American Institute of Mathematical Sciences

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

Improved estimates for nonoscillatory phase functions

James Bremer 1, and  Vladimir Rokhlin 2,

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.
Keywords: Special functions, ordinary differential equations, Bessel functions, phsae functions., asymptotic expansions.
Mathematics Subject Classification: 34E99, 33E0.
Citation: James Bremer, Vladimir Rokhlin. Improved estimates for nonoscillatory phase functions. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

R. Bellman, Stability Theory of Differential Equations,, Dover Publications, (1953).   Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.4310/MAA.1995.v2.n2.a4.  Google Scholar

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

[8]

G. B. Folland, Real Analysis: Modern Techniques and Their Application,, 2nd edition, (1999).   Google Scholar

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

[10]

L. Grafakos, Classical Fourier Analysis,, Springer, (2014).  doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[11]

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

[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.  doi: 10.1016/j.jcp.2015.02.028.  Google Scholar

[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.  doi: 10.1016/j.acha.2014.12.002.  Google Scholar

[14]

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

[15]

L. Hörmader, The Analysis of Linear Partial Differential Operators II,, 2nd edition, (1990).   Google Scholar

[16]

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

[17]

F. Neuman, Global Properties of Linear Ordinary Differential Equations,, Kluwer Academic Publishers, (1991).   Google Scholar

[18]

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

[19]

W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill, (1976).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.09.030.  Google Scholar

[21]

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

[22]

N. Trefethen, Approximation Theory and Approximation Practice,, Society for Industrial and Applied Mathematics, (2013).   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

R. Bellman, Stability Theory of Differential Equations,, Dover Publications, (1953).   Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.4310/MAA.1995.v2.n2.a4.  Google Scholar

[7]

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

[8]

G. B. Folland, Real Analysis: Modern Techniques and Their Application,, 2nd edition, (1999).   Google Scholar

[9]

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

[10]

L. Grafakos, Classical Fourier Analysis,, Springer, (2014).  doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[11]

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

[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.  doi: 10.1016/j.jcp.2015.02.028.  Google Scholar

[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.  doi: 10.1016/j.acha.2014.12.002.  Google Scholar

[14]

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

[15]

L. Hörmader, The Analysis of Linear Partial Differential Operators II,, 2nd edition, (1990).   Google Scholar

[16]

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

[17]

F. Neuman, Global Properties of Linear Ordinary Differential Equations,, Kluwer Academic Publishers, (1991).   Google Scholar

[18]

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

[19]

W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill, (1976).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.09.030.  Google Scholar

[21]

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

[22]

N. Trefethen, Approximation Theory and Approximation Practice,, Society for Industrial and Applied Mathematics, (2013).   Google Scholar

[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.  Google Scholar

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