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Journals / Sakarya University Journal of Science / 2023 / Cilt: 27 Sayı: 6

Approximate Solutions of Nonlinear Boundary Value Problems by Collocation Methods Compared to Newer Methods

Birkan DURAK
Hasan Ömür ÖZER
Aziz SEZGİN
Lütfi Emir SAKMAN
Pages
1345–1354
Publication Date
2023-12-18
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The overall performance of collocation method and the newer methods are compared on a number of problems, which were previously used to benchmark the newer methods.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:A large variety of new methods are being developed for fast and efficient solutions of nonlinear boundary value problems.
📊
Key Findings & Data:The overall performance of collocation method and the newer methods are compared on a number of problems, which were previously used to benchmark the newer methods.
💡
Conclusion & Impact:It is concluded that, at least for the problems considered, the collocation method performs as successfully as the newer methods.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Collocation method · nonlinear differential equations · boundary value problems

Abstract

A large variety of new methods are being developed for fast and efficient solutions of nonlinear boundary value problems. Some of these methods are, Adomian decomposition (ADM), differential transform (DTM), least squares vector machines (LSSVMM), and multiple variational iteration (MVIM). A natural question arises as to how efficient and simple to use these newer methods are compared to classical methods. One of the simplest and widely applicable classical methods is the collocation method. The overall performance of collocation method and the newer methods are compared on a number of problems, which were previously used to benchmark the newer methods. It is concluded that, at least for the problems considered, the collocation method performs as successfully as the newer methods.

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Cite This Article

DURAK, B., ÖZER, H. Ö., SEZGİN, A., SAKMAN, L. E. (2023). Approximate Solutions of Nonlinear Boundary Value Problems by Collocation Methods Compared to Newer Methods. Sakarya University Journal of Science, Cilt: 27 Sayı: 6, 1345–1354. https://ajindex.com/articles/1022323

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References (14)

  1. A. Ü. Keskin, “Boundary Value Problems for Engineers with MATLAB Solutions”, 1st ed., Switzerland AG, Switzerland, Springer Nature 2019
  2. A. M. Wazwaz, “A comparison between adomian decomposition method and taylor series method in the series solutions,” Applied Mathematics and Computation, vol. 97, no. 1, pp. 37-44, 1998
  3. J. A. Sánchez Cano, “Adomian decomposition method for a class of nonlinear Problems,” International Scholarly Research Notices, vol. 2011, Article ID 709753
  4. M. O. Kaya, “Free vibration analysis of a rotating timoshenko beam by differential transform method,” Aircraft Engineering and Aerospace Technology, vol. 78, no. 3, pp. 194–203, 2006
  5. A. Gökdoğan, M. Merdan, A. Yildirim, “Adaptive multi-step differential transformation method to solving nonlinear differential equations,” Mathematical and Computer Modelling, vol. 55, no. 3–4, pp. 761-769, 2012
  6. J. H. He, “Variational iteration method - a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999
  7. S. Ghafoori, M. Motevalli, M. G. Nejad, F. Shakeri, D.D. Ganji, M. Jalaal, “Efficiency of differential transformation method for nonlinear oscillation: comparison with HPM and VIM,” Current Applied Physics, vol. 11, no. 4, 2011
  8. M. A. Noor, S. T. Mohyud-Din, “An efficient method for fourth-order boundary value problems,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 1101-1111, 2007
  9. M. Ahsan, S. Farrukh, “A new type of shooting method for nonlinear boundary value problems,” Alexandria Engineering Journal, vol. 52, no. 4, pp. 801-805, 2013
  10. S. A. Khuri, A. Sayfy, “An iteration method for boundary value problems,” Nonlinear Science. Letter. A, vol. 8, no. 2, pp. 178-186, 2017
  11. S. Chakraverty, N. R. Mahato, P. Karunakar, T.D. Rao, “Advanced Numerical and Semi-Analytical Methods for Differential Equations,” first ed., John Wiley & Sons, Inc., USA, 2019
  12. R. H. Gallagher, J. T. Oden, C. Taylor, O. C. Zienkiewicz, “Finite elements in fluids. volume 2 - Mathematical Foundations, Aerodynamics and Lubrication,” first ed., John Wiley & Sons, Inc., UK, 1975
  13. Y. Han, M. Shufang, L. Yanbin, S. Hongquan, “Convergence and Stability in Collocation Methods of Equation 𝑢 ′ (𝑡) = 𝑎𝑢(𝑡) + 𝑏(𝑢[𝑡]).,” Journal of Applied Mathematics, vol. 2012, Article ID: 125926
  14. R. Amin, S. Nazir, I. García-Magariño, “A Collocation Method for Numerical Solution of Nonlinea