Trans. of the Society of Instrument and Control Engineers Vol.E-6, No.1, 1/8 (2007) A Numerical Algorithm of Discrete Fractional Calculus by using Inhomogeneous Sampling Data Fujio Ikeda∗ This paper presents an efficient numerical method to realize discrete models of fractional derivatives and inte- grals which imply derivatives and integrals of arbitrary real order. This approach is based on a class of Stieltjes integrals transferred from the Riemann-Liouville definition. It is to calculate on inhomogeneous sampling periods which are getting longer as the operation points go back toward the initial time. It leads to the effective quality which has low computational costs and enough accuracy. The calculation times and precision of the proposed procedure are compared with those of a conventional procedure for a practical numerical simulation and the effectiveness of this procedure is verified. Key Words: fractional derivative, fractional integral, Stieltjes integral, discrete model, numerical algorithm It has been common to employ the G1-Algorithm 1) 1. Introduction or any of the many other similar approaches based on Ordinarily, derivatives of functions are described in the Gr¨unwald-Letnikov differintegral, which is well suited terms of integers, for example, the first derivative of posi- for constructing algorithms suitable for numerical analy- 5)∼11) 12) tion yielding speed, the second derivative yielding accel- sis . Recently, Podlubny described both a geo- eration. The concept has been extended to non-integer metrical interpretation and physical meaning of FD. Ma 13) derivatives, which are known as fractional derivatives and Hori employed these viewpoints in their proposal (FD). Fractional integrals (FI) have also been defined, and for a numerical solution method that has a clear physical these together make up fractional calculus (FC) 1). The analogue. However, these direct discrete procedures all concept of FC was known at the time of Leibniz during suffer from high calculation costs imposed by the require- the 17th century, but it has received little theoretical con- ment of previous history. Practical implementations of sideration since then, because its physical meaning was these algorithms will need to get negotiate this problem questionable and its mathematics was complicated. How- by reducing the computational burden. 4) ever, in recent years much research has been performed The Short Memory Principle , which fixes the range of in physics, rheology, fractals, control systems and other calculations at some limited history and neglects whatever fields where fractional differential equations made up of occurred earlier, is one possible approach for reducing the FD and FI have been reported to show several advan- computations. However, the shorter the remaining calcu- tages 2)∼4). lation domain, the greater the error this principle intro- In order to handle FC operators numerically, the sys- duces, so the researcher is forced to trade-off calculation tems must be transformed into discrete-time system mod- time against precision. els. It is easy to obtain very accurate and concise models This paper describes an inhomogeneous sampling algo- of integer-order differential operators using Tustin trans- rithm (IS-Algorithm) as an essential solution for the issue 14), 15) forms or other well-known trapezoidal integration meth- of high computational costs . It is based on the ods. In contrast, FC operators are global with respect Riemann-Liouville integral, a class of Stieltjes integrals, to time, so there is a strict requirement for the calcula- which were used by Podlubny in the above publication to tions to include the entire time history of the operators. obtain some of his geometrical interpretations. In con- If the size of the time step (i.e., sampling period) is small, ventional methods, numerical integration is carried out in however, the satisfaction of this requirement necessitates the so-called Riemann sum over the variable t (“actual” the use of large amounts of memory, which significantly time) at step ∆τ, i.e., in a discretized model with a con- increases the computational expense of the procedure. stant sampling period. The present method, in contrast, employs the Stieltjes integral, performed over variable Tq ∗ Department of Mechanical Engineering, Kochi National T College of Technology, Monobe 200-1, Nankoku, Kochi 783- (“transformed” time) at a constant step ∆ . The rela- 8508, JAPAN. tion between “actual” time and “transformed” time is as TR 0001/07/E-601–0001 c 2007 SICE 2 T. SICE Vol.E-6 No.1 January 2007 ∞ follows: When the “actual” time steps are constant, the −t z−1 Γ(z) = e t dt , (z>0). (2) “transformed” time is interpreted to pass at a gradually 0 decreasing rate. Conversely, in this method, when the Eq.