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The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

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Abstract

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.

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References

  1. G. Hariharan, K. Kannan, Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 15, 284–292 (2009)

    Article  Google Scholar 

  2. A. Kolmogoroff, I. Petrovskii, N. Piscounoff, Etude de l’equation de la diffusion aveccroissance de la quantiti de matiere et son application a un probleme biologique. Moscow University. Bull. Math. 1, 1–25 (1937)

    Google Scholar 

  3. A.M. Wazwaz, A. Gorguis, An analytic study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154(3), 609–620 (2004)

    Article  Google Scholar 

  4. A. Meena, A. Eswari, L. Rajendran, Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations. J. Math. Chem. 48(2), 179–186 (2010). doi:10.1007/s10910-009-9659-5

    Article  CAS  Google Scholar 

  5. S.E.C.E.R.J. Aydin, Solving time-fractional reaction-diffusion equation by reduced differential transform method. J. Comput. Eng. Technol. 3(1), 19–22 (2012)

    Google Scholar 

  6. A. Meena, L. Rajendran, Mathematical modeling of amperometric and potentiometric biosensorsand system of non-linear equations - Homotopy perturbation approach. J. Electroanal. Chem. 644, 50–59 (2010)

    Article  CAS  Google Scholar 

  7. V. Ananthaswamy, A. Eswari, L. Rajendran, Nonlinear reaction-diffusion process in a thin membrane and Homotopy analysis method. Int. J. Autom. Control Eng. 2(1), 10–18 (2013)

    Google Scholar 

  8. X.Y. Wang, Exact and explicit solitary wave solutions for the generalized Fisher’s equation. Phys. Lett. A 131(4/5), 277–279 (1988)

    Google Scholar 

  9. M.B.A. Monsour, Travelling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Appl. Math. Model. 32, 240–247 (2008)

    Article  Google Scholar 

  10. D. Olmos, A pseudo-spectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193, 219–242 (2006)

    Article  Google Scholar 

  11. A.M. Wazwaz, Travelling wave solution of generalized form of burgers-Huxley eqaution. Appl. Math. Comput. 169, 639–656 (2005)

    Article  Google Scholar 

  12. A. Khan et al., On approaimate solutions for the time-fractional reaction diffusion equation of Fisher type. Int. J. Phys. Sci. 6(10), 2483–2496 (2011)

    CAS  Google Scholar 

  13. L. Rajendran, R. Senthamarai, Traveling-wave solution of non-linear coupled reaction diffusion equation arising in mathematical chemistry. J. Math. Chem. 46(2), 550–561 (2009). doi:10.1007/s10910-008-9479

    Article  CAS  Google Scholar 

  14. R. Baronas, F. Ivanauskas, J. Kulys, Modeling dynamics of amperometric biosensors in batch and flow injection analysis. J. Math. Chem 32(2), 225–237 (2002)

    Article  CAS  Google Scholar 

  15. T. Mavoungou, Y. Cherruault, Numerical study of Fisher’s equation by Adomian-decomposition method. Math. Comput. Model. Int. J. 19, 89–95 (1994)

    Article  Google Scholar 

  16. A.M. Wazwaz, Partial Differental Equations: Methods and Applications (Balkema Publishers, Amsterdam, 2002)

    Google Scholar 

  17. W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563–568 (1996)

    Google Scholar 

  18. Zeppetella Ablowitz, Explicit solutions of the Fisher’s equation for a special wave speed. Bull. Math. 41, 835–840 (1979)

    Google Scholar 

  19. S. Puri, Singular perturbation analysis of the Fisher’s equation. Phy. Rev. A 43, 7031–7039 (1991)

    Google Scholar 

  20. G.F. Carey, Yun Shen, Least square finite element method for Fisher’s reaction diffusion equation. Num. Method Part. Diff. Equ. 11, 175–186 (1995)

    Article  Google Scholar 

  21. K. Al-khaled, Numerical study of Fisher’s equation-diffusion equation by the Sinc-collocation method. J. Comput. Appl. Math. 137(2), 245–255 (2001)

    Article  Google Scholar 

  22. R.C. Mittal, Ram Jiwari, Numerical study of fisher’s equation by using differential quadrature method. Int. J. Inf. Syst. Sci. 5, 143–160 (2009)

