Moreover, some numerical solutions are given to illustrate the accuracy and implementation of the method
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this study, a collocation method based on Bernstein polynomials is developed for solution of the nonlinear ordinary differential equations with variable coefficients, under the mixed conditions.
📊
Key Findings & Data:These equations are expressed as linear ordinary differential equations via quasilinearization method iteratively.
💡
Conclusion & Impact:Moreover, some numerical solutions are given to illustrate the accuracy and implementation of the method
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this study, a collocation method based on Bernstein polynomials is developed for solution of the nonlinear ordinary differential equations with variable coefficients, under the mixed conditions. These equations are expressed as linear ordinary differential equations via quasilinearization method iteratively. By using the Bernstein collocation method, solutions of these linear equations are approximated. Combining the quasilinearization and the Bernstein collocation methods, the approximation solution of nonlinear differential equations is obtained. Moreover, some numerical solutions are given to illustrate the accuracy and implementation of the method
🔖
Cite This Article
DASCİOGLU, A. A., ISLER, N. (2013). BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS. Mathematical and Computational Applications, Cilt: 18 - Sayı: 3, 293–300. https://ajindex.com/articles/279818
B. Ahmad, R. A. Khan and S. Sivasundaram, Generalized Quasilinearization Method for Nonlinear Functional Differential Equations, Journal of Applied Mathematics and Stochastic Analysis 16, 33-43, 2003
R. E. Belmann and R. E. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier Publishing Company, New York, 1965
F. Calio, F. Munoz and E. Marchetti, Direct and iterative methods for the numerical solution of mixed integral equations, Applied Mathematics and Computation 216, 3739-3746, 2010
A. Charles and Jr. Baird, Modified Quasilinearization Technique for the Solution of Boundary Value Problems for Ordinary Differential equations, Journal of Optimization Theory and Applications 3, 1969.5. Z. Dricia, F. A. McRaeb and J. V. Devic, Quasilinearization for functional differential equations with retardation and anticipation, Nonlinear Analysis 70, 1763- 1775, 2009
R. T. Farouki, and V.T. Rajan, Algorithms for Polynomials in Bernstein Form, Computer Aided Geometric Design 5, 1-26, 1988
G. G. Lorentz, Bernstein polynomials, Chelsea Publishing, New York, N.Y., 1986
K. Maleknejad and E. Najai, Numerical solution of nonlinear volterra integral equations using the idea of quasilinearization, Commun Nonlinear Sci Numer Simulat 16, 93-100, 2011
V. B. Mandelzweig and F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications 141, 268-281, 2001
S. G. Pandit, Quadratically converging iterative schemes for nonlinear volterra integral equations and an application, Journal of Applied Mathematics and Stochastic Analysis 10, 169-178, 1997
J. I. Ramos, Piecewise quasilinearization techniques for singular boundary-value problems, Computer Physics Communications 158, 12-25, 2004
E. L. Stanley, Quasilinearization and Invariant Imbedding, Academic Press, New York, 1968
P. Wang, Y. Wu, and B. Wiwatanapaphee, An Extension of Method of Quasilinearization for Integro-Diferential Equations, International Journal of Pure and Applied Mathematics 54, 27-37, 2009.