On Rayleigh-Ritz and Collocation Methods for Solving Second Order Boundary Value Problems of Ordinary Differential Equations
Abstract
In this article, the Rayleigh-Ritz method is compared with the collocation method in solving second order boundary value problems of ordinary differential equations with the associated boundary conditions. The trial solution for the Rayleigh-Ritz method has to be chosen in such a way that the linearly independent functions must satisfy the boundary conditions. The collocation method, on the other hand make use of any basis function as the trial solution. The trial solution is then made to satisfy the differential equation and the boundary conditions at some interior points in the solution interval. Results obtained using the two methods show that the collocation method is simpler, easier and more accurate than the Rayleigh-Ritz method.
References
Fairweather, G. (1978). Finite element Galerkin methods for differential equations. Marcel Dekker, New York.
Gerald, C. F., & Wheatley, P. O. (2004). Applied numerical analysis (7th Edition). Chegg Inc.
Davis, M. E. (1984). Boundary-Value Problems for Ordinary Differential Equations: Finite Element Methods. Retrieved from https://authors.library.caltech.edu/25061/5/NumMethChE84-Ch3-BVPforODE-FEM.pdf
Ali, H., & Islam, M. S. (2017). Generalized Galerkin finite element formulation for the numerical solutions of second order nonlinear boundary value problems. J. Bangladesh Math. Soc., 37, 147-159. https://doi.org/10.3329/ganit.v37i0.35733
Arora, S., Dhaliwal, S. S., & Kukreja, V. K. (2005). Solution of two-point boundary value problems using orthogonal collocation on finite elements. Appl. Math. Comput., 171, 358-370. https://doi.org/10.1016/j.amc.2005.01.049
Cryer, C. W. (1973). The numerical solution of boundary value problems for second-order functional differential equations by finite differences. Numer. Math., 20, 288-299. https://doi.org/10.1007/BF01407371
Rao, S. S. (2010). The finite element method in engineering. Elsevier.
Islam, M. S., Ahmed, M., & Hossain, M. A. (2010). Numerical solutions of IVP using finite element method with Taylor series. GANIT: Journal of Bangladesh Mathematical Society, 30, 51-58. https://doi.org/10.3329/ganit.v30i0.8503
Jain, M. K., Iyengar, S. R. K., & Jain, R. K. (2012). Numerical methods for scientific and engineering computation (Sixth Edition). New Age International Publishers.
Villadsen, J., & Stewart, W. E. (1967). Solution of boundary value problems by orthogonal collocation. Chem. Eng. Sci., 22, 1483-1501. https://doi.org/10.1016/0009-2509(67)80074-5
Jang, B. (2008). Two-point boundary value problems by the extended Adomian decomposition method. J. Comput. Appl. Math., 219(1), 253-262. https://doi.org/10.1016/j.cam.2007.07.036
Aminikhah, H., & Hemmatnezhad, M. (2011). An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations. Applied Mathematics Letters, 24(9), 1502-1508. https://doi.org/10.1016/j.aml.2011.03.032
Burden, R. L., & Faires, J. D. (2010). Numerical analysis. Brooks/Cole, USA.
Musa, H., Suleiman, M. B., & Ismail, F. (2015). An implicit 2-point block extended backward differentiation formulae for solving stiff IVPs. Malaysian Journal of Mathematical Sciences, 9(1), 35-51.
Sagir, A. M. (2014). Numerical treatment of block method for the solution of ordinary differential equations. Int. J. Bioeng Life Sci., 8(2), 259-263.
Potta, A. U., & Alabi, T. J. (2015). Block method with one hybrid point for the solution of first-order initial value problems of ordinary differential equations. Int. J. Pure Appl. Math., 103(3), 511-521. https://doi.org/10.12732/ijpam.v103i3.12
Zavalani, G. (2015). A Galerkin finite element method for two-point boundary value problems of ordinary differential equations. Applied and Computational Mathematics, 4(2), 64-68. https://doi.org/10.11648/j.acm.20150402.15
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