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
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[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, (2011).

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

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, (2011).

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

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