Fractional Differential Equations in Matlab

分数阶微积分

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Fractional Derivatives, Fractional

Integrals, and Fractional Differential

Equations in Matlab

Ivo Petráš

Technical University of Košice

Slovak Republic

1. Introduction

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary

differentiation and integration to non-integer (arbitrary) order. The subject is as old as the

calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented

this kind of calculation. In a letter to L’Hospital in 1695 Leibniz raised the following question

(Miller and Ross, 1993): “Can the meaning of derivatives with integer order be generalized to

derivatives with non-integer orders?" The story goes that L’Hospital was somewhat curious

about that question and replied by another question to Leibniz. “What if the order will

be 1/2?" Leibniz in a letter dated September 30, 1695 replied: “It will lead to a paradox,

from which one day useful consequences will be drawn." The question raised by Leibniz for

a fractional derivative was an ongoing topic in the last 300 years. Several mathematicians

contributed to this subject over the years. People like Liouville, Riemann, and Weyl made

major contributions to the theory of fractional calculus. The story of the fractional calculus

continued with contributions from Fourier, Abel, Leibniz, Grünwald, and Letnikov.

Nowadays, the fractional calculus attracts many scientists and engineers. There are several

applications of this mathematical phenomenon in mechanics, physics, chemistry, control

theory and so on (Caponetto et al., 2010; Magin, 2006; Monje et al., 2010; Oldham and Spanier,

1974; Oustaloup, 1995; Podlubny, 1999). It is natural that many authors tried to solve the

fractional derivatives, fractional integrals and fractional differential equations in Matlab.

A few very good and interesting Matlab functions were already submitted to the MathWorks,

Inc. Matlab Central File Exchange, where they are freely downloadable for sharing among

the users. In this chapter we will use some of them. It is worth mentioning some addition to

Matlab toolboxes, which are appropriate for the solution of fractional calculus problems. One

of them is a toolbox created by CRONE team (CRONE, 2010) and another one is the Fractional

State–Space Toolkit developed by Dominik Sierociuk (Sierociuk, 2005). Last but not least we

should also mention a Matlab toolbox created by Dingyü Xue (Xue, 2010), which is based on

Matlab object for fractional-order transfer function and some manipulation with this class of

the transfer function. Despite that the mentioned toolboxes are mainly for control systems,

they can be “abused" for solutions of general problems related to fractional calculus as well.

www.intechopen.com

4 Will-be-set-by-IN-TECH

where Γ(.) is the gamma function. The Caputo deﬁnition of fractional derivatives can be

written as

f (t)=

Γ(n − α)



(n)

(τ)

(t − τ)

α−n+1

dτ,for(n −1 < α < n). (10)

In this chapter we will consider mainly the GL, the RL, and the Caputo deﬁnitions. This

consideration is based on the fact that, for a wide class of functions, the three best known

deﬁnitions - GL, RL, and Caputo - are equivalent under some conditions (Podlubny, 1999).

As it was already mentioned, under the homogeneous initial conditions the

Riemann-Liouville and the Caputo derivative are equivalent. Let us denote the

Riemann-Liouville fractional derivative as

t (t) and the Caputo deﬁnition as

f (t),

then the relationship between them is:

t (t)=

f (t)+

n−1

∑

k=0

(t − a)

k−α

Γ(k − α + 1)

(k)

(a),

for f

(k)

(a)=0 (k = 0, 1, . . . , n − 1).

The initial conditions for the fractional-order differential equations with the Caputo

derivatives are in the same form as for the integer-order differential equations. It is an

advantage because applied problems require deﬁnitions of fractional derivatives, where there

are clear interpretations of initial conditions, which contain f

(a), f

′

(a), f

′′

(a),etc.

