Abstract
The multiplicative version of Adams Bashforth–Moulton algorithms for the numerical solution of multiplicative differential equations is proposed. Truncation error estimation for these numerical algorithms is discussed. A specific problem is solved by methods defined in multiplicative sense. The stability properties of these methods are analyzed by using the standart test equation.
Similar content being viewed by others
References
Bashirov, A.E., Misirli Kurpinar, E., Ozyapici, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337, 36–48 (2008)
Aniszewska, D.: Multiplicative runge-kutta methods. Nonlinear Dyn. 50, 265–272 (2007)
Grossman, M.: Bigeometric Calculus, a System with a Scale-free Derivative. Archimedes Foundation, Rockport (1983)
Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove, Massachusats (1972)
Riza, M., Ozyapici, A., Misirli, E.: Multiplicative Finite Difference Methods. Q. Appl. Math. 67, 745–754 (2009)
Suli, E., Mayers, D.F.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003)
Misirli, E., Ozyapici, A.: Exponential approximations on multiplicative calculus. Proc. Jangjeon Math. Soc. 12, 227–236 (2009)
Dahlquist, G.G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)
Ehle, B.L.: On Pade approximations to the exponential function and a-stable methods for the numerical solution of initial value problems. Report 2010, University of Waterloo (1969)
Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Herman, Paris (1938).
Aniszewska, D., Rybaczuk, M.: Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems. Nonlinear Dyn. 54, 345–354 (2008)
Rybaczuk, M., Stoppel, P.: The fractal growth of fatigue defects in materials. Int. J. Fract. 103, 71–94 (2000)
Nottale, L.: Scale, relativity and fractal spacetime: applications to quantum physics, cosmology and chaotic systems. Chaos Soliton Fract. 7, 877–938 (1996)
DAmbrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 54, 169–193 (2010)
Khaliq, A.Q.M., Twizell, E.H.: Stability regions for one-step multiderivative methods in PECE mode with application to stiff systems. Int. J. Comput. Math. 17, 323–338 (1985)
Prothero, A., Robinson, A.: On the stability and the accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974)
Lambert, J.D.: Computational Methods in Ordinary Differential Equations, xv+278 pp. Wiley, Chichester (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Misirli, E., Gurefe, Y. Multiplicative Adams Bashforth–Moulton methods. Numer Algor 57, 425–439 (2011). https://doi.org/10.1007/s11075-010-9437-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-010-9437-2