August  2016, 36(8): 4101-4131. doi: 10.3934/dcds.2016.36.4101

Improved estimates for nonoscillatory phase functions

1. 

Department of Mathematics, University of California, Davis, Davis, CA 95616, United States

2. 

Department of Computer Science, Yale University, New Haven, CT 06511, United States

Received  May 2015 Published  March 2016

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations are used in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.
Citation: James Bremer, Vladimir Rokhlin. Improved estimates for nonoscillatory phase functions. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4101-4131. doi: 10.3934/dcds.2016.36.4101
References:
[1]

G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999. doi: 10.1017/CBO9781107325937.

[2]

R. Bellman, Stability Theory of Differential Equations, Dover Publications, 1953.

[3]

O. Borůvka, Linear Differential Transformations of the Second Order, The English University Press, 1971.

[4]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger Publishing Company, 1955.

[5]

A. O. Daalhuis, Hyperasymptotic solutions of second-order linear differential equations. II, Methods and Applications of Analysis, 2 (1995), 198-211. doi: 10.4310/MAA.1995.v2.n2.a5.

[6]

A. O. Daalhuis and F. W. J. Olver, Hyperasymptotic solutions of second-order linear differential equations. I, Methods and Applications of Analysis, 2 (1995), 173-197. doi: 10.4310/MAA.1995.v2.n2.a4.

[7]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1993. doi: 10.1007/978-3-642-58016-1.

[8]

G. B. Folland, Real Analysis: Modern Techniques and Their Application, 2nd edition, Wiley-Interscience, 1999.

[9]

M. Goldstein and R. M. Thaler, Bessel functions for large arguments, Mathematical Tables and Other Aids to Computation, 12 (1958), 18-26. doi: 10.2307/2002123.

[10]

L. Grafakos, Classical Fourier Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[11]

L. Grafakos, Modern Fourier Analysis, Springer, 2009. doi: 10.1007/978-0-387-09434-2.

[12]

Z. Heitman, J. Bremer and V. Rokhlin, On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime, Journal of Computational Physics, 290 (2015), 1-27. doi: 10.1016/j.jcp.2015.02.028.

[13]

Z. Heitman, J. Bremer, V. Rokhlin and B. Vioreanu, On the asymptotics of Bessel functions in the Fresnel regime, Applied and Computational Harmonic Analysis, 39 (2015), 347-356. doi: 10.1016/j.acha.2014.12.002.

[14]

L. Hörmader, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, 1990. doi: 10.1007/978-3-642-61497-2.

[15]

L. Hörmader, The Analysis of Linear Partial Differential Operators II, 2nd edition, Springer, 1990.

[16]

E. Kummer, De generali quadam aequatione differentiali tertti ordinis,, Progr. Evang. Köngil. Stadtgymnasium Liegnitz., (). 

[17]

F. Neuman, Global Properties of Linear Ordinary Differential Equations, Kluwer Academic Publishers, 1991.

[18]

F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.

[19]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.

[20]

J. Segura, Bounds for the ratios of modified Bessel functions and associated Turán-type inequalities, Journal of Mathematics Analysis and Applications, 374 (2011), 516-528. doi: 10.1016/j.jmaa.2010.09.030.

[21]

R. Spigler and M. Vianello, The phase function method to solve second-order asymptotically polynomial differential equations, Numerische Mathematik, 121 (2012), 565-586. doi: 10.1007/s00211-011-0441-9.

[22]

N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics, 2013.

[23]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Volume I: Fixed-point Theorems, Springer-Verlag, 1986. doi: 10.1007/978-1-4612-4838-5.

show all references

References:
[1]

G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999. doi: 10.1017/CBO9781107325937.

[2]

R. Bellman, Stability Theory of Differential Equations, Dover Publications, 1953.

[3]

O. Borůvka, Linear Differential Transformations of the Second Order, The English University Press, 1971.

[4]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger Publishing Company, 1955.

[5]

A. O. Daalhuis, Hyperasymptotic solutions of second-order linear differential equations. II, Methods and Applications of Analysis, 2 (1995), 198-211. doi: 10.4310/MAA.1995.v2.n2.a5.

[6]

A. O. Daalhuis and F. W. J. Olver, Hyperasymptotic solutions of second-order linear differential equations. I, Methods and Applications of Analysis, 2 (1995), 173-197. doi: 10.4310/MAA.1995.v2.n2.a4.

[7]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1993. doi: 10.1007/978-3-642-58016-1.

[8]

G. B. Folland, Real Analysis: Modern Techniques and Their Application, 2nd edition, Wiley-Interscience, 1999.

[9]

M. Goldstein and R. M. Thaler, Bessel functions for large arguments, Mathematical Tables and Other Aids to Computation, 12 (1958), 18-26. doi: 10.2307/2002123.

[10]

L. Grafakos, Classical Fourier Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[11]

L. Grafakos, Modern Fourier Analysis, Springer, 2009. doi: 10.1007/978-0-387-09434-2.

[12]

Z. Heitman, J. Bremer and V. Rokhlin, On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime, Journal of Computational Physics, 290 (2015), 1-27. doi: 10.1016/j.jcp.2015.02.028.

[13]

Z. Heitman, J. Bremer, V. Rokhlin and B. Vioreanu, On the asymptotics of Bessel functions in the Fresnel regime, Applied and Computational Harmonic Analysis, 39 (2015), 347-356. doi: 10.1016/j.acha.2014.12.002.

[14]

L. Hörmader, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, 1990. doi: 10.1007/978-3-642-61497-2.

[15]

L. Hörmader, The Analysis of Linear Partial Differential Operators II, 2nd edition, Springer, 1990.

[16]

E. Kummer, De generali quadam aequatione differentiali tertti ordinis,, Progr. Evang. Köngil. Stadtgymnasium Liegnitz., (). 

[17]

F. Neuman, Global Properties of Linear Ordinary Differential Equations, Kluwer Academic Publishers, 1991.

[18]

F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.

[19]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.

[20]

J. Segura, Bounds for the ratios of modified Bessel functions and associated Turán-type inequalities, Journal of Mathematics Analysis and Applications, 374 (2011), 516-528. doi: 10.1016/j.jmaa.2010.09.030.

[21]

R. Spigler and M. Vianello, The phase function method to solve second-order asymptotically polynomial differential equations, Numerische Mathematik, 121 (2012), 565-586. doi: 10.1007/s00211-011-0441-9.

[22]

N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics, 2013.

[23]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Volume I: Fixed-point Theorems, Springer-Verlag, 1986. doi: 10.1007/978-1-4612-4838-5.

[1]

M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121

[2]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39

[3]

Leon Ehrenpreis. Special functions. Inverse Problems and Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639

[4]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41

[5]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

[6]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21

[7]

Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59

[8]

Genghua Li, Shengjie Li, Manxue You. Asymptotic analysis of scalarization functions and applications. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022046

[9]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[10]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160

[11]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301

[12]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[13]

Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055

[14]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1

[15]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691

[16]

Gümrah Uysal. On a special class of modified integral operators preserving some exponential functions. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2021044

[17]

Pascale Charpin, Jie Peng. Differential uniformity and the associated codes of cryptographic functions. Advances in Mathematics of Communications, 2019, 13 (4) : 579-600. doi: 10.3934/amc.2019036

[18]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[19]

Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223

[20]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (156)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]