Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

A NEW APPROACH FOR NUMERICAL SOLUTION OF LINEAR AND NON-LINEAR SYSTEMS

  • ZEYBEK, HALIL (Department of Applied Mathematics, Faculty of Computer Science, Abdullah Gul University) ;
  • DOLAPCI, IHSAN TIMUCIN (Department of Mechanical Engineering, Faculty of Engineering, Celal Bayar University)
  • Received : 2016.05.24
  • Accepted : 2016.09.19
  • Published : 2017.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, Taylor matrix algorithm is designed for the approximate solution of linear and non-linear differential equation systems. The algorithm is essentially based on the expansion of the functions in differential equation systems to Taylor series and substituting the matrix forms of these expansions into the given equation systems. Using the Mathematica program, the matrix equations are solved and the unknown Taylor coefficients are found approximately. The presented numerical approach is discussed on samples from various linear and non-linear differential equation systems as well as stiff systems. The computational data are then compared with those of some earlier numerical or exact results. As a result, this comparison demonstrates that the proposed method is accurate and reliable.

Keywords

References

  1. I.H. Abdel-Halim Hassan, Application to differential transformation method for solving systems of di erential equations, Appl. Math. Model. 32 (2008), 2552-2559. https://doi.org/10.1016/j.apm.2007.09.025
  2. J. Biazar and H. Ghazvini, He's variational iteration method for solving linear and non-linear systems of ordinary differential equations, Appl. Math. Comput. 191 (2007), 287-297.
  3. B. Bulbul and M. Sezer, A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation, Appl. Math. Lett. 24 (2011), 1716-1720. https://doi.org/10.1016/j.aml.2011.04.026
  4. M. Gulsu and M. Sezer, The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. Comput. 168 (2005), 76-88.
  5. M. Gulsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82 (2005), 629-642. https://doi.org/10.1080/00207160512331331156
  6. M. Gulsu and M. Sezer, A Taylor polynomial approach for solving differential-difference equations, J. Comput. and Appl. Math. 186 (2006), 349-364. https://doi.org/10.1016/j.cam.2005.02.009
  7. M. Gulsu and M. Sezer, On the solution of Riccati equation by the Taylor matrix method, Appl. Math. Comput. 176 (2006), 414-421.
  8. M. Gulsu, M. Sezer and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid legendre and taylor polynomials, J. Franklin Inst. 343 (2006), 647-659. https://doi.org/10.1016/j.jfranklin.2006.03.015
  9. M. Gulsu and M. Sezer, Approximations to the solution of linear Fredholm integro differential-difference equation of high order, J. Franklin Inst. 343 (2006), 720-737. https://doi.org/10.1016/j.jfranklin.2006.07.003
  10. N. Guzel and M. Bayram, On the numerical solution of stiff systems, Appl. Math. Comput. 170 (2005), 230-236.
  11. R.P. Kanwal and K.C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol. 20 (1989), 411-414. https://doi.org/10.1080/0020739890200310
  12. D. Kaya, A reliable method for the numerical solution of the kinetics problems, Appl. Math. Comput. 156 (2004), 261-270.
  13. C. Kesan, Taylor polynomial solutions of linear differential equations, Appl. Math. Comput. 142 (2003), 155-165.
  14. N. Kurt and M. cevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mechanics Research Communications 35 (2008), 530-536. https://doi.org/10.1016/j.mechrescom.2008.05.001
  15. S. Nas, S. Yalcnbas and M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Int. J. Math. Educ. Sci. Technol. 31 (2000), 213-225. https://doi.org/10.1080/002073900287273
  16. M. Sezer, Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol. 25 (1994), 625-633. https://doi.org/10.1080/0020739940250501
  17. M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol. 27 (1996), 821-834. https://doi.org/10.1080/0020739960270606
  18. M. Sezer, A. Karamete and M. Gulsu, Taylor polynomial solutions of systems of linear differential equations with variable coefficients, Int. J. Comput. Math. 82 (2005), 755-764. https://doi.org/10.1080/00207160512331323336
  19. M. Sezer and M. Gulsu, A new polynomial approach for solving difference and Fredholm integro-difference equations with mixed argument, Appl. Math. Comput. 171 (2005), 332-344.
  20. M. Sezer and M. Gulsu, Polynomial solution of the most general linear Fredholm integro differential-difference equation by means of Taylor matrix method, Int. J. Complex Variables 50 (2005), 367-382.
  21. M. Sezer, B. Tanay and M. Gulsu, A polynomial approach for solving high-order linear complex differental equations with variable coefficients in disc, Erciyes Universitesi Fen Bilimleri Enstitusu Dergisi 25 (2009), 374-389.
  22. S. Yalcinbas and M. Sezer, The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), 291-308.
  23. S. Kome, M.T. Atay, A. Eryilmaz, C. Kome and S. Piipponen, Comparative numerical solutions of stiff ordinary differential equations using Magnus series expansion method, New Trends in Mathematical Sciences 3 (2015), 35-45.
  24. M.T. Atay, A. Eryilmaz and S. Kome, Magnus series expansion method for solving nonhomogeneous stiff systems of ordinary differential equations, Kuwait J. Sci. 43 (2016), 25-38.