In addition, for approximation purpose we consider the Laplace transform method of the

fractional differintegral

. The Laplace transform of the join operator of order α for the GL,

RL and Caputo deﬁnitions, under the zero initial conditions is:

f (t); s} = s

F(s). (11)

Laplace transform technique is considered as an efﬁcient way in solving differential equations

with integer order and fractional order as well. For differential equations with fractional

order, the Laplace transform technique works effectively only for relatively simple equations,

because of the difﬁculties of calculating inversion of Laplace transforms. This problem was

solved by applying a numerical inverse Laplace transform algorithms in fractional calculus

(Sheng et al., 2011). Three numerical inverse Laplace transform algorithms in Matlab, named

invlap(), gavsteh(),andnilt(), were described there and tested.

The Laplace transforms for several known Mittag-Lefﬂer type functions are summarized as

follows (Magin, 2006; Podlubny, 1999):

L{E

(−λt

)} =

α−1

+ λ

, L{t

α−1

α,α

(−λt

)} =

+ λ

, (12)

L{t

β−1

α,β

(−λt

)} =

α−β

+ λ

, L{E

(t, ±λ; α, β)} =

k!s

α−β

∓λ )

k+1

2.3 Numerical methods for fractional calculus

There are many numerical methods appropriate for the fractional calculus computation. They

were described in several works and a good survey can be found in (Vinagre et al., 2000; 2003).

In the above-mentioned papers are described the time domain and frequency methods in

242

Engineering Education and Research Using MATLAB

www.intechopen.com

Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab 5

the form of IIR and FIR approximations together with illustrative examples. Some of the

mentioned frequency methods in both forms of approximation have been realized as the

Matlab routines in Duarte Valerio’s toolbox called ninteger (see detail review in (Valério, 2005)).

Also created in this toolbox was a Simulink block nid for fractional derivative and integral,

where the order of derivative/integral and method of its approximation can be selected.

2.3.1 Grünwald-Letnikov method

For numerical calculation of fractional-order derivatives we can use the relation (13) derived

from the GL deﬁnition (8). This approach is based on the fact that for a wide class of functions

the three deﬁnitions - GL (8), RL (9), and Caputo’s (10) - are equivalent if f

(a)=0. The relation

for the explicit numerical approximation of q-th derivative at the points kh, (k = 1,2,...) has

the following form (Dorˇcák, 1994; Podlubny, 1999):

(k−L

/h)

f (t) ≈ h

−q

∑

j=0

(−1)





f (t

k−j

)=h

−q

∑

j=0

(q)

f (t

k−j

), (13)

where L

is the “memory length", t

= kh, h is the time step of calculation and c

(q)

(j =

0, 1, . . . , k) are binomial coefﬁcients. For their calculation we can use for instance the following

expression (Dorˇcák, 1994):

(q)

= 1, c

(q)



1 −

1 + q



(q)

j−1

. (14)

The binomial coefﬁcients c

(q)

(j = 0,1,...,k) can be also expressed by using a factorial.

Writing the factorial as a gamma function (2), it allows the binomial coefﬁcient to be

generalized to non-integer arguments. We can write the relations:

(−1)





=(−1)

Γ(q + 1)

Γ(j + 1)Γ(q − j + 1)

Γ(j − q)

Γ(−q)Γ(j + 1)

. (15)

Obviously, for this simpliﬁcation we pay a penalty in the form of some inaccuracy. If f (t) ≤

M, we can easily establish the following estimate for determining the memory length L

providing the required accuracy ǫ:

≥



ǫ|Γ(1 −q)|



1/q

. (16)

The above-described numerical method is called Power Series Expansion (PSE) of a generating

function. It is important to note that PSE leads to approximation in the form of polynomials,

that is, the discretized fractional operator is in the form of FIR ﬁlter, which has only zeros

(Chen and Moore, 2002; Vinagre et al., 2003).

The resulting discrete transfer function, approximating fractional-order operators, can be

expressed in z-domain as follows:

±r

G(z)=

Y(z)

F(z)





±r

PSE





1 − z

−1



±r



≈ T

∓r

−1

), (17)

243

Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab

www.intechopen.com

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