On highly oscillatory problems arising in electronic engineering
ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 785-804.

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.

DOI : 10.1051/m2an/2009024
Classification : 65L05, 65T99
Mots-clés : high oscillation, quadrature, ordinary differential equations 
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     title = {On highly oscillatory problems arising in electronic engineering},
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Condon, Marissa; Deaño, Alfredo; Iserles, Arieh. On highly oscillatory problems arising in electronic engineering. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 785-804. doi : 10.1051/m2an/2009024. http://archive.numdam.org/articles/10.1051/m2an/2009024/

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, (1964). | MR

[2] 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. | MR

[3] E. Dautbegovic, M. Condon and C. Brennan, An efficient nonlinear circuit simulation technique. IEEE Trans. Microwave Theory Tech. 53 (2005) 548-555.

[4] P.J. Davis and P. Rabinowitz, Methods of Numerical Integration. Second Edition, Academic Press, Orlando, USA (1984). | MR | Zbl

[5] V. Grimm and M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A: Math. Gen. 39 (2006) 5495-5507. | MR | Zbl

[6] S. Haykin, Communications Systems. Fourth Edition, John Wiley, New York, USA (2001).

[7] D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44 (2006) 1026-1048. | MR | Zbl

[8] A. Iserles, On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42 (2002a) 561-599. | MR | Zbl

[9] A. Iserles, Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Num. Anal. 43 (2002b) 145-160. | MR | Zbl

[10] A. Iserles and S.P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation. BIT 44 (2004) 755-772. | MR | Zbl

[11] A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives. Proc. Royal Soc. A 461 (2005) 1383-1399. | MR | Zbl

[12] A. Iserles and S.P. Nørsett, From high oscillation to rapid approximation I: Modified Fourier expansions. IMA J. Num. Anal. 28 (2008) 862-887. | MR

[13] 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).

[14] M. Khanamirian, Quadrature methods for systems of highly oscillatory ODEs. Part I. BIT 48 (2008) 743-761. | MR | Zbl

[15] C.A. Micchelli and T.J. Rivlin, Quadrature formulæ and Hermite-Birkhoff interpolation. Adv. Maths 11 (1973) 93-112. | MR | Zbl

[16] S. Olver, Moment-free numerical integration of highly oscillatory functions. IMA J. Num. Anal. 26 (2006) 213-227. | MR | Zbl

[17] R. Pulch, Multi-time scale differential equations for simulating frequency modulated signals. Appl. Numer. Math. 53 (2005) 421-436. | MR | Zbl

[18] 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. | MR | Zbl

[19] C.J. Weisman, The Essential Guide to RF and Wireless. Second Edition, Prentice-Hall, Englewood Cliffs, USA (2002).

[20] R. Wong, Asymptotic Approximations of Integrals. SIAM, Philadelphia (2001). | MR | Zbl

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