- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
International Journal of Differential Equations
Volume 2013 (2013), Article ID 857410, 11 pages
Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations
1Department of Applied Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
2Okayama University of Science, Okayama 700-0005, Japan
Received 7 December 2012; Accepted 8 January 2013
Academic Editor: Norio Yoshida
Copyright © 2013 Mervan Pašić and Satoshi Tanaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- C. Tricot, Curves and Fractal Dimension, Springer, New York, NY, USA, 1995.
- M. Pašić, “Fractal oscillations for a class of second order linear differential equations of Euler type,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 211–223, 2008.
- M. K. Kwong, M. Pašić, and J. S. W. Wong, “Rectifiable oscillations in second-order linear differential equations,” Journal of Differential Equations, vol. 245, no. 8, pp. 2333–2351, 2008.
- M. Pašić and J. S. W. Wong, “Rectifiable oscillations in second-order half-linear differential equations,” Annali di Matematica Pura ed Applicata: Series 4, vol. 188, no. 3, pp. 517–541, 2009.
- M. Pašić and S. Tanaka, “Fractal oscillations of self-adjoint and damped linear differential equations of second-order,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 2281–2293, 2011.
- Y. Naito, M. Pašić, S. Tanaka, and D. Žubrinić, “Fractal oscillations near domain boundary of radially symmetric solutions of p-Laplace equations, fractal geometry and dynamical systems in pure and applied mathematics,” Contemporary Mathematics American Mathematical Society. In press.
- W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, Mass, USA, 1965.
- P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass, USA, 2nd edition, 1982.
- F. Neuman, “A general construction of linear differential equations with solutions of prescribed properties,” Applied Mathematics Letters, vol. 17, no. 1, pp. 71–76, 2004.
- F. Neuman, “Structure of solution spaces via transformation,” Applied Mathematics Letters, vol. 21, no. 5, pp. 529–533, 2008.
- P. Borgnat and P. Flandrin, “On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation,” Applied and Computational Harmonic Analysis, vol. 15, no. 2, pp. 134–146, 2003.
- E. J. Candès, P. R. Charlton, and H. Helgason, “Detecting highly oscillatory signals by chirplet path pursuit,” Applied and Computational Harmonic Analysis, vol. 24, no. 1, pp. 14–40, 2008.
- S. Jaffard and Y. Meyer, “Wavelet methods for pointwise regularity and local oscillations of functions,” Memoirs of the American Mathematical Society, vol. 123, no. 587, pp. 1–110, 1996.
- Y. Meyer and H. Xu, “Wavelet analysis and chirps,” Applied and Computational Harmonic Analysis, vol. 4, no. 4, pp. 366–379, 1997.
- G. Ren, Q. Chen, P. Cerejeiras, and U. Kaehle, “Chirp transforms and chirp series,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 356–369, 2011.
- M. Képesi and L. Weruaga, “Adaptive chirp-based time-frequency analysis of speech signals,” Speech Communication, vol. 48, no. 5, pp. 474–492, 2006.
- L. Weruaga and M. Képesi, “The fan-chirp transform for non-stationary harmonic signals,” Signal Processing, vol. 87, no. 6, pp. 1504–1522, 2007.
- E. Barlow, A. J. Mulholland, A. Nordon, and A. Gachagan, “Theoretical analysis of chirp excitation of contrast agents,” Physics Procedia, vol. 3, no. 1, pp. 743–747, 2009.
- M. H. Pedersen, T. X. Misaridis, and J. A. Jensen, “Clinical evaluation of chirp-coded excitation in medical ultrasound,” Ultrasound in Medicine and Biology, vol. 29, no. 6, pp. 895–905, 2003.
- T. Paavle, M. Min, and T. Parve, “Using of chirp excitation for bioimpedance estimation: theoretical aspects and modeling,” in Proceedings of the 11th International Biennial Baltic Electronics Conference (BEC'08), pp. 325–328, Tallinn, Estonia, October 2008.
- M. Pašić, “Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 724–738, 2007.
- K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Hoboken, NJ, USA, 1999.
- P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1995.
- K. J. Falconer, “On the Minkowski measurability of fractals,” Proceedings of the American Mathematical Society, vol. 123, no. 4, pp. 1115–1124, 1995.
- C. Q. He and M. L. Lapidus, Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Function, vol. 127 of Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, USA, 1997.
- M. L. Lapidus and M. van Frankenhuijsen, Fractal geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2006.
- M. Pašić, “Minkowski-Bouligand dimension of solutions of the one-dimensional -Laplacian,” Journal of Differential Equations, vol. 190, no. 1, pp. 268–305, 2003.
- J. S. W. Wong, “On rectifiable oscillation of Euler type second order linear differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2007, no. 20, pp. 1–12, 2007.
- M. Pašić, “Rectifiable and unrectifiable oscillations for a generalization of the Riemann-Weber version of Euler differential equation,” Georgian Mathematical Journal, vol. 15, no. 4, pp. 759–774, 2008.
- M. Pašić and S. Tanaka, “Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 27–42, 2011.
- M. Pašić, D. Žubrinić, and V. Županović, “Oscillatory and phase dimensions of solutions of some second-order differential equations,” Bulletin des Sciences Mathématiques, vol. 133, no. 8, pp. 859–874, 2009.
- L. Korkut and M. Resman, “Fractal oscillations of chirp-like functions,” Georgian Mathematical Journal, vol. 19, no. 4, pp. 705–720, 2012.