THERMAL SCIENCE

International Scientific Journal

External Links

NUMERICAL SOLUTIONS OF A CLASS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS IN HERMITE SERIES

ABSTRACT
The purpose of this paper is to present a Hermite polynomial approach for solving a high-order ordinary differential equation with nonlinear terms under mixed conditions. The method we used is a matrix method based on collocation points together with truncated Hermite series and reduces the solution of equation to solution of a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Hermite coefficients. In addition, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing result in literature.
KEYWORDS
PAPER SUBMITTED: 2018-12-15
PAPER REVISED: 2018-12-30
PAPER ACCEPTED: 2019-01-10
PUBLISHED ONLINE: 2019-03-09
DOI REFERENCE: https://doi.org/10.2298/TSCI181215047G
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 1, PAGES [S339 - S351]
REFERENCES
  1. Gavalas, G. R., Nonlinear Differential Equations of Chemically Reacting Systems, Springer-Verlag New York Inc., 2013
  2. Fucik, S. and Kufner, A., Nonlinear Differential Equations, Elsevier, 2014
  3. Ordokhani, Y. and Mohtashami, M. J., Approximate Solution of Nonlinear Fredholm Integro-Differential Equations with time delay by using Taylor Method, Journal of Science (Kharazmi University), 9(2010), 1, pp. 73-84
  4. Yuksel, G., et al., Chebyshev polynomial solutions of a class of second-order nonlinear ordinary differential equations, Journal of Advanced Research in Scientific Computing, 3(2011), 4, pp. 11-24
  5. Guler, C., A new numerical algorithm for the abel equation of the second kind, International Journal of Computer Mathematics, 84(2007), 1, pp. 109-119
  6. Gurbuz, B. And Sezer, M., Laguerre polynomial approach for solving Lane-Emden type functional differential equations, Applied Mathematics and Computation, 242(2014), pp. 255-264
  7. Bulbul, B. and Sezer, M., Numerical solution of duffing equation by using an improved taylor matrix method, Journal of Applied Mathematics, 2013(2013)
  8. Gulsu, M. and Sezer, M., On the solution of Riccati equation by Taylor matrix method, Applied Mathematics and Computation, 176(2006), 2, pp. 414-421
  9. Balci, M. A. and Sezer, M., Hybrid euler taylor matrix method for solving of generalized linear fredholm integro-differential difference equations, Applied Mathematics and Computation, 273(2016), pp. 33-41
  10. Rawashdeh, M. S. and Maitama, S., Solving nonlinear ordinary differential equations using the NDM, Journal of Applied Analysis and Computation, 5(2015), 1, pp. 77-88
  11. Bülbül, B. and Sezer, M., A new approach to numerical solution of nonlinear Klein Gordon equation, Mathematical Problems in Engineering, 2013(2013)
  12. Guler, C. and Kaya, S. O., A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations, Mathematical Problems in Engineering, 2018(2018)
  13. Gokmen, E., et al., A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays, Journal of Computational and Applied Mathematics, 311(2017), pp. 354-363
  14. Saaty, T.L., Modern Nonlinear Equations, New York, 1981
  15. Hale, J., Theory of Functional Differential Equations, Springer-Verlag, New York, 1987
  16. Yalçınbaş, S., et al., A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348 (2011), 6, pp. 1128-1139
  17. Gokmen, E. and Sezer, M., Approximate solution of a model describing biological species living together by Taylor collocation method, New Trends in Mathematical Sciences, 3(2015), 2, pp. 147-158
  18. Akgonullu, N., et al., A Hermite Collocation Method for the Approximate Solutions of Higher-Order Linear Fredholm Integro-Differential Equations, Numerical Methods for Partial Differential Eqautions, 27(2010), 6, pp. 1708-1721
  19. Hristov, J., An approximate analytical (integral-balance) solution to a nonlinear heat diffusion equation, Thermal Science, 19(2015), 2, pp. 723-733
  20. Tian, Y., et al. A new method for solving a class of heat conduction equations, Thermal Science, 19(2015), 4, pp. 1205-1210

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence