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A simple efficient method for solving sixth-order nonlinear boundary value problems

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Abstract

In this paper, we propose a simple efficient method for a sixth-order nonlinear boundary value problem. It is based on the reduction of the problem to an operator equation for the right-hand-side function. The existence and uniqueness of a solution and its positivity are established. An iterative method for finding the solution is investigated. A numerical realization of the iterative method with the use of a difference scheme of sixth-order accuracy shows the efficiency and advantages of the proposed method over some other methods.

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References

  • Aceto L, Ghelardoni P, Magherini C (2009) Boundary value methods as an extension of numerovs method for Sturm–Liouville eigenvalue estimates. Appl Numer Math 59:1644–1656

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  • Al-Hayani W (2011) Adomian decomposition method with Greens function for sixth-order boundary value problems. Comput Math Appl 61:1567–1575

    Article  MathSciNet  Google Scholar 

  • Boutayeb A, Twizell EH (1992) Numerical methods for the solution of special sixth order boundary-value problems. Int J Comput Math 45:207–223

    Article  Google Scholar 

  • Chandrasekhar S (1981) Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, p 1961 (Reprinted by Dover Books, New York 1981)

  • Dang QA, Dang QL, Ngo TKQ (2017) A novel efficient method for fourth order nonlinear boundary value problems. Numer Algorithms 76:427–439

    Article  MathSciNet  Google Scholar 

  • Glatzmaier GA (1985) Numerical simulations of stellar convection dynamics at the base of the convection zone. Fluid Dyn 31:137–150

    Article  Google Scholar 

  • He JH (2003) Variational approach to the sixth order boundary value problems. Appl Math Comput 143:537–538

    MathSciNet  MATH  Google Scholar 

  • Khalid M, Sultana M, Zaidi F (2014) Numerical solution of sixth-order differential equations arising in astrophysics by neural network. Int J Comput Appl 107:1–6

    Google Scholar 

  • Lang FG, Xu XP (2015) An effective method for numerical solution and numerical derivatives for sixth order two-point boundary value problems. Comput Math Math Phys 55(5):811–822

    Article  MathSciNet  Google Scholar 

  • Mohyud-Din ST, Noor MA, Waheed A (2009) Variation of parameters method for solving sixth-order boundary value problems. Commun Korean Math Soc 24(4):605–615

    Article  MathSciNet  Google Scholar 

  • Noor MA, Mohyud-Din ST (2008) Homotopy perturbation method for solving sixth-order boundary value problems. Comput Math Appl 55:2953–2972

    Article  MathSciNet  Google Scholar 

  • Noor MA, Noor KI, Mohyud-Din ST (2009) Variational iteration method for solving sixth-order boundary value problems. Commun Nonlinear Sci Numer Simul 14:2571–2580

    Article  MathSciNet  Google Scholar 

  • Pandey PK (2013) Fourth order finite difference method for sixth order boundary value problems. Comput Math Math Phys 53:57–62

    Article  MathSciNet  Google Scholar 

  • Protter MH, Weinberger HF (1984) Maximum principles in differential equations. Springer, New York

    Book  Google Scholar 

  • Ramadan MA, Lashien IF, Zahra WK (2008) A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems. Int J Comput Math 85:759–770

    Article  MathSciNet  Google Scholar 

  • Toomore J, Zahn JP, Latour J, Spiegel EA (1976) Stellar convection theory II: Single mode study of the second convection zone in an A-type star. Astrophys J 207:545–563

    Article  Google Scholar 

  • Twizell EH, Boutayeb A (1990) Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Benard layer Eigenvalue problems. Proc R Soc Lond A 431:433–450

    Article  Google Scholar 

  • Wazwaz AM (2001) The numerical solution of sixth order boundary value problems by the modified decomposition method. Appl Math Comput 118:311–325

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the reviewers for their helpful comments for improving the quality of the paper. This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the Grant number 102.01-2017.306.

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Authors and Affiliations

  1. Centre for Informatics and Computing, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Dang Quang A

  2. Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Dang Quang Long

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  1. Dang Quang A
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  2. Dang Quang Long
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Correspondence to Dang Quang A.

Additional information

Communicated by Elbert Macau, Antônio Fernando Bertachini de Almeida Prado and Othon Cabo Winter.

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Quang A, D., Quang Long, D. A simple efficient method for solving sixth-order nonlinear boundary value problems. Comp. Appl. Math. 37 (Suppl 1), 16–26 (2018). https://doi.org/10.1007/s40314-018-0643-1

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  • DOI: https://doi.org/10.1007/s40314-018-0643-1

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