Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-17T11:25:25.904Z Has data issue: false hasContentIssue false

On highly oscillatory problems arising in electronic engineering

Published online by Cambridge University Press:  08 July 2009

Marissa Condon
School of Electronic Engineering, Dublin City University, Dublin 9, Ireland.
Alfredo Deaño
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK.
Arieh Iserles
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK.
Get access


In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.

Research Article
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, (1964).
D. Cohen, T. Jahnke, K. Lorenz and C. Lubich, Numerical integrators for highly oscillatory Hamiltonian systems: a review, in Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke Ed., Springer-Verlag (2006) 553–576.
Dautbegovic, E., Condon, M. and Brennan, C., An efficient nonlinear circuit simulation technique. IEEE Trans. Microwave Theory Tech. 53 (2005) 548555. CrossRef
P.J. Davis and P. Rabinowitz, Methods of Numerical Integration. Second Edition, Academic Press, Orlando, USA (1984).
Grimm, V. and Hochbruck, M., Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A: Math. Gen. 39 (2006) 54955507. CrossRef
S. Haykin, Communications Systems. Fourth Edition, John Wiley, New York, USA (2001).
Huybrechs, D. and Vandewalle, S., On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44 (2006) 10261048. CrossRef
Iserles, A., On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42 (2002a) 561599. CrossRef
Iserles, A., Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Num. Anal. 43 (2002b) 145160.
Iserles, A. and Nørsett, S.P., On quadrature methods for highly oscillatory integrals and their implementation. BIT 44 (2004) 755772. CrossRef
Iserles, A. and Nørsett, S.P., Efficient quadrature of highly oscillatory integrals using derivatives. Proc. Royal Soc. A 461 (2005) 13831399. CrossRef
Iserles, A. and Nørsett, S.P., From high oscillation to rapid approximation I: Modified Fourier expansions. IMA J. Num. Anal. 28 (2008) 862887. CrossRef
M.C. Jeruchim, P. Balaban and K.S. Shanmugan, Simulation of Communication Systems, Modeling, Methodology and Techniques. Second Edition, Kluwer Academic/Plenum Publishers, New York, USA (2000).
Khanamirian, M., Quadrature methods for systems of highly oscillatory ODEs. Part I. BIT 48 (2008) 743761. CrossRef
Micchelli, C.A. and Rivlin, T.J., Quadrature formulæ and Hermite-Birkhoff interpolation. Adv. Maths 11 (1973) 93112. CrossRef
Olver, S., Moment-free numerical integration of highly oscillatory functions. IMA J. Num. Anal. 26 (2006) 213227. CrossRef
Pulch, R., Multi-time scale differential equations for simulating frequency modulated signals. Appl. Numer. Math. 53 (2005) 421436. CrossRef
J. Roychowdhury, Analysing circuits with widely separated time scales using numerical PDE methods. IEEE Trans. Circuits Sys. I, Fund. Theory Appl. 48 (2001) 578–594.
C.J. Weisman, The Essential Guide to RF and Wireless. Second Edition, Prentice-Hall, Englewood Cliffs, USA (2002).
R. Wong, Asymptotic Approximations of Integrals. SIAM, Philadelphia (2001).