Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2015, Article ID 594685, 10 pages
http://dx.doi.org/10.1155/2015/594685
Research Article

Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

1Department of Mathematics, Faculty of Sciences, Gazi University, 06100 Ankara, Turkey
2Department of Mathematics, Faculty of Sciences and Arts, Bozok University, 66200 Yozgat, Turkey
3Department of Mathematics, Faculty of Sciences and Arts, Batman University, 72060 Batman, Turkey

Received 1 August 2014; Revised 28 October 2014; Accepted 12 November 2014

Academic Editor: Allaberen Ashyralyev

Copyright © 2015 Uğur Kadak and Muharrem Özlük. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. Volterra and B. Hostinsky, Operations Infinitesimales Lineares, Gauthier-Villars, Paris, France, 1938.
  2. M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, Mass, USA, 1972.
  3. M. Grossman, Bigeometric Calculus, Archimedes Foundation, Rockport, Mass, USA, 1983.
  4. M. Grossman, The First Nonlinear System of Differential and Integral Calculus, Mathco, 1979.
  5. A. Ozyapıcı and E. Kurpınar, “Exponential approximation on multiplicative calculus,” in Proceedings of the 6th ISAAC Congress, p. 471, 2007.
  6. A. Özyapıcı and E. Kurpinar, “Notes on multiplicative calculus,” in Proceedings of the 20th International Congress of the Jangjeon Mathematical Society, vol. 67, p. 80, 2008.
  7. M. Riza, A. Özyapici, and E. Misirli, “Multiplicative finite difference methods,” Quarterly of Applied Mathematics, vol. 67, no. 4, pp. 745–754, 2009. View at Google Scholar · View at MathSciNet
  8. M. Rıza and B. Eminaga, “Bigeometric calculus—a modelling tool,” http://arxiv.org/abs/1402.2877.
  9. A. E. Bashirov, E. M. Kurpınar, and A. Ozyapıcı, “Multiplicative calculus and its applications,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 36–48, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. Bashirov and G. Bashirova, “Dynamics of literary texts and diffusion,” Online Journal of Communication and Media Technologies, vol. 1, no. 3, pp. 60–82, 2011. View at Google Scholar
  11. L. Florack and H. van Assen, “Multiplicative calculus in biomedical image analysis,” Journal of Mathematical Imaging and Vision, vol. 42, no. 1, pp. 64–75, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. D. Aniszewska, “Multiplicative Runge-Kutta methods,” Nonlinear Dynamics, vol. 50, no. 1-2, pp. 265–272, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. A. Uzer, “Multiplicative type complex calculus as an alternative to the classical calculus,” Computers & Mathematics with Applications, vol. 60, no. 10, pp. 2725–2737, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. A. Bashirov and M. Riza, “On complex multiplicative differentiation,” TWMS Journal of Applied and Engineering Mathematics, vol. 1, no. 1, pp. 75–85, 2011. View at Google Scholar · View at MathSciNet
  15. E. Misirli and Y. Gurefe, “Multiplicative Adams Bashforth-Moulton methods,” Numerical Algorithms, vol. 57, no. 4, pp. 425–439, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. F. Çakmak and F. Başar, “Some new results on sequence spaces with respect to non-Newtonian calculus,” Journal of Inequalities and Applications, vol. 2012, article 228, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. F. Çakmak and F. Başar, “Certain spaces of functions over the field of non-Newtonian complex numbers,” Abstract and Applied Analysis, vol. 2014, Article ID 236124, 12 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Tekin and F. Başar, “Certain sequence spaces over the non-Newtonian complex field,” Abstract and Applied Analysis, vol. 2013, Article ID 739319, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. U. Kadak, “Determination of the Köthe-Toeplitz duals over the non-Newtonian complex field,” The Scientific World Journal, vol. 2014, Article ID 438924, 10 pages, 2014. View at Publisher · View at Google Scholar
  20. U. Kadak and H. Efe, “Matrix transformations between certain sequence spaces over the non-Newtonian complex field,” The Scientific World Journal, vol. 2014, Article ID 705818, 12 pages, 2014. View at Publisher · View at Google Scholar
  21. U. Kadak and H. Efe, “The construction of Hilbert spaces over the Non-Newtonian field,” International Journal of Analysis, vol. 2014, Article ID 746059, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, Cengage Learning, 9th edition, 2012.
  23. F. Costabile and A. Napoli, “A class of collocation methods for numerical integration of initial value problems,” Computers & Mathematics with Applications, vol. 62, no. 8, pp. 3221–3235, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. A. Ashyralyev and Y. Özdemir, “On numerical solution of multipoint NBVP for hyperbolic-parabolic equations with Neumann condition,” AIP Conference Proceedings, vol. 1470, pp. 80–83, 2012. View at Google Scholar
  25. A. Ashyralyev and B. Hicdurmaz, “On the numerical solution of fractional Schrödinger differential equations with the Dirichlet condition,” International Journal of Computer Mathematics, vol. 89, no. 13-14, pp. 1927–1936, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. Ashyralyev and A. Sirma, “A note on the numerical solution of the semilinear Schrödinger equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2507–e2516, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus