Article Contents
Article Contents

Rational approximations of semigroups without scaling and squaring

• 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.
Mathematics Subject Classification: Primary: 65R10, 47A58, 44A10, 41A20, 41A25.

 Citation:

•  [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 Verlag, 2011. [3] P. Brenner and V. Thomée, On rational approximation of semigroups, SIAM J. Numer. Anal., 16 (1979), 683-694.doi: 10.1137/0716051. [4] B. L. Ehle, $A$-stable methods and Padé approximations to the exponential function, SIAM J. Math. Anal., 4 (1973), 671-680.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, Fall 2012. [7] R. Hersh and T. Kato, High-accuracy stable difference schemes for wellposed initial value problems, SIAM J. Numer. Anal., 16 (1979), 670-682.doi: 10.1137/0716050. [8] 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. [9] P. Jara, F. Neubrander and K. Özer, Rational inversion of the Laplace transform, Journal of Evolution Equations, to appear. doi: 10.1007/s00028-012-0139-1. [10] Mihály 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 $A$-stable rational functions, New Zealand J. Math., 36 (2007), 41-56. [13] Koray Özer, "Laplace Transform Inversion and Time-Discretization Methods for Evolution Equations," Ph.D. thesis, Louisiana State University, 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., New York, 1950. [16] Armin Reiser, "Time Discretization for Evolution Equations," Diplomarbeit, Louisiana State University and Universität Tübingen, 2008. [17] D. V. Widder, "The Laplace Transform," Princeton University Press, 1946. [18] Teresa Sandmaier, "Implizite und Explizite Approximationsverfahren," Wissenschaftliche Arbeit, Universität Tübingen, 2010.