December  2020, 13(12): 3565-3579. doi: 10.3934/dcdss.2020238

On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups

1. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

2. 

School of Engineering, Computing and Construction Management, Roger Williams University, Bristol, RI 02809-2921, USA

3. 

Department of Mathematics and Statistics, Winona State University, Winona, MN 55987-0838, USA

Received  December 2018 Revised  August 2019 Published  January 2020

The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Padé approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomée Approximation Theorem of Semigroup Theory.

Citation: Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238
References:
[1]

T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Monographs in Mathematics, Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7.  Google Scholar

[3]

P. Brenner and V. Thomée, On rational approximation of semigroups, SIAM J. Numer. Anal., 16 (1979), 683-694.  doi: 10.1137/0716051.  Google Scholar

[4]

M. Egert and J. Rozendaal, Convergence of subdiagonal Padé approximations of $C_0$-semigroups, J. Evol. Equ., 13 (2013), 875-895.  doi: 10.1007/s00028-013-0207-1.  Google Scholar

[5]

B. L. Ehle, $ \mathscr{A}$-stable methods and Padé approximations to the exponential function, SIAM J. Math. Anal., 4 (1973), 671-680.  doi: 10.1137/0504057.  Google Scholar

[6] J. A. Goldstein, Semigroups of Operators and Applications, Oxford University Press, 1985.   Google Scholar
[7]

E. HairerS. P. Nørsett and G. Wanner, Order stars and stability theorems, BIT Numerical Mathematics, 18 (1978), 475-489.  doi: 10.1007/BF01932026.  Google Scholar

[8]

R. Hersh and T. Kato, High-accuracy stable difference schemes for well-posed initial value problems, SIAM J. Numer. Anal., 16 (1979), 670-682.  doi: 10.1137/0716050.  Google Scholar

[9]

P. Jara, Rational approximation schemes for bi-continuous semigroups, J. Math. Anal. Appl., 344 (2008), 956-968.  doi: 10.1016/j.jmaa.2008.02.068.  Google Scholar

[10]

M. Kovács, On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups, Ph.D thesis, Louisiana State University, 2004.  Google Scholar

[11]

M. Kovács, On the convergence of rational approximations of semigroups on intermediate spaces, Math. Comp., 76 (2007), 273-286.  doi: 10.1090/S0025-5718-06-01905-3.  Google Scholar

[12]

M. Kovács and F. Neubrander, On the inverse Laplace-Stieltjes transform of $ \mathscr{A}$-stable rational functions, New Zealand J. Math., 36 (2007), 41-56.   Google Scholar

[13]

F. NeubranderK. Özer and T. Sandmaier, Rational approximation of semigroups without scaling and squaring, Discrete and Continuous Dynamical Systems, 33 (2013), 5305-5317.  doi: 10.3934/dcds.2013.33.5305.  Google Scholar

[14]

M. H. Padé, Sur répresentation approchée d'une fonction par des fractionelles, Ann. de l'Ecole Normale Superieure, 9 (1892), 3-93.  doi: 10.24033/asens.378.  Google Scholar

[15]

O. Perron, Die Lehre von den Kettenbrüchen, Chelsea Pub. Co., New York, 1950.  Google Scholar

[16]

A. Reiser, Time Discretization for Evolution Equations, Diplomarbeit, Louisiana State University and Universität Tübingen, 2008. Google Scholar

[17]

T. Sandmaier, Implizite und Explizite Approximationsverfahren, Wissenschaftliche Arbeit, Universität Tübingen, 2010. Google Scholar

[18] D. V. Widder, The Laplace Transform, Princeton University Press, 1941.   Google Scholar
[19]

L. Windsperger, Operational Methods for Evolution Equations, Ph. D thesis, Louisiana State University, 2012. Google Scholar

show all references

References:
[1]

T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Monographs in Mathematics, Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7.  Google Scholar

[3]

P. Brenner and V. Thomée, On rational approximation of semigroups, SIAM J. Numer. Anal., 16 (1979), 683-694.  doi: 10.1137/0716051.  Google Scholar

[4]

M. Egert and J. Rozendaal, Convergence of subdiagonal Padé approximations of $C_0$-semigroups, J. Evol. Equ., 13 (2013), 875-895.  doi: 10.1007/s00028-013-0207-1.  Google Scholar

[5]

B. L. Ehle, $ \mathscr{A}$-stable methods and Padé approximations to the exponential function, SIAM J. Math. Anal., 4 (1973), 671-680.  doi: 10.1137/0504057.  Google Scholar

[6] J. A. Goldstein, Semigroups of Operators and Applications, Oxford University Press, 1985.   Google Scholar
[7]

E. HairerS. P. Nørsett and G. Wanner, Order stars and stability theorems, BIT Numerical Mathematics, 18 (1978), 475-489.  doi: 10.1007/BF01932026.  Google Scholar

[8]

R. Hersh and T. Kato, High-accuracy stable difference schemes for well-posed initial value problems, SIAM J. Numer. Anal., 16 (1979), 670-682.  doi: 10.1137/0716050.  Google Scholar

[9]

P. Jara, Rational approximation schemes for bi-continuous semigroups, J. Math. Anal. Appl., 344 (2008), 956-968.  doi: 10.1016/j.jmaa.2008.02.068.  Google Scholar

[10]

M. Kovács, On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups, Ph.D thesis, Louisiana State University, 2004.  Google Scholar

[11]

M. Kovács, On the convergence of rational approximations of semigroups on intermediate spaces, Math. Comp., 76 (2007), 273-286.  doi: 10.1090/S0025-5718-06-01905-3.  Google Scholar

[12]

M. Kovács and F. Neubrander, On the inverse Laplace-Stieltjes transform of $ \mathscr{A}$-stable rational functions, New Zealand J. Math., 36 (2007), 41-56.   Google Scholar

[13]

F. NeubranderK. Özer and T. Sandmaier, Rational approximation of semigroups without scaling and squaring, Discrete and Continuous Dynamical Systems, 33 (2013), 5305-5317.  doi: 10.3934/dcds.2013.33.5305.  Google Scholar

[14]

M. H. Padé, Sur répresentation approchée d'une fonction par des fractionelles, Ann. de l'Ecole Normale Superieure, 9 (1892), 3-93.  doi: 10.24033/asens.378.  Google Scholar

[15]

O. Perron, Die Lehre von den Kettenbrüchen, Chelsea Pub. Co., New York, 1950.  Google Scholar

[16]

A. Reiser, Time Discretization for Evolution Equations, Diplomarbeit, Louisiana State University and Universität Tübingen, 2008. Google Scholar

[17]

T. Sandmaier, Implizite und Explizite Approximationsverfahren, Wissenschaftliche Arbeit, Universität Tübingen, 2010. Google Scholar

[18] D. V. Widder, The Laplace Transform, Princeton University Press, 1941.   Google Scholar
[19]

L. Windsperger, Operational Methods for Evolution Equations, Ph. D thesis, Louisiana State University, 2012. Google Scholar

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