Von der Carl-Friedrich-Gauß-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 Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background 1. Referent: Prof. Dr. Kai Diethelm 2. Referent: 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 Clinic Contents 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-Leffler 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 definition . 107 5.1.2 Diethelm's fractional backward differences based on quadrature . 112 5.1.3 Lubich's fractional 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 wm . 134 5.3.2 Computation of the starting weights wm,j . 136 i ii 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 216 Introduction It seems like one day very useful consequences will be drawn form this paradox, since there are little paradoxes without usefulness. Leib- niz in a letter [117] to L'Hospital on the signif- icance of derivatives of order 1/2. Fractional Calculus The field 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 field dealing with various aspects in differ- ent ways. Possibly the easiest access to the idea of the non-integer differential and integral operators studied in the field of fractional calculus is given by Cauchy's well known repre- sentation of an n-fold integral as a convolution integral x xn 1 x1 n − J y(x) = y(x0)dx0 . dxn 2dxn 1 · · · − − Z0 Z0 Z0 1 x 1 = y(t)dt, n N, x R+, (n 1)! (x t)1 n 2 2 − Z0 − − where Jn is the n-fold integral operator with J0y(x) = y(x). Replacing the discrete factorial (n 1)! with Euler's continuous gamma function G(n), which satisfies (n 1)! = G(n) for − − n N, one obtains a definition of a non-integer order integral, i.e. 2 x a 1 1 J y(x) = y(t)dt, a, x R+. G(a) (x t)1 a 2 Z0 − − Several important aspects of fractional calculus originate from non-integer order deriva- tives, which can simplest be defined as concatenation of integer order differentiation and fractional integration, i.e. a n n a a n a n D y(x) = D J − y(x) or D y(x) = J − D y(x), ∗ where n is the integer satisfying a n < a + 1 and Dn, n N, is the n-fold differential ≤ 2 operator with D0y(x) = y(x). The operator Da is usually denoted as Riemann-Liouville 1 2 INTRODUCTION differential operator, while the operator Da is named Caputo differential operator. The fact that there is obviously more than one way∗to define non-integer order derivatives is one of the challenging and rewarding aspects of this mathematical field. Because of the integral in the definition of the non-integer order derivatives, it is ap- parent that these derivatives are non-local operators, which explains one of their most significant 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 mate- rials 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 frac- tional calculus: Because of their non-local property fractional derivatives can be used to construct simple material models and unified principles. Prominent examples for diffusion processes are given in the textbook by Oldham and Spanier [110] and the paper by Olm- stead 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 field 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], Pod- lubny [121] and Podlubny et. al [124]. Additionally a number of surveys with collections of applications can be found e.g. in Gorenflo and Mainardi [59], Mainardi [102] or Podlubny [122]. The utilization of the memory effect of fractional derivatives in the construction of sim- ple material models or unified principles comes with a high cost regarding numerical solv- ability. 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 lit- erature: 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 integral as defined 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 equa- tions numerically. In addition several papers deal with numerical methods to solve differ- ential 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 or- der. 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 methods 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 chosen in such a way that not only experts in the field of fractional calculus can understand the presented results within this thesis, but also readers with knowledge obtained e.g. in the first semesters of a mathematical or engineering study course can com- prehend the benefits and the problems of efficient numerical methods for fractional differ- ential equations and their analytical background. For this reason this thesis is structured as follows: The thesis begins in Chapter 1 with a brief historical review of the theory of fractional calculus and its applications. The theory of non-integer order differentiation and inte- gration is almost as old as classical calculus itself, but nevertheless there seems to be an astonishing lack of knowledge of this field in most mathematicians. A look at the historical development can in parts explain the absence of this field in today's standard mathematics textbooks on calculus and in addition give the reader not familiar with this field a good ac- cess to the topics addressed in this thesis. Moreover, the possession of an understanding of the historical development of any mathematical field often can give significant additional insight in an otherwise only theoretical presentation. In Chapter 2 some well known analytical and numerical results on classical calculus are stated. One reason behind this is due to the fact that those results are needed for several proofs of theorems in later chapters and thus they are stated here for completeness.