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The convergence properties of the Green’s function method for third order functional differential equations

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Abstract

The Green’s function method is a powerful technique to investigate the existence and uniqueness of the solution for two-point boundary value problems. The main task of this paper is to establish the maximal order of convergence of the Green’s function method applied in the approximation of the solution for third- and fourth-order two-point boundary value problems with deviating argument. The developed iterative method uses a complete cubic spline interpolation procedure being able to provide both discrete and continuous approximation of the solution. Some numerical examples confirm the obtained theoretical results and illustrate the accuracy of the algorithm.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  • Agarwal RP (1973) Non-linear two point boundary value problems. Indian J Pure Appl Math 31(9–10):757–769

    MathSciNet  MATH  Google Scholar 

  • Agarwal RP (1986) Boundary value problems for high order differential equations. World Scientific, Singapore

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Aktaş MF, Tiryaki A, Zafer A (2010) Oscillation criteria for third-order nonlinear functional differential equations. Appl Math Lett 23:756–762

    Article  MathSciNet  MATH  Google Scholar 

  • Barnett NS, Dragomir SS (2002) A perturbed trapezoid inequality in terms of the fourth derivative. Korean J Comput Appl Math 9(1):45–60

    Article  MathSciNet  MATH  Google Scholar 

  • Bernis F, Peletier LA (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J Math Anal 27(2):515–527

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Calagar HN, Calagar SH, Twizell EH (1999) The numerical solution of third order boundary value problems with fourth degree B-spline. Int J Computer Math 71:373–381

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Dang QA, Dang QL (2021b) Simple numerical methods of second and third-order convergence for solving a fully third-order nonlinear boundary value problem. Numer Algorithms 87(1):1479–1499

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir SS, Cerone P, Sofo A (2000) Some remarks on the trapezoid rule in numerical integration. Indian J Pure Appl Math 31(5):475–494

    MathSciNet  MATH  Google Scholar 

  • El-Danaf TS (2008) Quartic nonpolynomial spline solutions for third order two-point boundary value problem. Int J Math Comput Sci 2(9):637–640

    Google Scholar 

  • Fazal-i-Haq H, Iltaf A, Arshed A (2011) Haar wavelets based numerical method for third-order boundary and initial value problems. World Appl Sci J 13(10):2244–2251

    Google Scholar 

  • Hakl R (2012a) On a two-point boundary value problem for third-order linear functional differential equations. Part I. Math Appl (Brno) 1:57–78

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Hale JK (1977) Retarded functional differential equations: basic theory. Theory of functional differential equations. Springer, New York, pp 36–56

    Google Scholar 

  • Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  • Hall CA, Meyer WW (1976) Optimal error bounds for cubic spline interpolation. J Approx Theory 16:105–122

    Article  MathSciNet  MATH  Google Scholar 

  • Jadlovská I, Chatzarakis GE, Džurina J, Grace SR (2021) On sharp oscillation criteria for general third-order delay differential equations. Mathematics 9:1675

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Khuri SA, Louhichi I (2021) A new fixed point iteration method for nonlinear third-order BVPs. Int J Comput Math 98(11):2220–2232

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Khuri SA, Sayfy A (2018) Numerical solution of functional differential equations: a Green’s function-based iterative approach. Int J Comput Math 95(10):1937–1949

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Lv X, Gao J (2017) Treatment for third-order nonlinear differential equations based on the Adomian decomposition method. LMS J Comput Math 20(2):1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Padhi S, Pati S (2014) Theory of third-order differential equations. Spriger, New-Delhi

    Book  MATH  Google Scholar 

  • Pandey PK (2016) Solving third-order boundary value problems with quartic splines. Springer Plus 5:326

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Reutskiy SYu (2015) A new collocation method for approximate solution of the pantograph functional differential equations with proportional delay. Appl Math Comput 266:642–655

    Article  MathSciNet  MATH  Google Scholar 

  • Şenel MT (2018) On oscillation of third order functional differential equation. Electron J Math Anal Appl 6:103–108

    MathSciNet  MATH  Google Scholar 

  • Sun B (2021) Existence theory for positive iterative solutions to a type of boundary value problem. Symmetry 13:1585

    Article  Google Scholar 

  • Tunç C (2010) Bound of solutions to third-order nonlinear differential equations with bounded delay. J Frankl Inst 347:415–425

    Article  MathSciNet  MATH  Google Scholar 

  • Wazwaz MA, Raja MAZ, Syam MI (2017) Reliable treatment for solving boundary value problems of pantograph delay differential equation. Rom Rep Phys 69:102

    Google Scholar 

  • Zarin H, Roos H-G, Teofanov L (2018) A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem. Comput Appl Math 37:175–190

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors want to thank to anonymous reviewers for their valuable suggestions that improved the quality of this paper. The research of the first author was developed under the auspices of the institutional project No. 126/25.06.2021 sustained by University of Oradea.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Oradea, Universitatii 1, 410087, Oradea, Romania

    Alexandru Mihai Bica & Diana Curila

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  1. Alexandru Mihai Bica
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  2. Diana Curila
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Correspondence to Alexandru Mihai Bica.

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Communicated by Zhaosheng Feng.

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Bica, A.M., Curila, D. The convergence properties of the Green’s function method for third order functional differential equations. Comp. Appl. Math. 41, 352 (2022). https://doi.org/10.1007/s40314-022-02065-0

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