Sabatier Agrawal Machado Advances in Fractional Calculus Episode 2 pps

It has also been shown [7] THE CAPUTO FRACTIONAL DERIVATIVE 29... As is well known, in the solution of fractional differential equations, the initial conditions are specified in terms of

a t t

and

,

,0),,,,,()()

)(

1),,,,

()

dt

d t f d dt

t c m

m q

0),,,,

and the definition in Eq (4) is applied for (f, p,a,c,t) in Eq (7) The next section considers the extention to the Caputo fractional derivative

3 Initialization of the Caputo Fractional Derivative

The Caputo fractional derivative was introduced [8,9] to alleviate some of the difficulties associated with the RL approach to fractional differential equations when applied to the solution of physical problems and is defined by [8]:

)1

()()

()(

1)

t a

m m

t

C

m t

f

may be used for the history period, i.e., a t c, while f

function to be fractionally differintegrated, i.e., t c It has also been shown [7]

THE CAPUTO FRACTIONAL DERIVATIVE 29

As is well known, in the solution of fractional differential equations, the initial conditions are specified in terms of fractional derivatives in the RL approach, but, in terms of integer order derivatives with known physical interpretations in the Caputo approach [10] In view of the popularity of the Caputo formulation in applications of physical interest, the key question to be asked is: when viewed from the LH general initialization perspective, what “history” is inferred [11,12] for the Caputo derivative?

4 Relation Between the Initialized LH and Caputo Fractional Derivatives

As has been noted, the generalized initialization as applied to RL fractional

4.1

We first consider the case when 0 1, m 1, then

)()

( 0 1 0 (1 )

0D t f t D t D t f t

,0,

),0,,1,()() 1 (

0,0,),1(,)() 1 ( 0

dt

d t

f

Substituting explicitly for the quantities in curly brackets in Eq (11) yields

derivative, according to LH is given by Eq (7) and will be used in the following

Simple cases : 0 1 and 1 2

i e.,

examples, where for convenience, is used in the place of q, i.e.,

m p 0,mis a positive integer, and as before, for terminal initialization,

(h, m, a, c, t) 0 Hereafter t c corresponds to t 0 Three cases will be considered below

31

0 0

)1(

1)

()()1(

)0()1()()

1(1

0 0 0

t d f t

dt d

f t d t f t

f

D

a

t t

(13)

Rewriting the argument of the convolution integral as t f and using the definition of the Caputo derivative, Eq (9) with m 1 and 0 1, one can write the following expression relating the Caputo derivative to the initialized LH derivative for 0 1:

.0,

),0,),1(,()

1(

)0()

()

dt

d f t t f d

t

f

where the last integral in Eq (13) is restated as an LH initialization

For the case 1 2 , m 2 and the initialized LH derivative given by

0,0,0,),2(,)() 2 ( 0

dt

d dt

d t

f

yields on substituting explicit expressions for the quantities in the curly brackets

),0,),2(,()

()()2(

1

2 2 0

1

dt

d d f t

dt

d dt

(16)

THE CAPUTO FRACTIONAL DERIVATIVE

Recasting the convolution integral in Eq (16) by interchanging the arguments and carrying out the differentiation of the integral using Leibnitz’ rule yields the expression relating the Caputo derivative to the initialized LH derivative for the case1 2 as [11]:

.0),,0,),2(,()

1(

)0()

2(

)0()

(

2

2 1

0

dt

d f t f t t f d

t

f

The expressions in Eq (14) and Eq (17) can be generalized as shown below

Generalizing to the case when m 1 m we get

0,),0,),(,()

()

dt

d t f d

dt

d t

f

D

m

m m

t m

m

,),0,),(,()

1(

)0(

)()

()(

1)

(

1 0

0

1 0

t a m f dt

d k

f t

d f t

m t

f

D

m

m m

k

k k

t

m m t

(19)

or,

.1

,0,

),0,),(,(

)1(

)0()

