Cmjv04i02p0096

Page 1

J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

The Initialization Fractional Differential Equations of Caputo Fractional Derivative *

HARISH NAGAR and **ANIL KUMAR MENARIA *

Head, Department of Mathematics, Mewar University, Chittorgarh, INDIA. ** Research Scholars, Mewar University, Chittorgarh, INDIA. (Received on: February 24, 2013) ABSTRACT Recognizing the importance of proper initialization of a system, which is evolving in time according toa differential equation of fractional order, Lorenzo and Hartley developed the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations. Key words: Caputo fractional derivatives, initialization issues. MSC 2010: 30E99, 33C90, 33E99. 1. INTRODUCTION Lorenzo and Hartley (LH)1,2 have clearly established the importance of timedependent initialization function in taking into account the history of a system which evolves according to a differential equation

of fractional order. They have considered both the Riemann-Liouville (RL) and the Grunwald formulations of fractional calculus3-6 in developing the initialization function7. In section 2, we discuss about Initialization of the Riemann-Liouville fractional differintegral. In section 3 we state

Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

97

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

the Initialization of the Caputo fractional derivative8,9. And in the final section, we find the relation between the inilized Lorenzo and Hartley and Caputo fractional derivatives.

2. INITIALIZATION OF THE RIEMANNLIOUVILLE FRACTIONAL DIFFERINTEGRAL: ୲ୌ

Consider the following q order fractional integrals of f t integral starting at time t Îą and the second, starting at time t c , respectively:

a

dt− q f (t ) =

c

t > a,

And t

1 (t − Ď„ ) ( q −1) f (Ď„ ) dĎ„ , Γ(q ) âˆŤc

t > c,

(2) It is assumed that the function f t is zero for all t a the time interval between

a function Ďˆ to the integral starting at time t c so that the result of fractional integration starting at time t c is equal to that of the integral starting at time t a for all t , i.e., c

The generalized fractional derivative, for q and p real is defined by c

d t− q f (t ) +Ďˆ = a dt− q f (t ),

t > c, (3)

Or in other words, c 1 Ďˆ= (t − Ď„ ) ( q −1) f (Ď„ ) dĎ„ , âˆŤ Γ(q ) a

t > c. (4)

Of the two types of initializations described by LH7, only the terminal initialization, in which case the integral can only be initialized prior to the start time t c, will be considered here. Then the generalized fractional integral, for arbitrary, real, and nonnegative values of ν is defined by, c

d f (t ) = ( c d )( c d m t

−p t

) f (t ),

t > c, (6)

Where, m is an integer such that m 1 , . Furthermore,q 0 and t 0.

dt− v f (t ) = c dv− v f (t ) +Ďˆ ( f , −v, a, c, t ),

v ≼ 0,

(5)

t > c ≼ a, and f (t ) for all t ≤ a and where Ďˆ ( f , −v, a, c, t ) =

q t

d t− q f (t ) . Initialization consists in adding

t

1 (t − Ď„ )( q −1) f (Ď„ ) dĎ„ , Γ(q ) âˆŤa

(1) −q f (t ) = c dt

t a and t c being considered to be the �history� of the fractional integral

c

1 (t − Ď„ ) ( v −1) f (Ď„ ) d (Ď„ ), as defined in equation (4). Γ(v) âˆŤa

In terms of the conventional notation, c

dtq f (t ) =

dm dm d − q f (t ) m Ďˆ ( f , − p, a, c, t ), m c t dt dt

t > c,

(7)

h= d

−p t

f (t ).

c Where It is of course clear that f t may be considered to be a composite function, for example a function different than f t may be used for the

Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

history period, i.e.,a , while f t remains the function to be fractionally differintegrated, i. e. , t . It has also been shown7 that for terminal initialization of the integer derivative,

ψ ( h, m, a, c, t ) = 0,

t ≥ 0,

(8)

and the definition in equation (4) is applied for ψ ( f , − p , a, c, t ) in equation (7). The next section considers the extention to the Caputo fractional derivative. 3. INITIALIZATION OF THE CAPUTO FRACTIONAL DERIVATIVE The Caputo fractional derivative was introduced8,9 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 by8: c

α

dtα f (t ) =

t

1 (t − τ )m −α −1 f m (τ ) dτ , Γ (m − α ) ∫a

(m − 1 < α < m).

(9) 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 0

Dtα f (t ) =

98

Caputo approach10. 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 inferred11,12 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 derivative, according to LH is given by Eq. (7) and will be used in the following examples, where for convenience, α is used in the place of q, i.e., α m p 0, is a positive integer, and as before, for terminal initialization, ψ h, m, a, c, t 0. Hereafter t c corresponds to t 0. Two cases will be considered below. 4.1 Simple Cases: # $ % &'( % $ ) We first consider the case when 0 1, *. +. , 1, then 0

Dtα f (t ) = 0 Dt1 { 0 Dt1(1−α ) f (t )} =

d { 0 Dt−(1−α ) f (t )} +ψ (h,1, a, 0, t ), dt

t > 0,

(10) Noting that the initialization for the integer order derivative is zero

d { 0 Dt−(1−α ) f (t ) +ψ ( f , −(1 − α ), a, 0, t )} + 0, dt

t > 0,

Substituting explicitly for the quantities in curly brackets in equation (11) yields t 0  d   d  1 1 α −α D f ( t ) = ( t − τ ) f ( τ ) d τ + ( t − τ ) − α f (τ ) d τ  ,    0 t ∫ ∫ dt  Γ (1 − α ) 0  dt  Γ (1 − α ) α  Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

