Exponentially Convergent Algorithms for Abstract Differential Equations

189 Pages·2011·1.47 MB·English

by  Ivan Gavrilyuk

Frontiers in Mathematics
Advisory Editorial Board
Leonid Bunimovich (Georgia Institute of Technology, Atlanta)
Benoît Perthame (Université Pierre et Marie Curie, Paris)
Laurent Saloff-Coste (Cornell University, Ithaca)
Igor Shparlinski (Macquarie University, New South Wales)
Wolfgang Sprössig (TU Bergakademie Freiberg)
Cédric Villani (Institut Henri Poincaré, Paris)
Ivan Gavrilyuk
Volodymyr Makarov
Vitalii Vasylyk
Exponentially Convergent
Algorithms
for
Abstract Differential
Equations
Ivan Gavrilyuk Volodymyr Makarov
Staatliche Studienakademie Thüringen - Vitalii Vasylyk
Berufsakademie Eisenach Institute of Mathematics
University of Cooperative Education National Academy of Sciences of Ukraine
Am Wartenberg 2 3 Tereshchenkivska St.
99817 Eisenach 01601 Kyiv- 4
Germany Ukraine
[email protected] [email protected]
[email protected]
2010 Mathematics Subject Classification: 65J10, 65M70, 35K90, 35L90
ISBN 978-3-0348-0118-8 e-ISBN 978-3-0348-0119-5
DOI 10.1007/978-3-0348-0119-5
Library of Congress Control Number: 2011932317

