Multiplicative finite difference methods
Authors:
Mustafa Riza, Ali Özyapici and Emine Misirli
Journal:
Quart. Appl. Math. 67 (2009), 745-754
MSC (2000):
Primary 65L12; Secondary 65N06
DOI:
https://doi.org/10.1090/S0033-569X-09-01158-2
Published electronically:
May 14, 2009
MathSciNet review:
2588234
Full-text PDF Free Access
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Abstract: Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.
References
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- Michael Grossman and Robert Katz, Non-Newtonian calculus, Lee Press, Pigeon Cove, Mass., 1972. MR 0430173
- V. Volterra, B. Hostinsky, Operations Infinitesimales Lineares, Gauthier-Villars, Paris, 1938.
- Michael Grossman, Bigeometric calculus, Archimedes Foundation, Rockport, Mass., 1983. A system with a scale-free derivative. MR 695495
- Agamirza E. Bashirov, Emine Mısırlı Kurpınar, and Ali Özyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl. 337 (2008), no. 1, 36–48. MR 2356052, DOI https://doi.org/10.1016/j.jmaa.2007.03.081
- Marek Rybaczuk, Alicja Kȩdzia, and Witold Zieliński, The concept of physical and fractal dimension. II. The differential calculus in dimensional spaces, Chaos Solitons Fractals 12 (2001), no. 13, 2537–2552. MR 1851077, DOI https://doi.org/10.1016/S0960-0779%2800%2900231-9
- D. Aniszewska, Multiplicative Runge–Kutta methods, Nonlinear Dynamics 50 (1-2) (2007) 265–272.
- W. Kasprzak, B. Lysik, M. Rybaczuk, Dimensions, invariants models and fractals, in: Ukranian Society on Fracture Mechanics SPOLOM, Wroclaw-Lviv, Poland, 2004.
References
\expandafter\ifx\csname url\endcsname\def#1#1\expandafter\ifx\csname urlprefix\endcsname\def\urlprefix{URL }
- M. Grossmann, R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, Massachusetts, 1972. MR 0430173 (55:3180)
- V. Volterra, B. Hostinsky, Operations Infinitesimales Lineares, Gauthier-Villars, Paris, 1938.
- M. Grossmann, Bigeometric Calculus, A System with a Scale-free Derivative, Archimedes Foundation, Rockport, 1983. MR 695495 (84h:26003)
- A. Bashirov, E. Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications 337 (1) (2008) 36–48. MR 2356052 (2008k:26001)
- M. Ryabczuk, A. Kedzia, W. Zielinski, The concept of physical and fractal dimension II. The differential calculus in dimensional spaces, Chaos Solitons Fractals 12 (2001) 2537–2552. MR 1851077
- D. Aniszewska, Multiplicative Runge–Kutta methods, Nonlinear Dynamics 50 (1-2) (2007) 265–272.
- W. Kasprzak, B. Lysik, M. Rybaczuk, Dimensions, invariants models and fractals, in: Ukranian Society on Fracture Mechanics SPOLOM, Wroclaw-Lviv, Poland, 2004.
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Additional Information
Mustafa Riza
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimağusa – North Cyprus, via Mersin 10, Turkey
Email:
mustafa.riza@emu.edu.tr
Ali Özyapici
Affiliation:
Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Email:
ali.ozyapici@emu.edu.tr
Emine Misirli
Affiliation:
Department of Mathematics, Ege University, Bornova, Izmir, Turkey
Email:
emine.kurpinar@ege.edu.tr
Keywords:
Finite difference method,
multiplicative calculus,
Volterra calculus,
bigeometric calculus
Received by editor(s):
August 24, 2008
Published electronically:
May 14, 2009
Additional Notes:
The first author was supported by the A-Type Research Grant of Eastern Mediterranean University
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.
