THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Elif Nuray Yildirim

,

Ali Akgül

,

Mustafa Inc

REPRODUCING KERNEL FUNCTIONS AND HOMOGENIZING TRANSFORMS

ABSTRACT
A lot of problems of the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution and its effectiveness are as important as the modelling. In this paper, on the basis of reproducing kernel theory, the reproducing kernel functions have been obtained for solving some non-linear higher order differential equations. Additionally, for each problem the homogenizing transforms have been obtained.
KEYWORDS
reproducing kernel functions, partial differential equations, bounded linear operator, Hilbert spaces
PAPER SUBMITTED: 2020-06-03
PAPER REVISED: 2020-10-22
PAPER ACCEPTED: 2020-11-03
PUBLISHED ONLINE: 2021-01-24
DOI REFERENCE: https://doi.org/10.2298/TSCI200603002Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Special issue 1, PAGES [9 - 18]
REFERENCES
  1. A. Alvandi and M. Paripour. The combined reproducing kernel method and Taylor series to solve nonlinear Abel's integral equations with weakly singular kernel, Cogent Mathematics, 3, (2016).
  2. A. Freihat, R. Abu-Gdairi, H. Khalil, E. Abuteen, M. Al-Smadi, and R. A. Khan. Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems,American Journal of Applied Sciences, 13, 501-510, (2016).
  3. Abbasbandy, S. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Appl. Math. Comput, 172, 485-490 (2006).
  4. Adomian, G. Nonlinear Stochastic Operator Equations,Academic Press, San Diego (1986).
  5. Aronszajn, N. Theory of reproducing kernels,Trans. Amer. Math. Soc. 68, 337-404 (1950)
  6. B. S. H. Kashkari and M. I. Syam. Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation, Complexity ,1-7 (2018).
  7. Bergman, Stefan. The Kernel Function and Conformal Mapping, American Mathematical Society, New York (1950)
  8. Da¸sçıo˘glu A., Yaslan, H. The solution of high-order nonlinear ordinary differential equations by Chebshev series, Appl. Math. Comput. 217, 5658-5666 (2011)
  9. E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations, Tata McGraw-Hill Publishing, 1972.
  10. Fox, L. and Mayers, D. F. Numerical Solution of Ordinary Differential Equations, Chapman and Hall, 1987.
  11. G. Akram and H. U. Rehman. Numerical solution of eighth order boundary value problems in reproducing Kernel space,Numer. Algor.,62 (3),527-540 (2013).
  12. Hoppensteadt, F. Properties of solutions of ordinary differential equations with small parameters. Communications on Pure and Applied Mathematics, 24(6), 807-840 (1971).
  13. Horn, M.K. Fourth- and fifth-order, scaled Runge-Kutta algorithms for treating dense output, SI AM J. Numer. Analysis, 20,558-568 (1983).
  14. K. Atkinson, W. Han, and D. Stewart. Numeical solution of differential equations, JohnWiley and Sons, Inc., 2009.
  15. M. Cui and Y. Lin, Nonlinear numerical analysis in the reproducing Kernel space. New York: Nova Science Publishers, 2009.
  16. M. I. Syam, Q. M. Al-Mdallal, and M. Al-Refai. A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Communications in Numerical Analysis, 2, 217-232 (2017).
  17. Menahem Friedman and Abraham Kandel. Fundamentals of Computer Numerical Analysis, 409-422(1994).
  18. Öztürk, Y., Gülsu, M. 2016. The Approximate Solution of High-Order Nonlinear Ordinary Differential Equations by Improved Collocation Method with Terms of Shifted Chebyshev Polynomials,Int. J. Appl. Comput. Math 2, 519-531 (2016).
  19. Quinney, D. An Introduction to Numerical Solution of Differential Equations,Research Studies Press, 51, 1987.
  20. Ramos, J.I. Linearization techniques for singular initial-value problems of ordinary differential equations, Appl. Math. Comput. 161, 525-542 (2005).
  21. Sell, G.R. On the fundamental theory of ordinary differential equations, Journal of Differential Equations 1, 370-392 (1965).
  22. Shoichiro Nakamura. Applied Numerical Methods With Software,Prentice Hall, New Jersey, 303-308 (1981).
  23. W. Jiang and T. Tian. Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, Applied Mathematical Modelling,39(16), 4871-4876 (2015).
  24. Wazwaz, A.M. A new method for solving initial value problems in second-order ordinary differential equations,Appl. Math. Comput. 128, 45-57 (2002).
  25. Wazwaz, A.M. The numerical solution of fifth-order boundary value problems by the decomposition method, J. Comput. Appl. Math. 136(1-2), 259-270 (2001).
  26. X. Y. Li, B. Y.Wu, and R. T.Wang. Reproducing Kernel Method for Fractional Riccati Differential Equations, Abstract and Applied Analysis, 1-6 (2014), Available: dx.doi.org/10.1155/2014/970967.
  27. Y. Lu et al. Solving higher order nonlinear ordinary differential equations with least squares support vector machines,Journal of Industrial Management Optimization, 16(3), 1481-1502 (2020).
  28. Waeleh et al. Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code,Journal of Mathematics and Statistics,8(1),77-81 (2012).
PDF VERSION [DOWNLOAD]
Reproducing kernel functions and homogenizing transforms

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, 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