0 0

0

m m

t t a m f dt d

k

f t t f d t

f

D

m m

m k

k k t

range 0 1, Eq (20) simplifies to the Eq (14), and for the range1 2,

4.2 General case m 1 m

f (t) and its integer-order derivatives f

t 0 for 0 The details of the derivation can be found in ref [11] For the

33

5 Inferred History of the Caputo Derivative

It is important to determine the “history” inferred by use of the Caputo derivative of a functionf t This can be achieved by setting the Caputo derivative equal to the LH fractional derivative of the same order , and for the same function f t , for t 0

It follows from Eq (14) that the two derivatives will be equal for 0 1 if

)1(

)0()

,0,),1(,

side (RHS) For the terminal initialization considered in this note, it follows that

“history” would be given by a function f1 , satisfying the following equation:

)1(

)0()

()()1

(

f t

It is important to note that the left-hand side (LHS) of Eq (22), which is the

instant prior to t 0 Specifically, RHS of Eq (22) is not a function of

THE CAPUTO FRACTIONAL DERIVATIVE

it reduces to Eq (17) Expressions in Eq (14), Eq (17), and Eq (20) will now

be used to determine the history inferred by the use of the Caputo derivative

5.1 Simple cases : 0 1 and 1 2

required initialization, is only related to the value of the function evaluated

at t 0, on the RHS, and not to the function or its derivatives at any

may be represented by a polynomial in , that is

1

)1(

)0()

(

n

n n

(1

)0

d

a

n n

n

(24)

this result, we obtain

0),

0(

!

!1

!

!

)0

t f

t

j

t n i

n n j

a a t n

j

i n i n

(25)

Differentiating with respect to t gives

.0),

0(

1

!1

1)

1 1 1

t f

t

j

t n i

n j

a a

t i f

j

i n i n

(26)

the RHS, (the summation of the higher power terms, tn , cannot sum to a t

term), and because all derivatives f1n 0 0,n 1 we have

integrand of Eq (22) and interchanging the order of integration and summation

It is clear that only the n = 0 case on the LHS can match the exponent of t on

A general solution for the definite integral can be derived [11] and substituting

35

.0,

01

0

a t

Because the starting point of the initialization “a” does not occur on the RHS

of Eq (26), we must have a , to force the first term of Eq (27) to zero Therefore, for0 1 we have

0for

,0constantt

f t

The above arguments can be extended to the case when 1 2as outlined below It follows from Eq (17) that the Caputo derivative and the LH fractional derivative of the same order and the same function f (t)would be equal to each other if

.0,)1(

)0()

2(

)0()

()()

2

(

1 1 2

2

t f

t f

t d f t

(1

)0

t d t

n

(31)

Substituting the result of integration and performing the differentiation operation yields

), and from Eq (23), f

THE CAPUTO FRACTIONAL DERIVATIVE

Substituting as before from the McLaurin expansion in Eq (23), and inter- changing the order of integration and summation yields

0()1()0

(

1

1)

2()0

(

!1

!1

)1()

1

1 1 1

t f

t f

t

j

t n n

n f

i n n j

a a

t i i

(32)

It is clear that only the n 0and

match those of the terms on the RHS It is required therefore that f1(n)(0 ) 0the RHS and because 1 2 we must have a Thus we must have

0),

0()1()0(

)3)(

2(

)2)(

3)(

0()

2

(

)1)(

1 )

1 ( 1 1

t f

t f

t

t

f t f

(33)

Therefore we must have f1(0 ) f(0 ), f1(1)(0 ) f1(1)(0 ) It follows from Eq (23) that

t f f t

f() (0) (0) t 0 , (34)

for the case of 1 2 Both the results of Eq (29) and Eq (34) can be obtained as a special case of the more general result derived below

n 1cases can allow exponents of t that will

for all n 2 Because the starting point of the initialization a does not occur on

37

In this case [11] setting the two derivatives in Eq (20) to be equal yields

.1

,0,)1(

)0()

