Fractional derivatives fractional integrals and fractional differential equations in matlab
Answers
Answer:
Step-by-step explanation:
2. Fractional calculus fundamentals
2.1 Special functions
Here we should mention the most important function used in fractional calculus - Euler’s
gamma function, which is defined as
Γ(n) = ∞
0
t
n−1
e
−t
dt. (1)
This function is a generalization of the factorial in the following form:
Γ(n)=(n − 1)! (2)
This gamma function is directly implemented in the Matlab with syntax []=gamma().
Another function, which plays a very important role in the fractional calculus, was in fact
introduced in 1953. It is a two-parameter function of the Mittag-Leffler type defined as
(Podlubny, 1999):
Eα,β(z) =
∞
∑
k=0
z
k
Γ(αk + β)
, (α > 0, β > 0). (3)
The Mittag-Leffler function (3) can be expressed in the integral representation as
Eα,β
(z) = 1
2πi
C
t
α−β
e
t
t
α − z
dt, (4)
the contour C starts and ends at −∞ and circles around the singularities and branch points of
the integrand.
There are some relationships (given e.g. in (Djrbashian, 1993; Podlubny, 1999)):
E1,1(z) = e
z
, E1/2,1(
√
z) = 2
√
π
e
−z
erfc(−
√
z),
E2,1(z) = cosh(
√
z), E2,1(−z
2
) = cos(z), E1,2(z) = e
z − 1
z
.
For β = 1 we obtain the Mittag-Leffler function in one parameter (Podlubny, 1999):
Eα,1(z) =
∞
∑
k=0
z
k
Γ(αk + 1)
≡ Eα(z). (5)
For the numerical evaluation of the Mittag-Leffler function (3) with the default accuracy set
to 10−6
the Matlab routine [e]=mlf(alf,bet,c,fi), written by Podlubny and Kacenak
(2005) can be used. Another Matlab function f=gml_fun(a,b,c,x,eps0) for a generalized
Mittag-Leffler function was created by YangQuan Chen (Chen, 2008).
In Fig. 1(a) and Fig. 1(b) are plotted the well-known functions (e
z and cos(z)) computed via
the Matlab routine []=mlf() created for the evaluation of the Mittag-Leffler function.
Another important function Ek
(t, λ; μ, ν) of the Mittag-Leffler type was introduced by
(Podlubny, 1999). The function is defined by
Ek
(t, λ; μ, ν) = t
μk+ν−1E
(k)
μ,ν(λt
μ
), (k = 0, 1, 2, . . .), (