# diethelm06-COLLOQUE

### A survey of numerical methods in fractional calculus

Kai DIETHELM
Institut of Computational Mathematics, Technische Universität Braunschweig, Germany
Gesellschaft für Numerische Simulation mbH (GNS), Braunschweig, Germany

In many applications in science and engineering, one needs to find fractional derivatives or integrals of a given function or to solve a (partial or ordinary)fractional differential equation. However, normally it is impossible to find exact analytical expressions for the solution of the problem at hand. This is true already in situations where the given function (i.e. the function that needs to be differentiated or integrated, or the right-hand side of the differential equation) is of a relatively simple nature. And even if one is in the lucky position to have found an analytic solution, then this solution is likely to be of a very complicated form (typically an expression involving special functions or other types of infinite series), so that it cannot be handled conveniently. Thus there is a large demand for efficient and reliable numerical methods in fractional calculus.
In this talk, we shall therefore discuss a number of numerical methods that can be used to compute approximate solutions for problems of the types mentioned above. We will restrict our attention to problems involving fractional derivatives or integrals of Riemann-Liouville and Caputo type. Since most applications require the solution of fractional differential equations of the form
Dαy(t) = f(t, y(t))
(augmented by some properly chosen initial or boundary conditions), the main focus of the talk will be on this class of problems. Due to the lack of time, we will not be able to talk about multi-order fractional differential equations like
Dαny(t) = f(t, y(t), Dα1y(t), Dα2y(t), . . . , Dαn-1y(t)),
a class of equations that introduces a whole set of additional new difficulties [3, 4].
In particular, we shall compare algorithms based on the Grünwald-Letnikov approach [9, 10] (already discussed in the classical book of Oldham and Spanier [13]), algorithms based on Lubich’s fractional generalization of linear multistep methods [11, 12] (the most important subclass of which is formed by fractional backward differentiation formulas) and algorithms based on classical numerical integration techniques [2, 5, 6]. Finally we talk about approaches that have been tailored for certain very specific problems [1]. Among the specific questions to be discussed we shall have :
• What can we say about the (theoretical) order of convergence of the algorithms ?
• Can we reduce the inherent arithmetic complexity (which is mainly due to the non-locality of fractional differential operators) ? Podlubny [14] and Ford and Simpson [8] have dealt with approaches in this context.
• Given a formal mathematical (i.e. theoretical) description of a rapidly convergent algorithm, how can we implement it in a reliable way in practice ? It has been shown in [7] that this may be a highly difficult task.
• Based on the answers for these questions, we will identify some prototype algorithms for the most common problems. These algorithms are going to form the core of a software library that is presently under development at GNS.

Références

 [1] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29,145–155, 2002. [2] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electronic Transactions on Numerical Analysis, 5, 1–6, 1997. [3] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods, Computing, 71, 305-319, 2003. [4] K. Diethelm, N.J. Ford, Multi-order fractional differential equations and their numerical solution, Applied Mathematics and Computation, 154 (3), 621–640, 2004. [5] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36 (1), 31–52, 2004. [6] K. Diethelm, N.J. Ford, A.D. Freed, Y. Luchko, Algorithms for the fractional calculus : A selection of numerical methods, Computer Methods in Applied Mechanics and Engineering, 194 (6-8), 743–773, 2005. [7] K. Diethelm, J.M. Ford, N.J. Ford, M. Weilbeer, Pitfalls in fast numerical solvers for fractional differential equations. Journal of Computational and Applied Mathematics, 186 (2), 482–503, 2006. [8] N.J. Ford, A.C. Simpson, The numerical solution of fractional differential equations : Speed versus accuracy, Numerical Algorithms, 26 (4), 333–346, 2001. [9] A.K. Grünwald, Über “begrenzte” Derivationen und deren Anwendung, Zeitshrift für angewandte Mathematik und Physik, 12, 441-480, 1867. [10] A.V. Letnikov, Theory of differentiation with an arbitrary index, Math. Sb., 3, 1–66, 1868. (in Russian). [11] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Mathematics of Computation, 45 (172), 463–469, 1985. [12] C. Lubich, Discretized fractional calculus, SIAM Journal on Mathematical Analysis, 17 (3), 704–719, 1986. [13] K.B. Oldham, J. Spanier, The fractional calculus, Academic Press, New York, 1974 ; reprint : Dover, New York, 2006. [14] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.

Laboratoire de Mécanique des Structures et des Systèmes Couplés - LMSSC