,0,),(

,

0

m m

t k

f t t

a m

,0

,)1(

)0(

0

0

1 1

m m

t

k

f t d

f t

m m

m

(36)

Again representing f1 as a continuous function by Eq (23), gives

.1

,0,)1(

)0(

1

01

1

0

0

0 1 1

m m

t k

f t

d n

f t

n

n m

,0

,)1(

)0(1

m m

t

k

f t j

m

j n t

f m

m k

k k m

j

m j

n n

(38)

In general, the equality will only hold when

.1,,1,0,

00

5.2 General case: m 1 m

THE CAPUTO FRACTIONAL DERIVATIVE

, and hence, terms on the LHS, (after

the mth order differentiation,) with exponents greater than this must have zero

Placing these results into Eq (23) we have, for m 1 m

1 0

0,

)1(

)0()

n

n n

t a t

n

f t

5.3 Example

22

t t

21

2 1 2

0

2 / 1 2

1 2

t t

values of the integer-order derivatives evaluated at t = 0 It is also observed that

A simple example will illustrate the profound differences between the Caputo derivative and the LH initialization of the RL derivative Consider the semi-derivative of f with a history period starting at

which has removed the effect of the singularity at t = 0 Because 1/ 2, we

We now consider the LH initialization of the RL derivative for terminal initialization of the function f

with order higher than order m 1 will in general be discon-

tinuous at t = 0 Eq (40) yields Eq (29) and Eq (34) as special cases

39

,0,0,0,2,2/1,2

22

2 2 / 1 0

2 2 / 1 0 1 2 2

/

1

0

t t f

t d dt d

t D D t

D

t

t t t

(42)

,0,22

2

0 2

2 2

1

1 2

0 12

1 2

t d t

d t

dt

d t

D

t

(43) Integrating , collecting terms, and simplifying yields

2 1

2 / 3 2

2 / 1 0

15

)2(40

5.4

Here we examine application of the inferred history of the Caputo derivative developed in earlier sections to fractional differential equations to gain further insight into the initialization issues associated with the Caputo derivative

Case 1

Suppose we consider only fractional differential equations of the form

Cd f ( t ) y ( t ), (45)

The results of Eq (41) and Eq (44), are shown graphically in Figure 1 It is

Also shown in Figure 1 is the uninitialized LH semi-derivative of the

Caputo derivative in application

THE CAPUTO FRACTIONAL DERIVATIVE

Inferred Caputo Initialization

<−−− back to minus infinity

Chosen LH Initialization

LH Semi−Derivative started at t=0

LH Semi−Derivative

started at t=a=−2

<−− Historical Period Problem Space −−>

2

Use of the Caputo derivative and its inferred orderhistory, Eq (34), may

be acceptable if i) it is found that the history acceptable to the physics defining

the problem and ii) if it is acceptable to have discontinuity of the derivatives of

0

, t

t Thus for this case, the orders of the terms lie between different integers, that is for the 2 -order derivative the order is 1 2 2 and for the -order derivative it is 0 1 Thus based on the results of the

Fig 1. Initialization for Caputo and LH semi-derivatives of (t 2)

history, i.e., f

0

(

)

will satisfy both is f t f 0 , t 0 with f 0 0 But this forces the function being differentiated, that is, f t , t 0 to have f 0 0which in general may not be the case

If on the other hand, the individual terms in the differential equation of Eq (46) each have separate and independent histories This means that each differential term in the equation is disconnected from the other differential terms, and is acted on by its own individual input (history) in negative time Then at time zero, all of the individual differential terms are connected together, with the requirement that all of the individual positions, velocities, etc (including

Caputo derivative can be used in the description of Eq (46) if it is assumed that all derivative terms (elements) have separate and independent histories, are compatibly connected at timet 0, and that each derivative term according to the magnitude of its order has the history specified by Eq (40) That is, the inferred initialization of each term depends on the order of that term and the limitations of Case 1 apply