(1) above is used to define the fractional derivative “transformed” time steps are set constant, the “actual” (FD): The q-th (q ≥ 0,q ∈) derivative of function f(t) time steps do not remain constant in length, but become is given by longer as one moves backward in time. As a result, fewer points in the history are sampled than in the conventional dm Dpf t D−(m−p)f t ( ) = m ( ) methods, greatly reducing the computational costs. Also, dt m t since the influence of the damping characteristics in the 1 d m−p−1 = (t − τ) f(τ)dτ, (3) Γ(m − p) dtm general case of physical dynamic phenomena diminishes t0 as one goes backward in time, this method is less prone to m m − ≤ q<m lose computational precision, despite the large time steps where is a natural number satisfying 1 . 1) for calculations involving the distant past. The G-L definition has been used as the basis for the This paper is organized as follows. Section 2 provides G1-Algorithm, a typical example of conventional discrete the definitions of the FC employed in this paper and procedures, and may be applied for assessing any arbi- q q ∈ f t describes the geometrical interpretations of those oper- trary derivative. The -th ( ) FC of function ( )is ations and also summarizes conventional discrete proce- defined as n−1 dures. Section 3 shows the proposed discrete procedure q −q j q D f(t) = lim ∆τ (−1) f(t − j · ∆τ).(4) for the new FC based on the geometrical interpretation ∆τ→0 j t−t j=0 ∆τ= 0 introduced in Sec. 2. Section 4 compares the calculation n steps of the proposed method with that of conventional The above is a FD when q>0, and a FI when q<0, and procedures, and shows that the present procedure is ef- the binomial coefficients are given in the general form: fective at reducing the computational costs. An equation q q q q − ··· q − j , ( 1) ( +1) =1 j = j for estimating the error of the proposed procedure is also 0 ! (5) Γ(q − 1) derived, and methods for estimating computational pre- = . Γ(j + 1)Γ(q − j +1) cision are discussed. Section 5 compares the calculation 2. 2 Geometrical interpretation of the frac- times and precision of the proposed procedure with those tional integral of a conventional procedure for a practical numerical sim- Let us describe geometrical interpretation of the FI pre- ulation and verifies the effectiveness of this procedure. sented by Podlubny 12), which is a foundation of the pro- Finally, the results of this study are summarized. cedure suggested in this paper. The process here applies 2. Fractional Calculus to the FI, but it may equally be applied to justify the FD. The definition of the FI given by Eq.(1) can be re- 2. 1 Definition of fractional calculus written in the form of a Stieltjes integral 17), which ex- In contrast to integer-order differentials dn/dtn, presses a version of the ordinary Riemann form integra- fractional-order differentials (FC) are defined as operators tion of τ as an integration of the variable Tq(τ), itself a whose order has been extended to non-integer numbers. function of τ: 1), 4), 16) Several definitions have been proposed , but here t −q the well-known Riemann-Liouville (R-L) and Gr¨unwald- D f(t)= f(τ)dTq(τ). (6) t0 Letnikov (G-L) definitions are used. 4) The variable Tq(τ) is given by The R-L definition is used as the basis for the discrete 1 q q procedure proposed in this paper. The definitions differ Tq (τ)= {(t − t0) − (t − τ) } . (7) Γ(q +1) according to the sign in the FC, i.e., whether it is differ- τ entiation or integration. Let us first define the fractional If we consider the variable of integration in Eq.(1) as the “actual” time, then the variable of integration Tq(τ) integral (FI). The q-th (q>0,q ∈) FI of function f(t) in Eq.(6) can be interpreted as the “transformed” time, is given by while the degree of integration is here limited to 0 <q≤ 1 t −q 1 q−1 D f(t) = (t − τ) f(τ)dτ. (1) without loss of generality. Γ(q) t0 The geometrical interpretation of the Stieltjes inte- Here, Γ(z) is the Gamma function: gral in Eq.(6) is given in Figure 1. The function used T. SICE Vol.E-6 No.1 January 2007 3 also obtained if a power series expansion (PSE) is car- ried out for the fractional differential operator Dq (z−1)= D -0.5f (t) q (1 − z−1)/∆τ . An expression for the discretized model f (Ǽ -1 D f (t) obtained by z-transform in the above equation is given by n−1 q −1 −q j q −j D (z ) =∆τ (−1) z , (10) G1 j j=0 where z−1f(t)=f(t − ∆τ). The sampling period ∆τ in “actual” time τ is constant in conventional discretization procedures such as the G1- Algorithm as shown in Fig.1.