November  2013, 33(11&12): 5305-5317. doi: 10.3934/dcds.2013.33.5305

Rational approximations of semigroups without scaling and squaring

1. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States

2. 

Department of Mathematics, Roger Williams University, Bristol, RI 02809, United States

3. 

Mathematisches Institut, Universität Tübingen, Tübingen, 72076, Germany

Received  January 2012 Published  May 2013

We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
Citation: Frank Neubrander, Koray Özer, Teresa Sandmaier. Rational approximations of semigroups without scaling and squaring. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5305-5317. doi: 10.3934/dcds.2013.33.5305
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, (2011).   Google Scholar

[3]

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

[4]

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

[5]

J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985).   Google Scholar

[6]

William Harrison, Ph.D thesis,, Louisiana State University, (2012).   Google Scholar

[7]

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

[8]

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

[9]

P. Jara, F. Neubrander and K. Özer, Rational inversion of the Laplace transform,, Journal of Evolution Equations, ().  doi: 10.1007/s00028-012-0139-1.  Google Scholar

[10]

Mihály Kovács, "On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups,", Ph. D. Thesis, (2004).   Google Scholar

[11]

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

[12]

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

[13]

Koray Özer, "Laplace Transform Inversion and Time-Discretization Methods for Evolution Equations,", Ph.D. thesis, (2008).   Google Scholar

[14]

M. H. Padé, Sur répresentation approchée d'une fonction par des fractionelles,, Ann. de l'Ecole Normale Superieure, 9 (1892).   Google Scholar

[15]

O. Perron, "Die Lehre von den Kettenbrüchen,", Chelsea Pub. Co., (1950).   Google Scholar

[16]

Armin Reiser, "Time Discretization for Evolution Equations,", Diplomarbeit, (2008).   Google Scholar

[17]

D. V. Widder, "The Laplace Transform,", Princeton University Press, (1946).   Google Scholar

[18]

Teresa Sandmaier, "Implizite und Explizite Approximationsverfahren,", Wissenschaftliche Arbeit, (2010).   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, (2011).   Google Scholar

[3]

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

[4]

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

[5]

J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985).   Google Scholar

[6]

William Harrison, Ph.D thesis,, Louisiana State University, (2012).   Google Scholar

[7]

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

[8]

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

[9]

P. Jara, F. Neubrander and K. Özer, Rational inversion of the Laplace transform,, Journal of Evolution Equations, ().  doi: 10.1007/s00028-012-0139-1.  Google Scholar

[10]

Mihály Kovács, "On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups,", Ph. D. Thesis, (2004).   Google Scholar

[11]

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

[12]

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

[13]

Koray Özer, "Laplace Transform Inversion and Time-Discretization Methods for Evolution Equations,", Ph.D. thesis, (2008).   Google Scholar

[14]

M. H. Padé, Sur répresentation approchée d'une fonction par des fractionelles,, Ann. de l'Ecole Normale Superieure, 9 (1892).   Google Scholar

[15]

O. Perron, "Die Lehre von den Kettenbrüchen,", Chelsea Pub. Co., (1950).   Google Scholar

[16]

Armin Reiser, "Time Discretization for Evolution Equations,", Diplomarbeit, (2008).   Google Scholar

[17]

D. V. Widder, "The Laplace Transform,", Princeton University Press, (1946).   Google Scholar

[18]

Teresa Sandmaier, "Implizite und Explizite Approximationsverfahren,", Wissenschaftliche Arbeit, (2010).   Google Scholar

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