    Google Scholar 

  23. G. Hariharan, K. Kannan, Haar wavelet method for solving some nonlinear Parabolic equations. J. Math. Chem. 48(4), 1044–1061 (2010)

    Article  CAS  Google Scholar 

  24. F. Yin, J. Song, X. Cao, F. Lu, Couple of the variational iteration method and Legendre wavelets for nonlinear partial differential equations. J. Appl. Math., Article ID 157956 (2013)

  25. E. Hesameddini, S. Shekarpaz, Wavelet solutions of the Klein-Gordon equation. J. Mahani Math. Res. Centre 1(1), 29–45 (2012)

    Google Scholar 

  26. M. Razzaghi, S. Yousefi, The Legendre wavelets direct method for variational problems. Math. Comput. Simul. 53, 185–192 (2000)

    Article  Google Scholar 

  27. 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 

  28. F. Yin, J. Song, F. Lu, H. Leng, A coupled method of laplace transform and Legendre wavelets for Lane-Emden-Type differential equations. J. Appl. Math. 2012, Article ID 163821 (2012). doi:10.1155/2012/163821

  29. F. Mohammadi, M.M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. J. Frankl. Inst. 348, 1787–1796 (2011)

    Article  Google Scholar 

  30. H. Parsian, Two dimension Legendre wavelets and operational matrices of integration. Acta. Math. Acad. Paedagog. Nyiregyhziens 21, 101–106 (2005)

    Google Scholar 

  31. G.U.O. Ben-Yu, Z. **ao-Yang, \(^{ (2}\)Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions. Appl. Num. Math. 57(4), 455–471 (2007)

    Article  Google Scholar 

  32. M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, F. Mohammadi, Wavelet collocation method, for solving multiorder fractional differential equations. J. Appl. Math. 2012, Article ID 54240 (2012). doi:10.1155/2012/542401

  33. M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration. Int. J. Syst. Sci. 32, 495–502 (2001)

    Google Scholar 

  34. S.A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type. Appl. Math. Comput. 181, 1417–1442 (2006)

    Article  Google Scholar 

  35. K. Maleknejad, S. Sohrabi, Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets. Appl. Math. Comput. 186, 836–843 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  37. H. Jafari, M. Soleymanivaraki, M.A. Firoozjaee, Legendre wavelets for solving fractional differential equations. J. Appl. Math. 7(4), 65–70 (2011)

    Google Scholar 

  38. Yin Yang, Solving a nonlinear multi-order fractional differential equation using Legendre pseudo-spectral method. Appl. Math. 4, 113–118 (2013). doi:10.4236/am.2013.41020

    Article  Google Scholar 

  39. U. Lepik, Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput. 185, 695–704 (2007)

    Article  Google Scholar 

  40. Shi Zhi, Deng Li Yuan, Haar wavelet method for solving the convection-diffusion equation. Math. Appl. 21, 98–104 (2008)

    Google Scholar 

  41. 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 

  42. G. Hariharan, The homotopy analysis method applied to the Kolmogorov-Petrovskii-Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000 (2013). doi:10.1007/s10910-012-0132-5

    Article  CAS  Google Scholar 

  43. E. Hesameddini, S. Shekarpaz, Wavelet solutions of the second Painleve equation. Iran. J. Sci. Technol. IJST A4, 287–291 (2011)

    Google Scholar 

  44. E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University Press, Cambridge, 1990)

    Google Scholar 

  45. J. Sherratt, On the transition from initial data traveling waves in the Fisher-KPP equation. Dyn. Stab. Syst. 13(2), 167–174 (1998)

    Article  Google Scholar 

  46. A.G. Nikitin, R.J. Wiltshire, Symmetries of systems of nonlinear reaction -Diffusion equation. prcoceeding of institute of Mathematics of NAS of Ukraini, 30(1), 47–59 (2000).

  47. M. Bastani, D.K. Salkuyeh, A highly accurate method to solve Fishers equation. J. Phys. 78, 335–346 (2012)

    Google Scholar 

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Authors and Affiliations

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

    M. Mahalakshmi, G. Hariharan & K. Kannan

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  1. M. Mahalakshmi
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  2. G. Hariharan
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  3. K. Kannan
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Correspondence to G. Hariharan.

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Mahalakshmi, M., Hariharan, G. & Kannan, K. 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/s10910-013-0216-x

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  • DOI: https://doi.org/10.1007/s10910-013-0216-x

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