6 Summary

1 0

) (

.0,

1,

)1(

)0()

n

n n

t a m m

t n

f t

implying that integer order derivatives of f t with order higher than m 1

will in general be discontinuous at t 0 While it has been known for quite sometime that the Caputo derivative is more restrictive than the RL derivative

[13], it is now clear that the Caputo derivative can not represent general

initializations required for most analysis, physics, and engineering problems

fractional derivatives), have the same values at time zero Under this scenario

the initialization of each term reverts the situation of Case 1 Thus the

Using the LH initialization of the RL derivative it is shown that the

polynomial with maximum order of the polynomial being m 1 and given by

THE CAPUTO FRACTIONAL DERIVATIVE

-order Caputo derivative with m 1 m infers a history in the form of a

Applications of Differentiation and Integration to Arbitrary Order Academic

Press, New York

4

Fractional Differential Equations Wiley, New York

Derivatives: Theory and Applications Gordon and Breach Science

Publishers, Philadelphia, PA

6 Podlubny L (1999) Fractional Differential Equations Academic Press,

SanDiego, CA

Application in the Generalized Fractional Calculus NASA TM-1998

8 Caputo M (1969) Elasticita e Dissipazione Zanichelli, Bologna

9 Caputo M (1967) Linear Model of dissipation whose Q is almost frequency independent-II Geophys J R Astron Soc 13:529–539

Fractional Derivative NASA TM-2003

12 Achar BN, Narahari, Lorenzo CF, Hartley T (2005) Initialization Issues of

the Caputo Fractional Derivative, Proceedings of IDETC/CIE 2005, Sept

fractional evolution processes J Comp Appl Math 118:283–299

Lorenzo CF, Hartley TT (2000) Initialized fractional calculus Int J Appl Lorenzo CF, Hartley TT (2001) Initialization in Fractional Order Systems, in:

3 Oldham KB, Spanier J (1974) The Fractional Calculus- Theory and

Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and

5 Samko SG, Kilbas AA, Marichev OI (1993) Fractional Integrals and

7 Lorenzo CF, Hartley TT (1998) Initialization, Conceptualization, and

Gorenflo R, Mainardi F (1997) Fractional calculus, integral and differential

Achar BNN, Lorenzo CF, Hartley T (2003) Initialization and the Caputo

24–28, Long Beach CA DETC2005 pp 1–8

13 Mainardi F, Gorenflo R (2000) On Mittag-Leffler -type functions in

Proceedings of the European Control Conference, Porto, Portugal pp 1471–

COMPARISON OF FIVE NUMERICAL

SCHEMES FOR FRACTIONAL

dif-tional speeds for these algorithms are examined Numerical simulations exhibitthat the choice of a numerical scheme will depend on the problem consideredand the performance criteria selected

Mechanical Engineering, Southern Illinois University, Carbondale, IL 62901;Mechanical Engineering, Southern Illinois University, Carbondale, IL 62901;om@engr.siu.edu

The schemes considered are a linear, a quadratic, a cubic, a state-space for five different problems which include two linear 1-D, two nonlinear 1-Dinitial conditions (ICs) are considered The stability, accuracy, and computa-

non-Fractional differential equations, fractional derivatives, numerical schemes

in Physics and Engineering, 43– 60

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

to solve these equations These methods include Laplace and Fourier forms [3, 9, 10], eigenvector expansion [11], method based on Laguerre inte-gral formula [12], direct solution based on Gr¨unwald Letnikov approximation[3], truncated Taylor series expansion [13], diffusive representation method[14], approximate state-space representations [15, 16], and numerical methods[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] Sabatier and Malti [28] groupedthe FDEs appearing in the above applications into number of classes to de-velop a benchmark to evaluate performance of numerical algorithms Theyalso presented numerical techniques and results for each group of FDEs.This brief review of applications of FDs and various analytical and numer-ical techniques to solve FDEs is by no means complete Examples of manyother applications could be found in [3, 4, 6, 7, 8] and the references there in,and many other analytical and numerical schemes to solve the FDEs are cited

trans-in [21, 21, 26, 27]

