Skip to main content
SpringerLink
Log in

New higher-order implict method for approximating solutions of the initial value problems

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor’s approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that outperforms Taylor’s method in terms of accuracy. We derive an error bound for the Obreschkoff higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Taylor method. To substantiate our claims, numerical experiments are provided, which highlight the exceptional efficacy of our proposed method over the traditional Taylor method.

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

Similar content being viewed by others

Data availability

The data used to support the findings of this study are included in the article.

References

  1. Abdeljawad, T., Amin, R., Shah, K., Al-Mdallal, Q., Jarad, F.: 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)

    Article  Google Scholar 

  2. Appell, P.: Sur une classe de polynômes. Ann. Sci. Ecole. Norm. Sup. 9(2), 119–144 (1880)

    Article  MathSciNet  Google Scholar 

  3. Borri, A., Carravetta, F., Palumbo, P.: Quadratized Taylor series methods for ODE numerical integration. Appl. Math. Comput. 458, 128237 (2023)

    MathSciNet  Google Scholar 

  4. Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M.S., Sergeyev, Ya.D.: A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic. Math. Comput. Simul. 141, 24–39 (2017)

    Article  MathSciNet  Google Scholar 

  5. Arefin, M.A., Gain, B., Karim, R.: Accuracy analysis on solution of initial value problems of ordinary differential equations for some numerical methods with different step sizes. Int. Ann. Sci. 10(1), 118–133 (2021)

    Article  Google Scholar 

  6. Baeza, A., Boscarino, S., Mulet, P., Russo, G., Zorío, D.: Approximate Taylor methods for ODEs. Comput. Fluids 159, 156–166 (2017)

    Article  MathSciNet  Google Scholar 

  7. Baeza, A., Boscarino, S., Mulet, P., Russo, G., Zorío, D.: Reprint of: Approximate Taylor methods for ODEs. Comput. Fluids 169, 87–97 (2018)

    Article  MathSciNet  Google Scholar 

  8. Boas, R.P.: Generalized Taylor series, quadrature formulas, and a formula by Kronecker, short notes. SIAM Rev. 12(1), 116–119 (1970)

    Article  MathSciNet  Google Scholar 

  9. Alomari, M.W.: Two-point Ostrowski and Ostrowski–Grüss type inequalities with applications. J. Anal. 28(3), 623–661 (2020)

    Article  MathSciNet  Google Scholar 

  10. Alomari, M.W.: Two-point Ostrowski’s inequality. Results Math. 72(3), 1499–1523

  11. Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks/Cole, Cengage Learning, Boston (2011)

    Google Scholar 

  12. Carrillo, H., Macca, E., Parés, C., Russo, G., Zorío, D.: An order-adaptive compact approximation Taylor method for systems of conservation laws. J. Comput. Phys. 438(1), 110358 (2021)

    Article  MathSciNet  Google Scholar 

  13. Corliss, G.: Solving ordinary differential using Taylor series. ACM Trans. Math. Softw. 8(2), 114–144 (1983)

    Article  MathSciNet  Google Scholar 

  14. Dababneh, A., Zraiqat, A., Farah, A., Al-Zoubi, H., Abu Hammad, M.: Numerical methods for finding periodic solutions of ordinary differential equations with strong nonlinearity. J. Math. Comput. Sci. 11, 6910–6922 (2021)

    Google Scholar 

  15. Gadisa, G., Garoma, H.: Comparison of higher order Taylor’s method and Runge–Kutta methods for solving first order ordinary differential equations. J. Comput. Math. Sci. 8(1), 12–23 (2017)

    MathSciNet  Google Scholar 

  16. Haq, F., Shah, K., Al-Mdallal, Q.M., Jarad, F.: Application of a hybrid method for systems of fractional order partial differential equations arising in the model of the one-dimensional Keller–Segel equation. Eur. Phys. J. Plus 134, 1–11 (2019)

