Skip to main content
SpringerLink
Log in

Numerical solution of some stiff systems arising in chemistry via Taylor wavelet collocation method

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

Abstract

This paper presents the innovative Taylor wavelet collocation method (TWCM) for the stiff systems arising in chemical reactions. In this technique, first, we generated the functional matrix of integration (FMI) for the Taylor wavelets. Using this FMI, the Taylor wavelet collocation method is proposed to obtain the numerical approximation of stiff systems in the form of a system of ordinary differential equations (SODEs). This method converts the SODEs into a set of algebraic equations, which can be solved by the Newton–Raphson method. To demonstrate the simplicity and effectiveness of the presented approach, numerical results are obtained. Graphs and tables illustrate the created strategy's effectiveness and consistency. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique, and a comparison is made with the current results. Results reveal that the newly selected strategy is superior to previous approaches regarding precision and effectiveness in the literature. Most semi-analytical and numerical methods work based on controlling parameters, but this technique is free from controlling parameters. Also, it is easy to implement and consumes less time to handle the system. The suggested wavelet-based numerical method is computationally appealing, successful, trustworthy, and resilient. All computations have been made using the Mathematica 11.3 software. The convergence of this strategy is explained using theorems.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Data availability and materials

The data supporting this study's findings are available within the article.

References

  1. L.F. Shampine, S. Thompson, Stiff systems. Scholarpedia 2(3), 2855 (2007)

    Google Scholar 

  2. J. Carroll, A matricial exponentially fitted scheme for the numerical solution of stiff initial-value problems. Comput. Math. Appl. 26(4), 57–64 (1993)

    Google Scholar 

  3. G. Hojjati, M.R. Ardabili, S.M. Hosseini, A-EBDF: an adaptive method for the numerical solution of stiff systems of ODEs. Math. Comput. Simul. 66(1), 33–41 (2004)

    Google Scholar 

  4. J.R. Cash, the integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Comput. Math. Appl. 9(5), 645–657 (1983)

    Google Scholar 

  5. S.M. Hosseini, G. Hojjati, Matrix-free MEBDF method for the solution of stiff systems of ODEs. Math. Comput. Model. 29(4), 67–77 (1999)

    Google Scholar 

  6. C.H. Hsiao, Haar wavelet approach to linear stiff systems. Math. Comput. Simul. 64(5), 561–567 (2004)

    Google Scholar 

  7. N.M. Bujurke, C.S. Salimath, S.C. Shiralashetti, Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series. Nonlinear Dyn. 51, 595–605 (2008)

    Google Scholar 

  8. M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff systems of ordinary differential equations. Comput. Math. Appl. 54(7–8), 1055–1063 (2007)

    Google Scholar 

  9. G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 41(3), 373–398 (1983)

    Google Scholar 

  10. A. Prothero, A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28(125), 145–162 (1974)

    Google Scholar 

  11. S. Dhawan, J.A.T. Machado, D.W. Brzeziński, M.S. Osman, A Chebyshev wavelet collocation method for some types of differential problems. Symmetry. 13(4), 536 (2021)

    Google Scholar 

  12. M. Faheem, A. Raza, A. Khan, Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations. Math. Comput. Simul. 180, 72–92 (2021)

    Google Scholar 

  13. S. Kumbinarasaiah, K.R. Raghunatha, the applications of the Hermite wavelet method to nonlinear differential equations arising in heat transfer. Int. J. Thermofluids. 9, 100066 (2021)

    Google Scholar 

  14. S. Kumbinarasaiah, M. Mulimani, A study on the non-linear Murray equation through the Bernoulli wavelet approach. Int. J. Appl. Comput. Math. 9(3), 40 (2023)

    Google Scholar 

  15. S. Kumbinarasaiah, R.A. Mundewadi, Numerical solution of fractional-order integro-differential equations using the Laguerre wavelet method. J. Inf. Optim. Sci. 43(4), 643–662 (2022)

    Google Scholar 

  16. T. Abdeljawad, R. Amin, K. Shah, Q. Al-Mdallal, F. Jarad, Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method. Alex. Eng. J. 59(4), 2391–2400 (2020)

    Google Scholar 

  17. S. Erman, A. Demir, E. Ozbilge, solving inverse nonlinear fractional differential equations by generalized Chelyshkov wavelets. Alex. Eng. J. 66, 947–956 (2023)

    Google Scholar 

  18. S. Kumbinarasaiah, M. Mulimani, The Fibonacci wavelets approach for the fractional Rosenau-Hyman equations. Results Control Optim. 11, 100221 (2023)

    Google Scholar 

  19. X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method. Commun. Nonlinear Sci. Numer. Simul. 17(10), 3934–3946 (2012)

    Google Scholar 

  20. L.I. Yuanlu, solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear Sci. Numer. Simul. 15(9), 2284–2292 (2010)

    Google Scholar 

  21. A. Isah, C. Phang, Genocchi wavelet-like operational matrix and its application for solving nonlinear fractional differential equations. Open Physics. 14(1), 463–472 (2016)

