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1 Introduction, In recent years it has turned out that many phenomena in engineering physics. chemistry and other sciences can be described very successfully by models us. ing mathematical tools from fractional calculus i e the theory of derivatives. and integrals of fractional non integer order Some of the most prominent. examples are given in a book by Oldham and Spanier 41 diffusion pro. cesses and the classic papers of Bagley and Torvik 2 Caputo 4 and Caputo. and Mainardi 5 these three papers dealing with the modeling of viscoelas. tic materials as well as in the publications of Marks and Hall 34 signal. processing and Olmstead and Handelsman 42 also dealing with diffusion. problems More recent results are described e g in the works of Chern 6. finite element implementation of viscoelastic model Diethelm and Freed. ll viscoplastic material model Gaul Klein and Kempfle 19 descrip. tion of mechanical systems subject to damping Glockle and Nonnenmacher. 20 relaxation and reaction kinetics of polymers Gorenflo and Rutman 24. so called ultraslow processes Metzler et al 36 relaxation in filled polymer. networks Unser and Blu 52 53 splines and wavelets Podlubny 44 control. theory and Podlubny et al 47 heat propagation Surveys with collections. of applications can also be found in Gorenflo and Mainardi 23 Mainardi 32. Matignon and Montseny 35 Nonnenmacher and Metzler 39 and Podlubny. 45 Therefore it seems to be fair to say that there is a significant demand. for readily usable tools that numerically handle these mathematical problems. However to date such algorithms have been developed only to a rather lim. ited extent Thus it is the aim of this paper to identify the most important. problems for which specific algorithms are required t o present those schemes. that have been developed so far in a way that is directly accessible to the. applied scientist and to fill the remaining gaps with suitable new methods. The structure of this paper is as follows In the next three sections we briefly. recall the mathematical foundations of the fractional calculus We then turn. to a more detailed description and assessment of our numerical methods for. problems described by fractional order derivatives integrals and differential. equations After this we provide an algorithm for calculating the Mittag Leffler. function which appears in solutions to fractional order differential equations. Finally after a discussion of our algorithms we provide appendices where. additional useful results are laid out that are connected to the numerical issues. of fractional calculus indirectly such as e g a table of certain fractional. derivatives and some helpful auxiliary subroutines that can be utilized when. working with the fundamental algorithms described in the main part of the. 2 Riemann Liouville Fractional Integral, In the classical calculus of Newton and Leibniz Cauchy reduced the calculation. of an n fold integration of the function y x into a single convolution integral. possessing an Abel power law kernel,n I o z z len,y x dz nE N x E 1. where J is the n fold integral operator with Joy z y x N is the set of. positive integers and R is the set of positive reals Liouville and Riemann2. analytically continued Cauchy s result replacing the discrete factorial n l. with Euler s continuous gamma function r n noting that n l r n. thereby producing 30 45 Eqn A, where Ja is the Riemann Liouville integral operator of order a which com. mutes i e J JPy x JpJay z P P y z V Q P E Et Equation 2 is. the cornerstone of the fractional calculus although it may vary in its assign. ment of limits of integration In this document we take the lower limit t o be. zero and the upper limit to be some positive finite real Actually a can be. complex 51 but in the applications that are of interest to us Q is real. A brief history of the development of fractional calculus can be found in Ross. 50 and Miller and Ross 37 Chp 11 A survey of many emerging applications. of the fractional calculus in areas of science and engineering can be found in. the recent text by Podlubny 45 Chp lo,3 Caputo Type Fractional Derivative.

3 1 Fundamental definition, From this single definition for fractional integration one can construct several. definitions for fractional differentiation cf e g with Refs 45 51 The spe. cial operator 0 that we choose to use which requires the dependent variable. Riemann s pioneering work in the field of fractional calculus was done during. his student years but published posthumous forty four years after Liouville first. published in the field 50, y to be continuous and a times differentiable in the independent variable z. is defined by,D ay z J al aD aly z 3,lim D y z D y z. a n for any nE N 4, with D y z y x where a is the ceiling function giving the smallest. integer greater than or equal to a and where Q n means Q approaches n. from below The operator D n E N is the classical differential operator It is. accepted practice to call 0 the Caputo differential operator of order a after. Caputo 4 who was the among the first to use this operator in applications. and to study some of its properties Appendix A presents a table of Caputo. derivatives for some of the more common mathematical functions. The Caputo differential operator is a linear operator i e. D a ay bz z aD ay z bD az z 5,for arbitrary constants a and b that commutes viz.

if y z is sufficiently smooth and it possesses the desirable property that. DTc 0 for any constant c 7, The more common Riemann Liouville fractional derivative Da although lin. ear need not commute 45 pg 741 furthermore Dac DralJral ac. cz a I l a which depends on 5 Ross 50 attributes this startling fact. as the main reason why the fractional calculus has historically had a difficult. time being embraced by the mathematics and physics communities. Actually Liouville introduced the operator in his historic first paper on the topic. 30 y6 Eqn B Still nothing in Liouville s works suggests that he ever saw any. difference between D Jral aDraland Da DralJfal a with Da being his. accepted definition 30 first formula on pg 101 the Riemann Liouville differential. operator of order CY Liouville freely interchanged the order of integration and differ. entiation because the class of problems that he was interested in happened to be a. class where such an interchange is legal and he made only a few terse remarks about. the general requirements on the class of functions for which his fractional calculus. works 31 The accepted naming of the operator D after Caputo therefore seems. Rabotnov 49 pg 1291 introduced this same differential operator into the Russian. viscoelastic literature a year before Caputo s paper was published Regardless of this. fact operator D is commonly named after Caputo in the present day literature. The Riemann Liouville integral operator J a and the Caputo differential oper. ator 0 are inverse operators in the sense that, with y Dky O where LaJ is the floor function giving the largest integer. less than or equal to a The classic n fold integral and differential operators. of integer order satisfy like formulz viz D n J n y z y z and J n D n y z. y z x k n 1 k, O k y o n E N In this sense the Caputo derivative and Riemann. Liouville integral are analytic continuations of the well known n fold derivative. and integral from the classical calculus, Remark 1 Fractional derivatives do not satisfy the Leibniz product rule as it. is known in integer order calculus For example whenever the Caputo deriva. tive is restricted so that 0 a 1 its Leibniz product rule is given by. wherein unlike the Leibniz product rule for integer order derivatives the bi. nomial coeficients a a l a 2 a k,l k with z, a E and k E N do not become zero whenever k a because a N.

