Efficient numerical methods for fractional differential equations and their analytical background [Elektronische Ressource] / von Marc Weilbeer

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_carolo-wilhelmina_zu_braunschweig
• Publié le: 01 janvier 2005
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Von der Carl-Friedrich-Gau -Fakultat¤ fur¤ Mathematik und Informatik
der Technischen Universitat¤ Braunschweig
genehmigte Dissertation zur Erlangung des Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
von
Marc Weilbeer
Ef cient Numerical Methods for Fractional Differential Equations
and their Analytical Background
1. Referent: Prof. Dr. Kai Diethelm
2. Prof. Dr. Neville J. Ford
Eingereicht: 23.01.2005
Prufung:¤ 09.06.2005
Supported by the US Army Medical Research and Material Command
Grant No. DAMD-17-01-1-0673 to the Cleveland ClinicContents
Introduction 1
1 A brief history of fractional calculus 7
1.1 The early stages 1695-1822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Abel’s impact on fractional calculus 1823-1916 . . . . . . . . . . . . . . . . . . 13
1.3 From Riesz and Weyl to modern fractional calculus . . . . . . . . . . . . . . . 18
2 Integer calculus 21
2.1 Integration and differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Differential equations and multistep methods . . . . . . . . . . . . . . . . . . 26
3 Integral transforms and special functions 33
3.1 Integral transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Euler’s Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 The Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Mittag-Lef er function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Fractional calculus 45
4.1 Fractional integration and differentiation . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Riemann-Liouville operator . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.2 Caputo operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.3 Grunw¤ ald-Letnikov operator . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Fractional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Properties of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Fractional linear multistep methods . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Numerical methods 105
5.1 Fractional backward difference methods . . . . . . . . . . . . . . . . . . . . . . 107
5.1.1 Backward differences and the Grunw¤ ald-Letnikov de nition . . . . . . 107
5.1.2 Diethelm’s fractional backward differences based on quadrature . . . . 112
5.1.3 Lubich’s backward difference methods . . . . . . . . . . . . . 120
5.2 Taylor Expansion and Adomian’s method . . . . . . . . . . . . . . . . . . . . . 123
5.3 Numerical computation and its pitfalls . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.1 Computation of the convolution weights w . . . . . . . . . . . . . . . . 134m
5.3.2 Computation of the starting weights w . . . . . . . . . . . . . . . . . 136m,j
iii CONTENTS
5.3.3 Solving the fractional differential equations by formula (5.52) . . . . . 142
5.3.4 Enhancements of Lubich’s fractional backward difference method . . . 148
5.4 An Adams method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.5 Notes on improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Examples and applications 159
6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Diffusion-Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3 Flame propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7 Summary and conclusion 187
A List of symbols 195
B Some fractional derivatives 197
C Quotes 199
Bibliography 215
Index 216Introduction
It seems like one day very useful consequences
will be drawn form this paradox, since there
are little paradoxes without usefulness.
Leibniz in a letter [117] to L’Hospital on the
significance of derivatives of order 1/2.
Fractional Calculus
The eld of fractional calculus is almost as old as calculus itself, but over the last
decades the usefulness of this mathematical theory in applications as well as its merits
in pure mathematics has become more and more evident. Recently a number of textbooks
[105, 110, 122, 141] have been published on this eld dealing with various aspects in
different ways. Possibly the easiest access to the idea of the non-integer differential and integral
operators studied in the eld of fractional calculus is given by Cauchy’s well known
representation of an n-fold integral as a convolution integral
Z Z Zx x xn 1 1
nJ y(x) = y(x )dx . . . dx dx0 0 n 2 n 1
0 0 0
Z x1 1
= y(t)dt, n2 N, x2 R ,+1 n(n 1)! (x t)0
n 0where J is the n-fold integral operator with J y(x) = y(x). Replacing the discrete factorial
(n 1)! with Euler’s continuous gamma function G(n), which satis es (n 1)! = G(n) for
n2 N, one obtains a de nition of a non-integer order integral, i.e.