    Google Scholar 

  22. E. Keshavarz, Y. Ordokhani, M. Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Model. 38(24), 6038–6051 (2014)

    Google Scholar 

  23. S. Kumbinarasaiah, G. Manohara, G. Hariharan, Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane Emden equations. Math. Comput. Simul. 204, 133–165 (2022)

    Google Scholar 

  24. S. Kumbinarasaiah, G. Manohara, Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ T cells model. Results Control Optim. 10, 100197 (2023)

    Google Scholar 

  25. F. Mohammadi, C. Cattani, A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations. J. Comput. Appl. Math. 339, 306–316 (2018)

    Google Scholar 

  26. S. Kumbinarasaiah, Hermite wavelets approach for the multi-term fractional differential equations. J. Interdiscip. Math. 24(5), 1241–1262 (2021)

    Google Scholar 

  27. S. Kumbinarasaiah, W. Adel, Hermite wavelet method for solving nonlinear Rosenau-Hyman equation. Partial Differ. Equ. Appl. Math. 4, 100062 (2021)

    Google Scholar 

  28. M. Rehman, U. Saeed, Gegenbauer wavelets operational matrix method for fractional differential equations. J. Korean Math. Soc. 52(5), 1069–1096 (2015)

    Google Scholar 

  29. E. Keshavarz, Y. Ordokhani, M. Razzaghi, The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations. Appl. Numer. Math. 128, 205–216 (2018)

    Google Scholar 

  30. P.T. Toan, T.N. Vo, M. Razzaghi, Taylor wavelet method for fractional delay differential equations. Eng. Comput. 37, 231–240 (2021)

    Google Scholar 

  31. T.N. Vo, M. Razzaghi, P.T. Toan, Fractional-order generalized Taylor wavelet method for systems of nonlinear fractional differential equations with application to human respiratory syncytial virus infection. Soft. Comput. 26, 165–173 (2022)

    Google Scholar 

  32. S.C. Shiralashetti, S.I. Hanaji, Taylor wavelet collocation method for Benjamin–Bona–Mahony partial differential equations. Results Appl. Math. 9, 100139 (2021)

    Google Scholar 

  33. I. Dağ, A. Canıvar, A. Şahin, Taylor-Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2696–2708 (2011)

    Google Scholar 

  34. F. Li, H.M. Baskonus, S. Kumbinarasaiah, G. Manohara, W. Gao, E. Ilhan, An efficient numerical scheme for biological models in the frame of Bernoulli wavelets. Comput. Model. Eng. Sci. 137, 3 (2023)

    Google Scholar 

  35. S. Gümgüm, Taylor wavelet solution of linear and nonlinear Lane-Emden equations. Appl. Numer. Math. 158, 44–53 (2022)

    Google Scholar 

  36. G. Manohara, S. Kumbinarasaiah, Fibonacci wavelets operational matrix approach for solving chemistry problems. J. Umm Al-Qura Univ. Appl. Sci. (2023). https://doi.org/10.1007/s43994-023-00046-5

    Article  Google Scholar 

  37. G. Hariharan, K. Kannan, Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering. Appl. Math. Model. 38(3), 799–813 (2014)

    Google Scholar 

  38. G. Hariharan, R. Rajaraman, A new coupled wavelet-based method applied to the nonlinear reaction–diffusion equation arising in mathematical chemistry. J. Math. Chem. 51, 2386–2400 (2003)

    Google Scholar 

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

    Google Scholar 

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

    CAS  Google Scholar 

  41. S. Kumbinarasaiah, M. Mulimani, Fibonacci wavelets-based numerical method for solving fractional order (1 + 1)-dimensional dispersive partial differential equation. Int. J. Dyn. Control 11, 2232–2255 (2023)

    Google Scholar 

  42. M.M. Khalsaraei, A. Shokri, M. Molayi, The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs. J. Math. Chem. 58, 1987–2012 (2020)

    CAS  Google Scholar 

  43. Y. Öztürk, Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials. J. Taibah Univ. Sci. 12(2), 155–162 (2018)

    Google Scholar 

Download references

Funding

The author states that no funding is involved.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bengaluru, 560 056, India

    G. Manohara & S. Kumbinarasaiah

Authors
  1. G. Manohara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Kumbinarasaiah
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

KS proposed the main idea of this paper. KS and MG prepared the manuscript and performed all the steps of the proofs in this research. Both authors contributed equally and significantly to writing this paper. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to S. Kumbinarasaiah.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manohara, G., Kumbinarasaiah, S. Numerical solution of some stiff systems arising in chemistry via Taylor wavelet collocation method. J Math Chem 62, 24–61 (2024). https://doi.org/10.1007/s10910-023-01508-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-023-01508-1

Keywords

Mathematics Subject Classification

Search

Navigation