
- Previous Article
- CPAA Home
- This Issue
-
Next Article
Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity
The mathieu differential equation and generalizations to infinite fractafolds
1. | Department of Mathematics, Cornell University, Ithaca, NY 14853, USA |
2. | Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA |
3. | Department of Mathematics, The University of Hong Kong, Hong Kong |
4. | Department of Mathematics, Cornell University, Ithaca, NY 14853, USA |
One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions.
References:
[1] |
N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki,
The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.
doi: 10.1016/0024-3795(95)00699-0. |
[2] |
J. Avron and B. Simon,
The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.
doi: 10.1016/0003-4916(81)90005-1. |
[3] |
O. Ben-Bassat, R. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
[4] |
K. Dalrymple, R. Strichartz and J. Vinson,
Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: 10.1007/BF01261610. |
[5] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. |
[6] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
[7] |
A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007.
doi: 10.1137/1.9780898717822. |
[8] |
B. Hambly and T. Kumagai,
Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.
doi: 10.1112/S0024611599001744. |
[9] |
H. Hochstadt,
On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.
doi: 10.1137/0515005. |
[10] |
J. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
[11] |
M. Ionescu, L. Rogers and R. Strichartz,
Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.
doi: 10.4171/RMI/752. |
[12] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
[13] |
J. Kigami,
Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.
doi: 10.2307/2154402. |
[14] |
J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[15] |
D. Levy and J. Keller,
Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.
doi: 10.1002/cpa.3160160406. |
[16] |
W. Loud,
Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.
doi: 10.1016/0022-0396(75)90003-0. |
[17] |
R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. |
[18] |
H. Ruan and R. Strichartz,
Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.
doi: 10.4153/CJM-2009-054-5. |
[19] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. |
[20] |
R. Strichartz,
Fractals in the large, Can. J. Math., 50 (1996), 638-657.
doi: 10.4153/CJM-1998-036-5. |
[21] |
R. Strichartz,
Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.
doi: 10.1090/S0002-9947-03-03171-4. |
[22] |
R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006.
![]() ![]() |
[23] |
R. Strichartz,
Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.
doi: 10.4153/CJM-2009-055-9. |
[24] |
R. Strichartz and A. Teplyaev,
Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.
doi: 10.1007/s11854-012-0007-5. |
[25] |
A. Teplyaev,
Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: 10.1006/jfan.1998.3297. |
[26] |
M. Weinstein and J. Keller,
Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.
doi: 10.1137/0147062. |
show all references
References:
[1] |
N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki,
The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.
doi: 10.1016/0024-3795(95)00699-0. |
[2] |
J. Avron and B. Simon,
The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.
doi: 10.1016/0003-4916(81)90005-1. |
[3] |
O. Ben-Bassat, R. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
[4] |
K. Dalrymple, R. Strichartz and J. Vinson,
Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: 10.1007/BF01261610. |
[5] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. |
[6] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
[7] |
A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007.
doi: 10.1137/1.9780898717822. |
[8] |
B. Hambly and T. Kumagai,
Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.
doi: 10.1112/S0024611599001744. |
[9] |
H. Hochstadt,
On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.
doi: 10.1137/0515005. |
[10] |
J. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
[11] |
M. Ionescu, L. Rogers and R. Strichartz,
Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.
doi: 10.4171/RMI/752. |
[12] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
[13] |
J. Kigami,
Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.
doi: 10.2307/2154402. |
[14] |
J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[15] |
D. Levy and J. Keller,
Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.
doi: 10.1002/cpa.3160160406. |
[16] |
W. Loud,
Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.
doi: 10.1016/0022-0396(75)90003-0. |
[17] |
R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. |
[18] |
H. Ruan and R. Strichartz,
Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.
doi: 10.4153/CJM-2009-054-5. |
[19] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. |
[20] |
R. Strichartz,
Fractals in the large, Can. J. Math., 50 (1996), 638-657.
doi: 10.4153/CJM-1998-036-5. |
[21] |
R. Strichartz,
Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.
doi: 10.1090/S0002-9947-03-03171-4. |
[22] |
R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006.
![]() ![]() |
[23] |
R. Strichartz,
Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.
doi: 10.4153/CJM-2009-055-9. |
[24] |
R. Strichartz and A. Teplyaev,
Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.
doi: 10.1007/s11854-012-0007-5. |
[25] |
A. Teplyaev,
Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: 10.1006/jfan.1998.3297. |
[26] |
M. Weinstein and J. Keller,
Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.
doi: 10.1137/0147062. |
















































1 | 0.625056436387445 |
2 | 0.784428425813594 |
3 | 0.845139663995868 |
4 | 0.878143787704672 |
5 | 0.899154589232086 |
6 | 0.913800056779179 |
7 | 0.924632580597566 |
8 | 0.932988858616457 |
9 | 0.939640860147421 |
10 | 0.945067300763657 |
11 | 0.949581603650156 |
12 | 0.953397960289362 |
13 | 0.956667925443240 |
14 | 0.959502140424685 |
15 | 0.961982968671620 |
16 | 0.964174006346465 |
17 | 0.966133547589692 |
18 | 0.967239011961280 |
1 | 0.625056436387445 |
2 | 0.784428425813594 |
3 | 0.845139663995868 |
4 | 0.878143787704672 |
5 | 0.899154589232086 |
6 | 0.913800056779179 |
7 | 0.924632580597566 |
8 | 0.932988858616457 |
9 | 0.939640860147421 |
10 | 0.945067300763657 |
11 | 0.949581603650156 |
12 | 0.953397960289362 |
13 | 0.956667925443240 |
14 | 0.959502140424685 |
15 | 0.961982968671620 |
16 | 0.964174006346465 |
17 | 0.966133547589692 |
18 | 0.967239011961280 |
[1] |
Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013 |
[2] |
Ferenc Weisz. Cesàro summability and Lebesgue points of higher dimensional Fourier series. Mathematical Foundations of Computing, 2022, 5 (3) : 241-257. doi: 10.3934/mfc.2021033 |
[3] |
G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509 |
[4] |
Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020 |
[5] |
David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730 |
[6] |
Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118 |
[7] |
Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 |
[8] |
Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic and Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445 |
[9] |
Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic and Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036 |
[10] |
Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191 |
[11] |
P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure and Applied Analysis, 2020, 19 (1) : 47-84. doi: 10.3934/cpaa.2020004 |
[12] |
Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks and Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 |
[13] |
Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061 |
[14] |
John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 |
[15] |
Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic and Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017 |
[16] |
Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477 |
[17] |
Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020 |
[18] |
Jan Bouwe van den Berg, Gabriel William Duchesne, Jean-Philippe Lessard. Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach. Journal of Computational Dynamics, 2022, 9 (2) : 253-278. doi: 10.3934/jcd.2022005 |
[19] |
Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i |
[20] |
Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]