This paper presents a comparative study of the performance of five ent numerical schemes to solve FDEs, namely, the linear [21, 22], the quadratic[25], the cubic [26], the direct method based on the Gr¨un

differ-grator [15, 16] For completeness, the associated algorithms/procedures arediscussed briefly Details of these algorithms could be found in the referencescited above The issues investigated include the numerical stability, accuracy,

neous boundary conditions are considered

2 Notations and Definitions

order α > 0, which is given as

Iαy(t) = 1

Γ (α)

 t

0 (t − τ)α−1y(τ )dτ, (α > 0), (1)where Γ

Cauchy integral formula Here we take the lower limit of the integral as 0,operator Iα

IαIβy(t) = IβIαy(t) = Iα+βy(t) α, β > 0 (2)

These two derivatives are given as:

wald Letnikov proximation [3], and the state-space approximation of the fractional inte-

ap-examples, two linear one dimensional,two nonlinear dimensional, and and computational times for these algorithms Numerical results for five

one-We begin with the Riemann Liouville definition of the fractional integral of

is the gamma function For integer α > 0, Eq 1 is known as thehowever, a nonzero limit can also be taken It can be verified that the integral

commutes, i.e.,

We will largely deal with Caputo fractional derivatives (CFDs) However, wewill also come across the Riemann Liouville fractional derivatives (RLFDs).linear multidimensional are presented Both homogeneous and nonhomoge-

COMPARISON OF FIVE NUMERICAL SCHEMES 45Caputo Fractional Derivative (CFD)

n t

0 (t − τ)n−α−1y(τ )dτ (4)where α > 0, n is the smallest integer greater than or equal to α, and theoperator Dn is the ordinary differential operator These two derivatives arerelated by the formula

3 Statement of the Problem

We consider the following FDEs and the ICs

y(i)(0) = y(i)0 , i = 1, · · · , n − 1 (7)Observe that here we consider the FDE in terms of Caputo derivatives.This allows us to account for physical initial conditions Equations 6 and 7 areapplicable for both scalar and vector y In the discussions to follow, we usescalar y to derive an equation However, when solving a problem in which y

is a vector, we will use vector equivalent of the formulation without explicitlywriting these equations Note that [29] discusses the problem of finding thecorrect form of the initial conditions in a more general setting, not necessarilyassuming that the entire history of the process can be observed A similartreatment for the Caputo derivative is presented in [30]

Applying the operator Iαto Eq 6, and using Eqs 1, 2, 3, and 7, we obtain

y(t) = g(t) + 1

Γ (α)

 t

0 (t − τ)α−1f (τ, y(τ ))dτ, (8)where

4 The Numerical Schemes

In this section, we briefly review the five numerical schemes stated above.The first three schemes essentially attempts to solve Eq 8, the direct schemeapproximates the fractional derivative terms, and the state-space scheme isbased on the state-space approximation of a fractional integral operator

This scheme, presented by Diethelm, Ford, and Freed, is also called P (EC)Eand P (EC)M

correct, and M represents the iteration number [21, 22] The difference tween the P (EC)E and P (EC)ME schemes is that in the former scheme onlyone corrective step is taken whereas in the later scheme multiple correctivesteps are taken

be-Let T be the maximum simulation time To explain the scheme, dividethe time T into N equal parts, and let h = T /N be the time interval ofeach part The times at the grid points are given as tj = jh, j = 0, · · · , N.For simplicity in the discussion to follow, we use the following notations:y(tj) = y(jh) = yj, g(tj) = g(jh) = gj, and f (tj, y(tj)) = f (jh, y(jh)) = Fj.Note that in many numerical analysis papers y(tj) and yj represent the trueand the numerically computed value of y at tj No such distinction is madehere Where such distinction is necessary, the true and the computed valuesare explicitly identified

