Skip to main content
SpringerLink
Log in

On the maximal order of convergence of Green’s function method for solving two-point boundary value problems with deviating argument

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The Green’s function method is applied to second, third, and fourth order two-point boundary value problems with deviating argument. At each iterative step, the Picard-Green’s method is combined with suitable quadrature rule and interpolation procedures. The Hermite type cubic spline is applied to second and third order boundary value problems, while complete cubic splines are used at fourth order boundary value problems. The corrected trapezoidal quadrature rule is involved at third and fourth order boundary value problems with appropriate error bounds. The convergence of the method was proved, and the maximal order of convergence is obtained for each problem. The theoretical results are tested on some numerical examples.

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

Similar content being viewed by others

Data availability

Not applicable (it is not the case)

References

  1. Abushammala, M., Khuri, S.A., Sayfy, A.: A novel fixed point iteration method for the solution of third order boundary value problems. Appl Math Comput 271, 131–141 (2015)

    MathSciNet  Google Scholar 

  2. Agarwal, R.P.: Non-linear two point boundary value problems. Indian J Pure Appl Math 31, 757–769 (1973)

    MathSciNet  Google Scholar 

  3. Agarwal, R.P.: Boundary value problems for high order differential equations. World Scientific, Singapore (1986)

    Google Scholar 

  4. Agarwal, R.P., Chow, Y.M.: Finite difference methods for boundary-value problems of differential equations with deviating arguments. Comput Math Appl 12, 1143–1153 (1986)

    MathSciNet  Google Scholar 

  5. Akgűl, A., Kilicman, A.: Solving delay differential equations by an accurate method with interpolation. Abstr Appl Anal (2015) art. ID 676939

  6. Akgun, F.A., Rasulov, Z.: A new iteration method for the solution of third-order BVP via Green’s function. Demonstr Math 54, 425–435 (2021)

    MathSciNet  Google Scholar 

  7. Atkinson, K.E.: An introduction to numerical analysis, 2nd edn. John Wiley & Sons, New York (1989)

    Google Scholar 

  8. Aykut, A., Yildiz, B.: On a boundary value problem for a differential equation with variant retarded argument. Appl Math Comput 93, 63–71 (1998)

    MathSciNet  Google Scholar 

  9. Barnett, N.S., Dragomir, S.S.: A perturbed trapezoid inequality in terms of the fourth derivative. Korean J. Comput Appl Math 9, 45–60 (2002)

    MathSciNet  Google Scholar 

  10. Bartoszewski, Z.: A new approach to numerical solution of fixed point problems and its applications to delay differential equations. Appl Math Comput 215, 4320–4331 (2010)

    MathSciNet  Google Scholar 

  11. Bazm, S., Lima, P., Nemati, S.: Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type. J Comput Appl Math 398, art. no. 113628 (2021)

  12. Bernis, F., Peletier, L.A.: Two problems from draining flows involving third-order ordinary differential equations. SIAM J Math Anal 27, 515–527 (1996)

    MathSciNet  Google Scholar 

  13. Bica, A.: The error estimation in terms of the first derivative in a numerical method for the solution of a delay integral equation from biomathematics. Rev Anal Numér Théor Approx 34, 23–36 (2005)

    MathSciNet  Google Scholar 

  14. Bica, A.M., Curilă, M., Curilă, S.: Two-point boundary value problems associated to functional differential equations of even order solved by iterated splines. Appl Numer Math 110, 128–147 (2016)

    MathSciNet  Google Scholar 

  15. Bica, A.M., Curilă, M., Curilă, S.: About a numerical method of successive interpolations for two-point boundary value problems with deviating argument. Appl Math Comput 217, 7772–7789 (2011)

    MathSciNet  Google Scholar 

  16. Bica, A.M., Curilă (Popescu), D.: The convergence properties of the Green’s function method for third order functional differential equations. Comput Appl Math 41, art. no. 352 (2022)

  17. Bica, A.M., Curilă (Popescu), D.: Catmull-Rom spline approach and the order of convergence of Green’s function method for functional differential equations. Bull Transilv Univ Braşov Ser III 2(64) no. 2, 13–32 (2022)

  18. Calagar, H.N., Calagar, S.H., Twizell, E.H.: The numerical solution of third order boundary value problems with fourth degree B-Spline. Int J Comput Math 71, 373–381 (1999)

    MathSciNet  Google Scholar 

  19. Calio, F., Marchetti, E., Pavani, R., Micula, G.: A new deficient spline functions collocation method for second order delay differential equations. Pure Math Appl 13, 97–109 (2002)

    MathSciNet  Google Scholar 

  20. Chocholaty, P., Slahor, L.: A numerical method to boundary value problems for second order delay-differential equations. Numer Math 33, 69–75 (1979)

    MathSciNet  Google Scholar 

  21. Dragomir, S.S., Cerone, P., Sofo, A.: Some remarks on the trapezoid rule in numerical integration. Indian J Pure Appl Math 31, 475–494 (2000)

    MathSciNet  Google Scholar 

  22. El-Danaf, T.S.: Quartic nonpolynomial spline solutions for third order two-point boundary value problem. Int J Math Comput Sci 2, 637–640 (2008)

    Google Scholar 

  23. Dang, Q.A., Dang, Q.L.: A unified approach to study the existence and numerical solution of functional differential equation. Appl Numer Math 170, 208–218 (2021)