© Springer Basel AG 2011
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Preface
Thepresentbookisbasedontheauthors’worksduringrecentyearsonconstruct-
ing exponentially convergentalgorithms and algorithms without accuracysatura-
tionfor operatorequations and differentialequations with operatorcoefficients in
Banach space. It is addressed to mathematicians as well as engineering, graduate
and post-graduate students who are interested in numerical analysis, differential
equations and their applications. The authors have attempted to present the ma-
terial based on both strong mathematical evidence and the maximum possible
simplicity in describing algorithms and appropriate examples.
Our goal is to present accurate and efficient exponentially convergent meth-
ods for various problems, especially, for abstract differential equations with un-
boundedoperatorcoefficientsinBanachspace.Theseequationscanbeconsidered
asthemeta-modelsofthesystemsofordinarydifferentialequations(ODE)and/or
the partial differential equations (PDEs) appearing in various applied problems.
The framework of functional analysis allows us to get rather general but, at the
sametime, presenttransparentmathematicalresultsandalgorithmswhichcanbe
thenappliedtomathematicalmodelsoftherealworld.Theproblemclassincludes
initial value problems (IVP) for first-order differential equations with a constant
andvariableunboundedoperatorcoefficientinBanachspace(theheatequationis
asimpleexample),IVPsforfirst-orderdifferentialequationswithaparameterde-
pendent operatorcoefficient(parabolic partialdifferentialequations with variable
coefficients), IVPs for first-order differential equations with an operator coeffi-
cient possessing a variable domain (e.g. the heat equation with time-dependent
boundary conditions), boundary value problems for second-order elliptic differen-
tial equations with an operator coefficient (e.g. the Laplace equation) as well as
IVPsforsecond-orderstronglydampeddifferentialequations.We presentalsoex-
ponentially convergentmethods to IVP for first-order nonlinear differential equa-
tions with unbounded operator coefficients.
The efficiency of all proposed algorithms is demonstrated by numerical ex-
amples.
Unfortunately, it is almost impossible to provide a comprehensive bibliogra-
phy representingthe full setofpublications relatedto the topics ofthe book.The
list of publications is restricted to papers whose results are strongly necessary to
v
vi Preface
followthenarrative.Thusweaskthosewhowereunabletofindareferencetotheir
related publications to excuse us, even though those works would be appropriate.
Potential reader should have a basic knowledge of functional and numerical
analysis and be familiar with elements of scientific computing. An attempt has
beenmadetokeepthepresentedalgorithmsunderstandableforgraduatestudents
inengineering,physics,economicsand,ofcourse,mathematics.Thealgorithmsare
suitableforsuchinteractive,programmableandhighlypopulartoolsasMATLAB
and Maple which have been extensively employed in our numerical examples.
The authors are indebted to many colleagues who have contributed to this
work via fruitful discussions, but primarily to our families whose support is es-
pecially appreciated. We express our sincerest thanks to the German Research
Foundation (Deutsche Forschungsgemeinschaft)for financial support. It has been
a pleasure working with Dr. Barbara Hellriegel and with the Birkh¨auser Basel
publication staff.
Eisenach–Jena–Kyiv, I. Gavrilyuk, V. Makarov, V. Vasylyk
April, 2011
Contents
Preface v
1 Introduction 1
2 Preliminaries 9
2.1 Interpolation of functions . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Exponentially convergentquadrature rule . . . . . . . . . . . . . . 17
2.3 Estimates of the resolvent through fractional powers . . . . . . . . 19
2.4 Integration path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 The first-order equations 23
3.1 Exponentially convergentalgorithms for the
operator exponential with applications to
inhomogeneous problems in Banach spaces. . . . . . . . . . . . . . 24
3.1.1 Newalgorithmwithintegrationalongaparabolaandascale
of estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 New algorithm for the operator exponential with an
exponential convergence estimate including t=0 . . . . . . 29
3.1.3 Exponentially convergent algorithm II . . . . . . . . . . . . 35
3.1.4 Inhomogeneous differential equation . . . . . . . . . . . . . 39
3.2 Algorithms for first-order evolution equations . . . . . . . . . . . . 47
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 47
γ
3.2.2 Discrete first-order problem in the case γ <1 . . . . . . . . 52
3.2.3 Discrete first-order problem in the caseγγ ≤1 . . . . . . . . 58
3.3 Algorithm for nonlinear differential equations . . . . . . . . . . . . 71
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 A discretization scheme of Chebyshev type . . . . . . . . . 71
3.3.3 Error analysis for a small Lipschitz constant. . . . . . . . . 75
3.3.4 Modified algorithm for an arbitrary Lipschitz constant . . . 84
3.3.5 Implementation of the algorithm . . . . . . . . . . . . . . . 89
vii
viii Contents
3.4 Equations with an operator possessing a variable domain . . . . . 93
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Duhamel-like technique for first-order differential equations
in Banach space . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4.3 Existence and uniqueness of the solution of the integral
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4.4 Generalization to a parameter-dependent operator.
Existence and uniqueness result . . . . . . . . . . . . . . . . 102
3.4.5 Numerical algorithm . . . . . . . . . . . . . . . . . . . . . . 109
3.4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4.7 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 122
4 The second-order equations 127
4.1 Algorithm for second-order evolution equations . . . . . . . . . . . 128
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.1.2 Discrete second-order problem . . . . . . . . . . . . . . . . 128
4.2 Algorithm for a strongly damped differential equation . . . . . . . 135
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.2 Representation of the solution through the operator
exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.3 Representation and approximationof the operator
exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 144
4.3 Approximation to the elliptic solution operator . . . . . . . . . . . 147
4.3.1 Elliptic problems in cylinder type domains. . . . . . . . . . 147
4.3.2 New algorithm for the normalized operator sinh-family . . 149
4.3.3 Inhomogeneous differential equations . . . . . . . . . . . . . 155
Appendix: Tensor-product approximations of the operator exponential 167
Bibliography 173
Index 179
Chapter 1
Introduction
Theexponentialfunctiony =ex aswellasthe Eulernumber‘e’ playfundamental
roles in mathematics and possess many miraculous features expressing the fas-
cination of nature. The exponential function is an important component of the
solution of many equations and problems. For example, taking into account the
commutativepropertyofmultiplicationofnumbers,thesolutionoftheelementary
algebraic equation
AX +XB =C, A,B,C ∈R, A+B >0 (1.1)
can simply be found as
C
X = . (1.2)
A+B
At the same time, this solution can be represented in terms of the exponential
function as
(cid:2)
∞
X = e−At·C·e−Btdt.
0
Equation (1.1) is known as the Silvester equation and, when A = B, as the Lya-
punov equation. If A,B,C are matrices or operators, the first formula can no
longer be used (matrix multiplication is not commutative and we can not write
AX+XB =X(A+B)!)butthesecondoneremainstrue(undersomeassumptions
on A and B). This means that the latter can, after proper discretization, be used
incomputations.InthecaseofamatrixoraboundedoperatorA,theexponential
function e−At (the operator exponential) can be defined, e.g., through the series
(cid:3)∞
Aktk
e−At = (−1)k . (1.3)
k!
k=0
I.P. Gavrilyuk et al., Exponentially Convergent Algorithms for Abstract Differential Equations, 1
Frontiers in Mathematics, DOI 10.1007/978-3-0348-0119-5_1, © Springer Basel AG 2011