    Article  Google Scholar 

  17. Hummel, P.W., Seebeck, C.L.: A generalization of Taylor’s expansion. Am. Math. Mon. (MAA) 56(4), 243–247 (1949)

    Article  MathSciNet  Google Scholar 

  18. Jorba, Á., Zou, M.: A software package for the numerical integration of ODEs by means of high-order Taylor methods. Exp. Math. 14(1), 99–117 (2005)

    Article  MathSciNet  Google Scholar 

  19. Kythe, P.K., Schäferkotter, M.R.: Handbook of Computational Methods for Integration. Chapman & HallL/CRC, A CRC Press Company, London (2005)

    Google Scholar 

  20. Lakshminarasimhan, T.V.: On extensions of Taylor’s formula. Am. Math. Mon. 72(8), 877–881 (1965)

    Article  MathSciNet  Google Scholar 

  21. Matić, M., Pečarić, J., Ujević, N.: On new estimation of the remainder in generalized Taylor’s formula. Math. Inequal. Appl. 2(3), 343–361 (1999)

    MathSciNet  Google Scholar 

  22. Ndam, J.: Comparison of the solution of the Van der Pol equation using the modified Adomian decomposition method and truncated Taylor series method. J. Niger. Soc. Phys. Sci. 2(2), 106–114 (2020)

    MathSciNet  Google Scholar 

  23. Narayanan, K.L., Kavitha, J., Rani, R.U., Lyons, M.E., Rajendran, L.: Mathematical modelling of amperometric glucose biosensor based on immobilized enzymes: new approach of Taylor’s series method. Int. J. Electrochem. Sci. 17(10), 221064 (2022)

    Article  Google Scholar 

  24. Obreschkoff, N.: Neue Quadraturtormeln. Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. (4) (1940)

  25. Qi, F., Luo, Q.-M., Guo, B.-N.: Darboux’s formula with integral remainder of functions with two independent variables. Appl. Math. Comput. 199, 691–703 (2008)

    MathSciNet  Google Scholar 

  26. Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62(1), 26–29 (1955)

    MathSciNet  Google Scholar 

  27. Shah, K., Abdeljawad, T., Ali, A., Alqudah, M.A.: Investigation of integral boundary value problem with impulsive behavior involving non-singular derivative. Fractals 30(8), 1–15 (2022)

    Article  Google Scholar 

  28. Shah, K., Abdeljawad, T., Ali, A.: Mathematical analysis of the Cauchy type dynamical system under piecewise equations with Caputo fractional derivative. Chaos Solitons Fractals 161, 1–8 (2022)

    Article  MathSciNet  Google Scholar 

  29. Shah, K., Arfan, M., Ullah, A., Al-Mdallal, Q., Ansari, K.J., Abdeljawad, T.: Computational study on the dynamics of fractional order differential equations with applications. Chaos Solitons Fractals 157, 111955 (2022)

    Article  MathSciNet  Google Scholar 

  30. Wang, Z.X., Guo, D.R.: Special Functions, Translated from the Chinese by Guo and X. J. Xia. World Scientific, Teaneck (1989). https://doi.org/10.1142/0653

  31. Wang, K., Wang, Q.: Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations. J. Comput. Appl. Math. 260, 294–300 (2014)

    Article  MathSciNet  Google Scholar 

  32. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1940)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Information Technology, Jadara University, P.O. Box 733, Irbid, 21110, Jordan

    Mohammad W. Alomari

  2. Department of Mathematics, Al Zaytoonah University of Jordan, Amman, 11733, Jordan

    Iqbal M. Batiha

  3. Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

    Iqbal M. Batiha & Shaher Momani

  4. Department of Mathematics, The University of Jordan, Amman, 11942, Jordan

    Shaher Momani

Authors
  1. Mohammad W. Alomari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iqbal M. Batiha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shaher Momani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad W. Alomari.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

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

Alomari, M.W., Batiha, I.M. & Momani, S. New higher-order implict method for approximating solutions of the initial value problems. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02087-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12190-024-02087-3

Keywords

Mathematics Subject Classification

Search

Navigation