Working Material

Numerical method à la Crank-Nicolson for fractional diffusion equations

Santos Bravo Yuste

Author

Santos Bravo Yuste

Title

Numerical method à la Crank-Nicolson for fractional diffusion equations

Description

Solution of fractional diffusion equations by means of a generalization of the Crank-Nicolson method

Category

Working Material

Keywords

fractional diffusion equation, subdiffusion, numerical method, Crank-Nicolson method, finite difference method

URL

http://www.notebookarchive.org/2019-04-7ggkcgi/

DOI

https://notebookarchive.org/2019-04-7ggkcgi

Date Added

2019-04-16

Date Last Modified

2019-04-16

File Size

235.92 kilobytes

Supplements

Rights

CC BY 4.0

Cite this as: Santos Bravo Yuste, "Numerical method à la Crank-Nicolson for fractional diffusion equations" from the Notebook Archive (2019), https://notebookarchive.org/2019-04-7ggkcgi

Numerical method à la Crank-Nicolson for fractional diffusion equations

The equation


Numerical method


Example 1
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=x(1-x),u(0,t)=u(1,t)=0



Example 2
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0



Example 3 (normal diffusion)
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0
for γ=1, i.e.,
∂u
∂t
=K
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0


References


Numerical method à la Crank-Nicolson for fractional diffusion equations

The equation

Numerical method

Example 1 ∂u∂t=K1-γ0Dt2∂u∂2x,u(x,t=0)=x(1-x),u(0,t)=u(1,t)=0 

Example 2 ∂u∂t=K1-γ0Dt2∂u∂2x,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0 

Example 3 (normal diffusion) ∂u∂t=K1-γ0Dt2∂u∂2x,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0 for γ=1, i.e., ∂u∂t=K2∂u∂2x,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0

References

The equation


Numerical method


Example 1
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=x(1-x),u(0,t)=u(1,t)=0



Example 2
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0



Example 3 (normal diffusion)
∂u
∂t
=K
1-γ
0
D
t
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0
for γ=1, i.e.,
∂u
∂t
=K
2
∂
u
∂
2
x
,u(x,t=0)=δ(x),u(-∞,t)=u(∞,t)=0


References
