118 6 Initialized Differintegrals and Generalized Calculus
f(t) + + d−p
+ h(t) dm
+
q=(m−p) t>c
cDt qf(t) ψ( f, –p, a.c.t) ψ(h, m, a, c, t)
Fig. 6.3 Initialization of fractional derivative
In case of terminal charging, the fractional integral initialization part
ψ( f,−p,a,c,t)= 1 Γ( p)
c a
(t−τ)p−1f (τ)dτ for t >c (6.20)
Figure 6.3 demonstrates the initialization concept for fractional derivative
6.6 Initializing Fractional Differintegrals
i.e., un-initialized differintegral, as per standard notation, we write the same as:
ψ( f,q,a,c,t)=adtqf (t)−cdtqf (t). In this, substituting the GL series, we obtain the following:
ψ( f,q,a,c,t)= lim
N1→∞
⎧⎪
⎨
⎪⎩ t−a
N1
−q
Γ(−q)
N1−1 j=0
Γ( j−q)
Γ( j+1)f t−jt−a N1
⎫
⎪⎬
⎪⎭
− lim
N2→∞
⎧⎪
⎨
⎪⎩ t−c
N2
−q
Γ(−q)
N2−1 j=0
Γ( j−q)
Γ( j+1) f t− jt−c N2
⎫
⎪⎬
⎪⎭(6.24)
For all t >c and f (t)=0 for t <a.
After considerable manipulations, by adjusting delay element as equal, that is N2 = ((t − c)/(t − a))N1, and adjusting with ΔT = (t − a)/N1 and N3 =((c−a)/(t−a))N1
ψ( f,q,a,c,t)= lim
N1→∞
ΔT−q
Γ(−q) N3−1
j=0
Γ(N1−1−q−j )
Γ(N1−j ) f (t−[N1−1− j ]ΔT ) 7
(6.25)
6.7 Properties and Criteria for Generalized Differintegrals
One of the fundamental problems of fractional calculus is the requirement that the function and its derivatives be identically equal to zero at the start of initialization (i.e. start of differintegration process) at time t =c. This needed to assure compo- sition or index law holds implying thatcDvt cDtuf (t)=cDut cDtvf (t)=cDtu+vf (t).
It is difficult in engineering sciences to always require that the functions and its derivatives be at zero (rest) at initialization instants. This fundamentally implies that “there can be no initialization or composition is lost”. Thus, it is not in general true that f −dtd−Q−Q
dQf dtQ =0.
Therefore, while solving a fractional differential equation of the form ddtQQf =F, additional terms must be added, like
f − d−Q dt−Q
dQf
dtQ =C1tQ−1+C2tQ−2+. . .CmtQ−m, to achieve the most general solution:
f = d−QF
dt−Q +C1tQ−1+C2tQ−2+. . .+CmtQ−m.
These issues described says that the index law or the composition law is inadequate.
120 6 Initialized Differintegrals and Generalized Calculus Minimal set criteria have been thought fit to be applied for fractional (or gener- alized) calculus. They are listed as follows and are called Ross (1974) criteria:
i. If f (z) is the analytic function of the complex variable z, the differintegral
cDvz f (z) is the analytic function of z andv.
ii. The operatorcDvxf (x ) must produce the same result of differentiation, whenv is a positive integer.
iii. Ifvis a negative integer (sayv= −n), thencD−xnf (x ) must produce the same result of n-fold integration of function f (x ), andcD−xnf (x ) must vanish along with f(1), f(2), . . . , f(n−1), all the (n−1) derivatives at x=c.
iv. “Zero” operation leaves the function unchanged.
cD0xf (x )= f (x )
v. Linearity of the fractional (generalized) differintegral operator:
cD−xq[a f (x )+bg(x )]=acDx−qf (x )+bcD−xqg(x ) vi. The law of exponents for arbitrary order holds
cD−xucD−vx f (x )=cDx−u−vf (x )=cD−v−x u f (x ) The above notations are used by Ross.