i e the binomial s u m is now of infinite extent A similar infinite s u m exists. f o r the Leibniz product rule of the Riemann Liouville fractional derivative cf. Podlubny 45 pp 91 97,3 2 Alternative integral expressions. The Caputo derivative defined in Eq 3 can be expressed in a more explicit. notation as the integral, where the weak singularity caused by the Abel kernel of the integral operator. is readily observed This singularity can be removed through an integration. provided that the dependent variable y is continuous and 1 al times dif. ferentiable in the independent variable x over the interval of differentiation. integration 0 z In Eq 11 the power law kernel is bounded over the en. tire interval of integration whereas in Eq 10 the kernel is singular at the. upper limit of integration, The two representations in formulze 10 and 11 are quite useful for pen. and paper calculations but in order to obtain a numerical scheme for the. approximation of such fractional derivatives we found it even more helpful to. look at yet another representation that seems to have been introduced into. this context by Elliott 14 namely, This representation can also be obtained from Eq 10 using the method. of integration by parts but with the roles of the two factors interchanged. The advantage here is that the function y itself appears in the integrand in. stead of its derivative The disadvantage is that the singularity of the kernel. is now strong rather than weak and thus we have to interpret this integral as. a Hadamard type finite part integral This is cumbersome in pen and paper. calculations but as we shall see below it is not a problem to devise an al. gorithm that makes the computer do this job We provide a brief description. of such an algorithm in the following pages For more details the interested. reader is referred to Refs 7 14 and the references cited therein. 4 Caputo Type FDE s, Fractional material models a topic of interest to one of the authors ADF.

are an important application where systems of fractional order differential. equations FDE s arise that need to be solved in accordance with appropriate. initial and boundary conditions A number of other examples are well known. we refer the reader to the book by Podlubny 45 Chap 101 and the survey of. Mainardi 32 for more information,A FDE of the Caputo type has the form. satisfying a set of possibly inhomogeneous initial conditions. whose solution is sought over an interval 0 X I say where X E It turns. out that under some very weak conditions placed on the function f of the. right hand side a unique solution to Eqs 13 14 does exist 8. A typical feature of differential equations both classical and fractional is the. need t o specify additional conditions in order to produce a unique solution. For the case of Caputo FDE s these additional conditions are just the static. initial conditions listed in 14 which are akin to those of classical ODE S. and are therefore familiar to us In contrast for Riemann Liouville FDE s. these additional conditions constitute certain fractional derivatives and or. integrals of the unknown solution at the initial point x 0 27 which are. functions of x These initial conditions are not physical furthermore it is not. clear how such quantities are to be measured from experiment say so that. they can be appropriately assigned in an analysis If for no other reason the. need t o solve FDE s is justification enough for choosing Caputo s definition. i e D Jral aDral for fractional differentiation over the more commonly. used at least in mathematical analysis definition of Liouville and Riemann. viz D r a 1 r a l a, 5 Numerical Approximation of Caputo type Derivatives. 5 1 Fundamental ideas, Unlike ordinary derivatives which are point functionals fractional deriva. tives are hereditary functionals possessing a total memory of past states A. We explicitly note however the very recent paper by Podlubny 46 who attempts. to give highly interesting geometrical and physical interpretations for fractional. derivatives of both the Riemann Liouville and Caputo types These interpretations. are deeply related to the questions What precisely is time Is it absolute or not. And can it be measured correctly and accurately and if so how Thus we are. still a long way from a full understanding of the geometric and physical nature. of a fractional derivative let alone from an idea of how we can measure it in an. experiment but our mental image of what fractional derivatives and integrals look. like continues to improve, numerical algorithm for computing Caputo derivatives has been derived by. Diethelm 7 and is listed in Alg 1 Validity of its Richardson extrapolation. scheme for 1 a 2 or one similar to it has to date not been proven. or disproven Here yn denotes y x n while Y N represents y X where O X. is the interval of integration fractional differentiation with 0 5 x 5 X. This algorithm was arrived at by approximating the integral in Eq 12 with. a product trapezoidal method thereby restricting 0 a 2 Similar algo. rithms applicable to larger ranges of a can be constructed by using the general. procedure derived in Ref 7 if they become needed,Algorithm 1.

Computation of a Caputo fractional derivative 0 a 2 a 1. For interval 0 XI with grid x n h n 0 1 2 N where h X N. D yN h h r 2 a Cn Oa n N Y N n c E i N n khk k y 7. D y X D yN h O h2, using the quadrature weights derived from a product trapezoidal rule. n l 2n1 n 1 if 0 n N,1 a N N N l if n N,Refine if desired using Richardson extrapolation. Algorit hrns for the Fractional Calculus A Selection of Numerical Methods I Diethelmay1 N J Fordb A D Freedc Yu Luchkod a Institut fir Angewandte Mathematik Technische Universitat Braunschweig

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