    MathSciNet  Google Scholar 

  24. Dang, Q.A., Dang, Q.L.: Simple numerical methods of second and third-order convergence for solving a fully third-order nonlinear boundary value problem. Numer Algorithms 87, 1479–1499 (2021)

    MathSciNet  Google Scholar 

  25. Doha, E.H., Abd-Elhameed, W.M., Youssri, Y.H.: New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov-Galerkin method. J Advanced Research 6, 673–686 (2015)

    Google Scholar 

  26. Ge, Q., Zhang, X.: Numerical solution for third-order two-point boundary value problems with the barycentric rational interpolation collocation method. J Math Article ID 6698615 (2021)

  27. Hakl, R.: On a two-point boundary value problem for third-order linear functional differential equations. Part II, Math Appl (Brno) 1, 129–143 (2012)

  28. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer-Verlag, New York (1993)

    Google Scholar 

  29. Jankowski, T.: Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. Nonlinear Anal 73, 1289–1299 (2010)

    MathSciNet  Google Scholar 

  30. Keller, H.B.: Numerical Methods for Two-Point Boundary Value Problems. Dover Publications Inc., New York (1993)

    Google Scholar 

  31. Khan, A., Aziz, T.: The numerical solution of third order boundary value problems using quintic spline. Appl Math Comput 137, 253–260 (2003)

    MathSciNet  Google Scholar 

  32. Khuri, S.A., Sayfy, A.: Numerical solution of functional differential equations: a Green’s function-based iterative approach. Int J Computer Math 95, 1937–1949 (2018)

    MathSciNet  Google Scholar 

  33. Khuri, S.A., Sayfy, A.: Variational iteration method: Green’s functions and fixed point iterations perspective. Appl Math Lett 32, 28–34 (2014)

    MathSciNet  Google Scholar 

  34. Khuri, S.A., Louhichi, I.: A new fixed point iteration method for nonlinear third-order BVPs. Int J Comput Math 98, 2220–2232 (2021)

    MathSciNet  Google Scholar 

  35. Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  36. Lv, X., Gao, J.: Treatment for third-order nonlinear differential equations based on the Adomian decomposition method. LMS J Comput Math 20, 1–10 (2017)

    MathSciNet  Google Scholar 

  37. Lv, X., Cui, M.: Existence and numerical method for nonlinear third-order boundary-value problem in the reproducing kernel space. Bound Value Probl 2010, 1–13 (2010)

    MathSciNet  Google Scholar 

  38. Micula, G., Micula, S.: Handbook of Splines, Mathematics and Its Applications, vol. 462. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  39. Padhi, S., Pati, S.: Theory of Third-Order Differential Equations. Spriger, New-Delhi (2014)

    Google Scholar 

  40. Pandey, P.K.: Solving third-order boundary value problems with quartic splines. Springer Plus 5, 326 (2016)

    Google Scholar 

  41. Qu, R., Agarwal, R.P.: A subdivision approach to the construction of approximate solution of boundary-value problems with deviating arguments. Comput Math Appl 35, 121–135 (1998)

    MathSciNet  Google Scholar 

  42. Raja, M.A.Z.: Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl Soft Comput 24, 806–821 (2014)

    Google Scholar 

  43. Reutskiy, SYu.: A new collocation method for approximate solution of the pantograph functional differential equations with proportional delay. Appl Math Comput. 266, 6420–655 (2015)

    MathSciNet  Google Scholar 

  44. Sakai, M.: Numerical solution of boundary value problems for second order functional differential equations by the use of cubic splines. Mem Fac Sci Kyushu Univ Ser A 29, 113–122 (1975)

    MathSciNet  Google Scholar 

  45. Sun, Y., Wang, P.: Quasilinear iterative scheme for a fourth-order differential equation with retardation and anticipation. Appl Math Comput 217, 3442–3452 (2010)

    MathSciNet  Google Scholar 

  46. Sun, Y., Wang, P.: Iterative methods for a fourth-order differential equations with retardation and anticipation. Dyn Contin Discrete Impuls Syst Ser B, Appl Algorithms 17, 487–500 (2010)

    MathSciNet  Google Scholar 

  47. Sun, Y., Min Zhao, Li, S.: Monotone positive solution of nonlinear third-order two-point boundary value problem. Miskolc Math Notes 15, 743–752 (2014)

  48. Trif, D.: Direct operatorial tau method for pantograph-type equations. Appl Math Comput 219, 2194–2203 (2012)

    MathSciNet  Google Scholar 

  49. Wazwaz, M.A., Raja, M.A.Z., Syam, M.I.: Reliable treatment for solving boundary value problems of pantograph delay differential equation. Rom Rep Phys 69, 102 (2017)

    Google Scholar 

  50. Yao, Q.: Positive solutions for eigenvalue problems of fourth order elastic beam equations. Appl Math Lett 17, 237–243 (2004)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wants to thank to anonymous reviewers for their valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Oradea, Universitatii Street no.1, Oradea, 410087, Bihor, Romania

    Alexandru Mihai Bica

Authors
  1. Alexandru Mihai Bica
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

This work was entirely made by the corresponding author.

Corresponding author

Correspondence to Alexandru Mihai Bica.

Ethics declarations

Conflict of interest

The author declares no competing interests.

Ethical approval

Not applicable

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

Bica, A.M. On the maximal order of convergence of Green’s function method for solving two-point boundary value problems with deviating argument. Numer Algor 95, 943–978 (2024). https://doi.org/10.1007/s11075-023-01595-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01595-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation