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The Runge–Kutta method in geometric multiplicative calculus

Part of: Numerical analysis: Ordinary differential equations Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 August 2015

Mustafa Riza  and
Hatice Aktöre
Mustafa Riza
Affiliation:
Department of Physics, Eastern Mediterranean University, Gazimağusa, North Cyprus, via Mersin 10, Turkey email mustafa.riza@emu.edu.tr
Hatice Aktöre
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimağusa, North Cyprus, via Mersin 10, Turkey email hatice.aktore@emu.edu.tr

Abstract

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This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.

MSC classification

Primary: 65L06: Multistep, Runge-Kutta and extrapolation methods
Secondary: 34K28: Numerical approximation of solutions
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 539 - 554
Copyright
© The Author(s) 2015 

References

Aniszewska, D., ‘Multiplicative Runge–Kutta methods’, Nonlinear Dynam. 50 (2007) no. 1, 265–272.CrossRefGoogle Scholar
Baranyi, J. and Roberts, T. A., ‘A dynamic approach to predicting bacterial growth in food’, Int. J. Food Microbiol. 23 (1994) no. 3, 277–294.CrossRefGoogle ScholarPubMed
Baranyi, J. and Roberts, T. A., ‘Mathematics of predictive food microbiology’, Int. J. Food Microbiol. 26 (1995) no. 2, 199–218.Google Scholar
Bashirov, A. and Riza, M., ‘Complex multiplicative calculus’, Preprint, 2011, arXiv:1103.1462v1.Google Scholar
Bashirov, A. E., Kurpınar, E. M. and Özyapıcı, A., ‘Multiplicative calculus and its applications’, J. Math. Anal. Appl. 337 (2008) no. 1, 36–48.Google Scholar
Bashirov, A. E., Mısırlı, E., Tandoğdu, Y. and Özyapıcı, A., ‘On modeling with multiplicative differential equations’, Appl. Math. J. Chinese Univ. 26 (2011) no. 4, 425–438.Google Scholar
Bashirov, A. E. and Riza, M., ‘On complex multiplicative differentiation’, TWMS J. Appl. Eng. Math. 1 (2011) no. 1, 51–61.Google Scholar
Butcher, J. C., ‘A stability property of implicit Runge–Kutta methods’, BIT 15 (1975) no. 4, 358–361.Google Scholar
Córdova-Lepe, F., ‘The multiplicative derivative as a measure of elasticity in economics’, TMAT Revista Latinoamericana de Ciencias e Ingenieria 2 (2006) no. 3.Google Scholar
Dahlquist, G. G., ‘A special stability problem for linear multistep methods’, BIT 3 (1963) no. 1, 27–43.Google Scholar
Ehle, B. L., ‘On Pade approximations to the exponential function and A-stable methods for the numerical solution of initial value problems’, PhD Thesis, University of Waterloo, 1969.Google Scholar
Florack, L., ‘Regularization of positive definite matrix fields based on multiplicative calculus’, Scale space and variational methods in computer vision , Lecture Notes in Computer Science 6667 (eds Bruckstein, A. M., Haar Romeny, B. M., Bronstein, A. M. and Bronstein, M. M.; Springer, Berlin, 2012) 786–796.Google Scholar
Grossman, M., Bigeometric calculus, A system with a scale-free derivative (Archimedes Foundation, Rockport, MA, 1983).Google Scholar
Grossman, M. and Katz, R., Non-Newtonian calculus (Lee Press, Pigeon Cove, MA, 1972).Google Scholar
Huang, L., ‘Growth kinetics of Listeria monocytogenes in broth and beef frankfurters—determination of lag phase duration and exponential growth rate under isothermal conditions’, J. Food Sci. 73 (2008) no. 5, E235–E242.Google Scholar
Huang, L., ‘Growth kinetics of Escherichia coli O157:H7 in mechanically-tenderized beef’, Int. J. Food Microbiol. 140 (2010) no. 1, 40–48.CrossRefGoogle ScholarPubMed
Huang, L., ‘Optimization of a new mathematical model for bacterial growth’, Food Control 32 (2012) no. 1, 283–288.CrossRefGoogle Scholar
Khaliq, A. Q. M. and Twizell, E. H., ‘Stability regions for one-step multiderivative methods in PECE mode with application to stiff systems’, Int. J. Comput. Math. 17 (1985) no. 3–4, 323–338.CrossRefGoogle Scholar
Misirli, E. and Gurefe, Y., ‘Multiplicative Adams Bashforth–Moulton methods’, Numer. Algorithms 57 (2011) no. 4, 425–439.Google Scholar
Ozyapici, A. and Kurpinar, E. M., ‘Exponential approximation on multiplicative calculus’, 6th ISAAC Congress (2007) 471.Google Scholar
Ozyapici, A. and Kurpinar, E. M., ‘Notes on multiplicative calculus’, 20th International Congress of the Jangjeon Mathematical Society, vol. 67 (2008), available from http://icjms20.home.uludag.edu.tr/abstracts.pdf.Google Scholar
Özyapıcı, A., Riza, M., Bilgehan, B. and Bashirov, A. E., ‘On multiplicative and Volterra minimization methods’, Numer. Algorithms (2013) 1–14.Google Scholar
Prothero, A. and Robinson, A., ‘On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations’, Math. Comp. 28 (1974) no. 125, 145–162.Google Scholar
Riza, M. and Eminağa, B., ‘Bigeometric calculus—a modelling tool’, Preprint, 2014, arXiv:1402.2877.Google Scholar
Riza, M., Özyapici, A. and Misirli, E., ‘Multiplicative finite difference methods’, Quart. Appl. Math. 67 (2009) no. 4, 745.Google Scholar
Stoer, J. and Bulirsch, R., Introduction to numerical analysis , 2nd edn (Springer, New York, 1993).Google Scholar
Uzer, A., ‘Multiplicative type complex calculus as an alternative to the classical calculus’, Comput. Math. Appl. 60 (2010) no. 10, 2725–2737.Google Scholar
Volterra, V. and Hostinsky, B., Opérations infinitésimales linéaires (Gauthier-Villars, Paris, 1938).Google Scholar