Abstract
In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very effective method of dicretizing the differential equation system in question. Numerical experiments illustrate the effectiveness of the asymptotic method in contrast to the standard Runge–Kutta method.
Similar content being viewed by others
References
Bleistein, N., Handlesman, R.: Asymptotic Expansions of Integrals. Dover, New York (1975)
Chedjou, J.C., Fotsin, H.B., Woafo, P., Domngang, S.: Analog simulation of the dynamics of a Van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans. Circ. Syst. I: Fundam. Theory Appl. 48(6), 748–757 (2001)
D. Cohen. Analysis and numerical treatment of highly oscillatory differential equations. PhD thesis, University of Geneva (2004)
Cohen, D., Hairer, E., Lubich, C.: Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3, 327–345 (2003)
Condon, M., Deaño, A., Iserles, A.: On systems of differential equations with extrinsic oscillation. Discret. Cont. Dyn. Syst. 28, 1345–1367 (2010)
Condon, M., Deaño, A., Iserles, A.: On second order differential equations with highly oscillatory forcing terms. Proc. Royal Soc. A. 466, 1809–1828 (2010)
Condon, M., Deaño, A., Gao, J., Iserles, A.: Asymptotic solvers for ordinary differential equations with multiple frequencies. NA2011/11
Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1, 445–466 (1961)
Fodjouong, G.J., Fotsin, H.B., Woafo, P.: Synchronizing modified van der Pol-Duffing oscillators with offset terms using observer design: application to secure communications. Phys. Scr. 75, 638–644 (2007)
Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)
Hairer, E., Nørsett, S., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer, Berlin (2006)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)
Njah, A.N., Vincent, U.E.: Chaos synchronization between single and double wells Duffing-Van der Pol oscillators using active control. Chaos Solitons Fractals 37, 1356–1361 (2008)
Sanz-Serna, J.M.: Modulated Fourier expansions and heterogeneous multiscale methods. IMA J. Numer. Anal. 29, 595–605 (2009)
Slight, T.J., et al.: A Lienard oscillator resonant tunnelling diode-laser diode hybrid integrated circuit: model and experiment. IEEE J. Quantum Electron. 44(12), 1158–1163 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was supported by the Natural Science Foundation of China (NSFC) (Grant No. 11201370, 11171270) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090201120061) and the Fundamental Research Funds for the Central Universities.
Rights and permissions
About this article
Cite this article
Condon, M., Deaño, A., Gao, J. et al. Asymptotic solvers for second-order differential equation systems with multiple frequencies. Calcolo 51, 109–139 (2014). https://doi.org/10.1007/s10092-013-0078-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10092-013-0078-4
Keywords
- Highly oscillatory problems
- Second-order differential equations
- Modulated Fourier expansions
- Multiple frequencies
- Numerical analysis