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  December 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238
References:
[1]

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

[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.

[3]

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

[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.

[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.

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

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

[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.

[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.

[10]

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

[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.

[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. 

[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.

[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.

[15]

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

[16]

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

[17]

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

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

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

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

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

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

[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.

[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.

[10]

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

[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.

[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. 

[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.

[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.

[15]

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

[16]

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

[17]

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

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

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

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