Now assume that the approximate numerical values for y(t) have beendetermined at the grid points tj, j = 0, · · · , m, tj < T Assuming that y and

f (t, y(t)) vary linearly over each part and using Eq 8, ym+1is given as [21]

except that after first iteration the value evaluated using Eq 10 is used as thepredicted value for the subsequent iteration This is essentially equivalent to

a fixed point iteration The details of the algorithms can be found in [21, 22]

4.1 The linear scheme

E schemes, where P , E, and C stands for predict, evaluate, and

integral in Eq 8 is approximated using a product rectangular rule, and Eq 10

is used to correct the values In [22], this iteration is continued several times,

COMPARISON OF FIVE NUMERICAL SCHEMES 47

improve the accuracy of the results Here we take a slightly different approach.For linear case, we solve Eq 10 explicitly, and for nonlinear case we solve it

mas the starting guessfor ym+1

Note that in this class of schemes y and f (t, y(t)) are approximated usinglinear functions, and therefore we call them the linear schemes

In this scheme, N is taken as an even number, and y and f (t, y(t)) are imated over two adjacent parts using quadratic polynomials Assume that yj,

approx-j = 1, , 2m have already been computed Using Eq 8, the expressions for

y2m+1 and y2m+2 are given as

Since yj, j = 0, · · · , 2m are known, the first integrals in both Eqs 12 and

13 can be computed explicitly To compute the second integral in Eq 13,

f (t, y(t)) is approximated over [2mh, (2m + 2)h] in terms of F2m, F2m+1, and

(QIPs), which is 1 at node 2m + j and 0 at the two other nodes Substituting

Eq 14 into Eq 13, we obtain y2m+2in terms of F2m+1, and F2m+2 Note that

F2mis not included here as it can be computed directly from y2m To computethe second integral in Eq 12, f (t, y(t)) is approximated over [2mh, (2m + 1)h]

in terms of F2m, F2m+1/2and F2m+1using QIPs similar to the one used in Eq

14 Using Eq 14, F2m+1/2 is expressed in terms of F2m, F2m+1, and F2m+2.This leads to y2m+1 in terms of F2m+1, and F2m+2 Thus, we obtain twoequations in terms of two unknowns y2m+1and y2m+2, which are solved usingthe Newton

These authors also present a Richardson extrapolation-type scheme to further

using the Newton Raphson scheme for which we take y

4.2 The quadratic scheme

= 0, 1, and 2 are the quadratic interpolating polynomials

Raphson method The details of the algorithm can be found in [25]

In this scheme N

mated over three adjacent parts using cubic polynomials, and expressions aregenerated for y3m+1, y3m+2and y3m+3in terms of F3m+1, F3m+2and F3m+3.These expressions are solved using the Newton Raphson method as before.For brevity, the details of the algorithm is omitted here, and the readers arereferred to [26] where further details can be found

4.4 The direct scheme

To explain this scheme, assume that yj, j = 0, · · · , m have already been puted, and we want to compute ym+1

com-imated at tm+1using a Gr¨un Letnikov definition [3] This leads to

where the coefficients wα

j satisfy the following recurrence relationship,

w0α= 1, wαj = (1 −1 + αj )wj−1α , j = 1, 2, · · · (16)

Note that the CFD can be approximated directly using a slightly differentscheme (see [24]), the approach considered here is believed to be computation-ally efficient For nonlinear f (t, y(t)), Eq 15 leads to a nonlinear equation interms of ym+1

Note that if 1 < α < 2, then y1 is computed as

and phase lead filters Thus, the integral operator is written as [16],

4.3 The cubic scheme

is taken as a multiple of 3, y, and f (t, y(t)) are

approxi-wald−

which is solved using the Newton Raphson method as before

−

−

4.5 The state-space non-integer integrator

in terms of a set of integer-order integrators In the direct scheme, the CFD in Eq 6

is first replaced with the RLFD using Eq 5 and then the RLFD is