(11)

(12)

99

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

Recasting the convolution integral by interchanging the arguments and carrying

out the differentiation of the integral using Leibnitz rule, yields

1 t âˆ’Îą 1 d Ď„ âˆ’Îą f ' (t − Ď„ ) dĎ„ + (t − Ď„ ) âˆ’Îą f (Ď„ ) dĎ„ , t > 0, f (0) + 0 Dt f (t ) = âˆŤ âˆŤ Γ(1 − Îą ) 0 Γ(1 − Îą ) Γ(1 − Îą ) dt a (13) t

Îą

o

Rewriting the argument of the convolution integral as ିŕ°ˆ ′ and using the definition of the Caputo derivative, equation (9) with 1 0 1, one can Îą c Îą 0 Dt f (t ) = 0 Dt f (t )

t âˆ’Îą f (0) d + Ďˆ ( f , −(1 − Îą ), Îą , 0, t ), Γ(1 − Îą ) dt

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

0

DtÎą f (t ) =

write the following expression relating the Caputo derivative to the initialized LH derivative for 0 1:

t >0

(14)

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

d d  − (2 âˆ’Îą ) f (t ) + Ďˆ ( f , −(2 − Îą ), Îą , 0, t ) + 0}  ,  { 0 dt dt  dt 

t>0

(15)

yields on substituting explicit expressions for the quantities in the curly brackets Îą 0 Dt f (t ) =

t  d 2 d d  1 1âˆ’Îą ( t − Ď„ ) f ( Ď„ ) d Ď„     + 2 Ďˆ ( f , −(2 − Îą ), Îą , 0, t ), dt  dt  Γ(2 − Îą ) âˆŤ0   dt

Recasting the convolution integral in equation (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 case 1 2 as [11]:

t1âˆ’Îą f ' (0) t âˆ’Îą f (0) + +Ďˆ ( f , −(2 − Îą ), Îą , 0, t ), 0 Dt f (t ) = d f (t ) + Γ(2 − Îą ) Γ(1 − Îą ) Îą

c Îą 0 t

(16)

t > 0.

(17)

The expressions in equation (14) and equation (17) can be generalized as shown below. 4.2 General Case:

Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

100

Generalizing to the case when 1 , we get Îą 0 Dt f (t ) = Îą 0 Dt f (t ) =

dm dm − ( m âˆ’Îą ) d f t ( ) + Ďˆ ( f , −(m − Îą ), Îą , 0, t ), 0 t dt m dt m

t > 0,

(18)

m −1 k âˆ’Îą 1 t f k (0 + ) d m m âˆ’Îą −1 m ( t − Ď„ ) f ( Ď„ ) d Ď„ + + m Ďˆ ( f , − (m − Îą ), Îą , 0, t ), ∑ Γ (m − Îą ) âˆŤ0 dt k = 0 Γ ( k − Îą + 1) t

(19) m −1

0

k âˆ’Îą

k

+

m

t f (0 ) d + m Ďˆ ( f , −(m − Îą ), Îą , 0, t ), t > 0, m − 1 < Îą < m. dt k = 0 Γ ( k − Îą + 1)

DtÎą f (t ) = 0c dtÎą f (t ) + ∑

(20) Equation (20) expresses the LH order derivative 0 DtÎą f (t ) in terms of the order Caputo derivative and additional terms. The additional terms consist of a polynomial in with coefficients given by the values of the function and its integer-order derivatives áˆşŕŻžáˆť , all evaluated at 0ା and the LH initialization for a fractional derivative under the assumption of terminal initialization. The polynomial contains a term ( 0 term), which is singular at 0 for 0. The details of the derivation can be found in ref.11. For the range 0 1, equation (20) simplifies to the equation (14), and for the range 1 2, it reduces to equation (17). REFERENCES 1. Lorenzo CF, Hartley TT. Initialized fractional calculus. Int. J. Appl. Math. 3 (3): 249-265 (2000). 2. Lorenzo CF, Hartley TT. Initialization in Fractional Order Systems, in: Proceedings of the European Control Conference, Porto, Portugal pp. 14711476 (2001).

3. Oldham KB, Spanier J. The Fractional Calculus- Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974). 4. Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York (1993). 5. Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applicatios, Gordon and Breach Science Publishers, Philadelphia, PA (1993). 6. Podlubny L. Fractional Differential Equations. Academic Press, SanDiego, CA (1999). 7. Lorenzo CF, Hartley TT. Initialization, Conceptualization, and Application in the Generalized Fractional Calculus, NASA TM (1998). 8. Caputo M. Elasticita e Dissipazione. Zanichelli, Bologna (1969). 9. Caputo M. Linear Model of dissipation whose Q is almost frequency independent-II. Geophys, J. R. Astron. Soc. 13:529539 (1967). 10. Gorenflo R, Mainardi F. Fractional calculus, integral and differential equations of fractional order in: Fractals

Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

101

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.4 (2), 96-101 (2013)

and Fractional Calculus in Continuum Mechanics. Carpenteri, A, Mainardi, F (eds.), Springer, New York (1997). 11. Achar BNN, Lorenzo CF, Hartley T. Initialization and the Caputo Fractional Derivative, NASA TM (2003).

12. Achar BN, Narahari, Lorenzo CF, Hartley T. Initialization Issues of the Caputo Fractional Derivative, Proceedings of IDETC/CIE, 2428, Long Beach CA. DETC 2005. pp. 18 (2005).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 2, 30 April, 2013 Pages (80-134)

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.