An Ordinary Fractional Differential Equation

y

0

0.5

A

2

α

0.5

t	α = 1	α = 0.5
0	0.5	0.5
0.2	0.66484	0.56570
0.4	0.77534	0.64628
0.6	0.84940	0.69300
0.8	0.89905	0.72469
1.0	0.93233	0.74808
1.2	0.95464	0.76630
1.4	0.96960	0.78101
1.6	0.97962	0.79323
1.8	0.98634	0.80360
2.0	0.99084	0.81253

Fractional calculus generalizes ordinary calculus by letting differentiation and integration be of arbitrary order.

The definition of the fractional derivative is

-α

d

y(x)

d

-α

x

=

1

Γ(a)

x

∫

0

α-1

(x-t)

y(t)dt

,

for

α>0

and

x>0

, and

α

d

y(x)

d

α

x

=

m

d

m

x

m-α

d

y(x)

d

m-α

x

,

where

m

is any postive integer greater than

α

.

This Demonstration solves numerically the following ordinary fractional differential equation:

(1)

α

d

y

d

α

x

=A(1-y)

,

where

A∈

+

R

,

0<α<1

,

(2)

y(0)=β

.

Here

A

and

β

are parameters,

y

is a dependent variable, and

x

is an independent variable.

The discretization of equations (1) and (2) are

m

∑

j=0

α

C

j

y

m-1

-A(1-

y

m

)=0

,

y

0

=β

,

with

α

C

j

=

1

α

h

Γ(j-α)

Γ(-α)Γ(j+1)

, where

Γ

is the gamma function.