Z x1 1aJ y(x) = y(t)dt, a, x2 R .+1 aG(a) (x t)0
Several important aspects of fractional calculus originate from non-integer order
derivatives, which can simplest be de ned as concatenation of integer order differentiation and
fractional integration, i.e.
a n n a a n a nD y(x) = D J y(x) or D y(x) = J D y(x),
nwhere n is the integer satisfying a n < a + 1 and D , n 2 N, is the n-fold differential
0 aoperator with D y(x) = y(x). The operator D is usually denoted as Riemann-Liouville
12 INTRODUCTION
adifferential operator, while the operator D is named Caputo differential operator. The fact
that there is obviously more than one way to de ne non-integer order derivatives is one of
the challenging and rewarding aspects of this mathematical eld.
Because of the integral in the de nition of the non-integer order derivatives, it is
apparent that these derivatives are non-local operators, which explains one of their most
signi cant uses in applications: A non-integer derivative at a certain point in time or space
contains information about the function at earlier points in time or space respectively.
Thus non-integer derivatives possess a memory effect, which it shares with several
materials such as viscoelastic materials or polymers as well as principles in applications such
as anomalous diffusion. This fact is also one of the reasons for the recent interest in
fractional calculus: Because of their non-local property fractional derivatives can be used to
construct simple material models and uni ed principles. Prominent examples for diffusion
processes are given in the textbook by Oldham and Spanier [110] and the paper by
Olmstead and Handelsman [111], examples for modeling viscoelastic materials can be found
in the classic papers of Bagley and Torvik [10], Caputo [20], and Caputo and Mainardi
[21] and applications in the eld of signal processing are discussed in the publication [104]
by Marks and Hall. Several newer results can be found e.g. in the works of Chern [24],
Diethelm and Freed [39], Gaul, Klein and Kemplfe [57], Unser and Blu [143, 144],
Podlubny [121] and Podlubny et. al [124]. Additionally a number of surveys with collections of
applications can be found e.g. in Goren o and Mainardi [59], Mainardi [102] or Podlubny
[122].
The utilization of the memory effect of fractional derivatives in the construction of
simple material models or uni ed principles comes with a high cost regarding numerical
solvability. Any algorithm using a discretization of a non-integer derivative has, among other
things, to take into account its non-local structure which means in general a high storage
requirement and great overall complexity of the algorithm. Numerous attempts to solve
equations involving different types of non-integer order operators can be found in the
literature: Several articles by Brunner [14, 15, 16, 17, 18] deal with so-called collocation
methods to solve Abel-Volterra integral equations. In these equations the integral part is
essentially the non-integer order as de ned above. These, and additional results
can also be found in his book [19] on this topic. A book [83] by Linz and an article by Orsi
[112] e.g. use product integration techniques to solve Abel-Volterra integral equations as
well. Several articles by Lubich [92, 93, 94, 95, 96, 98], and Hairer, Lubich and Schlichte
[63], use so called fractional linear multistep methods to solve Abel-Volterra integral
equations numerically. In addition several papers deal with numerical methods to solve
differential equations of fractional order. These equations are similar to ordinary differential
equations, with the exception that the derivatives occurring in them are of non-integer
order. Approaches based on fractional formulation of backward difference methods can e.g.
be found in the papers by Diethelm [31, 38, 42], Ford and Simpson [53, 54, 55], Podlubny
[123] and Walz [146]. Fractional formulation of Adams-type methods are e.g. discussed in
the papers [36, 37] by Diethelm et al. Except the collocation by Brunner and the
product integration techniques by Linz and Orsi, most of the cited ideas are presented and
advanced in this thesis.INTRODUCTION 3
Outline of the thesis
In this thesis several aspects of fractional calculus will be presented ranging from its
history over analytical and numerical results to applications. The structure of this thesis
is deliberately