An Ordinary Fractional Differential Equation
An Ordinary Fractional Differential Equation
Fractional calculus generalizes ordinary calculus by letting differentiation and integration be of arbitrary order.
The definition of the fractional derivative is
-α
d
d
-α
x
1
Γ(a)
x
∫
0
α-1
(x-t)
for and , and
α>0
x>0
α
d
d
α
x
m
d
d
m
x
m-α
d
d
m-α
x
where is any postive integer greater than .
m
α
This Demonstration solves numerically the following ordinary fractional differential equation:
(1) y=A(1-y),
α
d
d
α
x
where ,
A∈
+
R
0<α<1
(2) .
y(0)=β
Here and are parameters, is a dependent variable, and is an independent variable.
A
β
y
x
The discretization of equations (1) and (2) are
m
∑
j=0
α
C
j
y
m-1
y
m
y
0
with =, where is the gamma function.
α
C
j
1
α
h
Γ(j-α)
Γ(-α)Γ(j+1)
Γ