Nonlinear Anal. Theory 4(2), 369–392 (1979)

    Article  Google Scholar 

  20. G. Rahamathunsia, L. Rajendran, Modeling of non-linear reaction–diffusion processes of amperometric polymer-modified electrodes. J. Theor. Comput. Chem. 7, 113–138 (2008)

    Article  Google Scholar 

  21. C.V. Pao, Monotone iterative methods for finite difference system of reaction–diffusion equations. Numerische Mathematik 46(4), 571–586 (1985)

    Article  Google Scholar 

  22. R. Baronas, F. Ivanauskas, J. Kulys, M. Sapagovas, Modeling of amperometric biosensors with rough surface of the enzyme membrane. J. Math. Chem. 34, 227–242 (2003)

    Article  CAS  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the referees for their valuable comments. This work was supported by the DST-SERB Project, Government of India (Project No. SB/FTP/MS-012/2013). Our hearty thanks are due to Prof. R. Sethuraman, Vice-Chancellor, SASTRA University, Dr. S. Vaidhyasubramaniam, Dean/Planning and development and Dr. S. Swaminathan, Dean/Sponsored research for their kind encouragement and for providing good research environment.

Author information

Authors and Affiliations

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

    M. Mahalakshmi & G. Hariharan

Authors
  1. M. Mahalakshmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Hariharan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Hariharan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahalakshmi, M., Hariharan, G. An efficient Chebyshev wavelet based analytical algorithm to steady state reaction–diffusion models arising in mathematical chemistry. J Math Chem 54, 269–285 (2016). https://doi.org/10.1007/s10910-015-0560-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-015-0560-0

Keywords

Search

Navigation