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

Year 2023, , 1345 - 1354, 18.12.2023

Birkan Durak , Hasan Ömür Özer , Aziz Sezgin , Lütfi Emir Sakman

https://doi.org/10.16984/saufenbilder.1342645

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.

Keywords

Collocation method, nonlinear differential equations, boundary value problems, approximate solution.

Thanks

The authors would like to thank Dr. Erol Uzal for his contributions.

References

• [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 Nonlinear Delay IntegroDifferential Equations for Wireless Sensor Network and Internet of Things,” Sensors, vol. 20, no. 7, 1962, pp. 1-11, 2020.
• [15] S. Moreno-Mart´ın, L. Ros, E. Celaya, “Collocation Methods for Second Order Systems,” in Conf. Robotics: Science and Systems, New York City, NY, USA, 2022, pp. 1-11.
• [16] H. Yarcı, “On the solutions of nonlinear boundary value problems,” M.S. Dissertation, Dokuz Eylül University, Turkey, 2008.
• [17] H. Jafari, V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no.2, pp. 700–706, 2006.
• [18] V. S. Ertürk, S. Momani, “Differential transform method for obtaining positive solutions for two-point nonlinear boundary value problems”, International Journal: Mathematical Manuscripts, vol. 1, no.1, pp. 65-72, 2007.
• [19] L. Yanfei, Y. Qingfei, L. Hongyi, S. Hongli, Y. Yunlei, H. Muzhou, “Solving higher order nonlinear ordinary differential equations with least squares support vector machines,” Journal of Industrial And Management Optimization, vol. 16, no. 3, pp. 1481- 1502, 2020.
• [20] A. Ghorbani, A. M. Wazwaz, “A multiple variational iteration method for nonlinear two-point boundary value problems with nonlinear conditions,” International Journal of Computational Methods, vol 18, no. 1, 2021.