It should be noted that there is a minor conflict contained in these criteria. Also a clear notation explanation should be given ascDqxf (x ) in the above criteria is an un-initialized differintegral. It is correct as the function itself starts at c,and before that, the same is zero. So at t =c,cDqx f (x )=cdqxf (x ). The criterias (ii) and (iii) call for backward compatibility and the criteria (vi) calls for index law to be holding vis-`a-vis integer order calculus.
The fundamental theorem of integer order calculus violates this “zero law” as:
d−mdmf (x ) = d0f (x ) = f (x ), for all f (x ) and for all m (integer). The fun- damental theorem states that
t c
f(t) = f (t)− f (c), and can be thus observed that the reversal of differentiation and integration differs from f (t) by f (c),that is by initialization (constant in integer order calculus). This failure in backward compatibility and index law is handled in the integer order calculus by constant of integration and by complimentary function for solution of differential equa- tions (in ad hoc manner). The law of exponents (index law) is demonstrated in Fig. 6.4.
The discussion in all the differintegrations is limited to the real domain. Under the condition of terminal charging of the uth andvth differintegrations,
cDt cu Dtvf (t)=cDt cv Dut f (t)=cDtu+vf (t) for t>0
During normal functioning
During initialization (terminal charging)
f(t) h(t)
+
adt−v
adt−u
t > a f(t)
aDt−u aDt−v
+ + +
f(t) cdt−v h(t) +cdt−u cDt−ut > c f(t)
cDt−v
ψ(h,–v,a,c,t) ψ(h,–u,a,c,t)
Fig. 6.4 Demonstration of index law
under the following conditions:
a. u <0,v <0 for continuous f (t)
b. u >0,v >0. For f (t) is m times differentiable.aDmt f (t) exists and is non-zero continuous function of t for t>a, where m is an integer larger than integer part [u] or [v].
c. u <0,v >0 same as (b)
6.7.1 Terminal Charging
Under the conditions of terminal charging, the above properties and criterias holds; this provides credibility to the initialized fractional (generalized) calcu- lus. Some conditions are however imposed, say on the linearity of fractional integrals. cD−vt (b f (t)+kg(t)) = bcD−vt f (t)+kcD−vt g(t),(t > c) holds only ifψ(b f +kg,−v,a,c,t)=bψ( f,−v,a,c,t)+kψ(g,−v,a,c,t).
Relative to the criteria of backward compatibility with the integer order calculus, the addition of the initialized function is clearly a generalization relative to inte- ger order calculus. In a strict senseψ(t) = 0 violates the criteria (ii); however, we are looking for generalization of integer order calculus, and it is clear that this generalization (i.e. addition of initialization function) will be very useful in many applications.
Relative to the criteria of zero-order property holds for terminal charging.
Relative to linearity holds for the terminal charging subject to the above said rule.
Relative to composition rule, the above (a) (b) and (c) should follow.
It is noted that f(k)(c)=0 for all k, no longer exists. This constraint has effec- tively been contained (shifted to) the requirement f (t)=0 for all t ≤a. This allows initialization of fractional differential equations.
In summary terminal charging case is backward compatible with integer order calculus and satisfies the applicable criteria established by Ross.
122 6 Initialized Differintegrals and Generalized Calculus
6.7.2 Side Charging
The case for side charging is less definitive. Criteria for backward compatibility is the same as the terminal charging case. Relative to zero property the condition ψ( f,−p,a,c,t)= Γ1( p)
c a
(t−τ)p−1f (τ)dτ =0=ψ(h,m,a,c,t) is required for side charging sinceψis arbitrary. When these conditions are not met, the zero-order operation on f (t) will return f (t)+g(t), i.e. the original function with extra time function ((g(t)), the effect of initialization. Relative to linearity, the side charging demands additional requirements about initialization.
These are not so much of an issue as it appears for practical applications. In the solution of fractional differential equations,ψ(t) will be chosen in the much the same manner as initialization are currently chosen for ordinary differential equations in integer order. This will imply the nature of f (t) from a to c. The new aspect is that to achieve a particular initialization for a given composition now requires attention to the initialization of the composing elements.