On the Numerical Solution of the 3-Dimensional Fractional Diffusion Equation in the shifted Grunwald estimate form
I. I. Gorial
Department of Mathematics, Address Ibn Al-Haithem College Education, Baghdad University, Email [email protected]
ICM 2012, 11-14 March, Al Ain
u(x,y,zR,t) =3(x,y,t), for x0 x  xR , y0 y yR and 0 t  Τ
ABSTRACT
In this paper, a numerical solution of the 3-dimensional fractional
diffusion equation has been presented. The algorithm for the numerical solution for this equation is based on implicit finite difference
method. The consistency, unconditional stability, and convergence
of the fractional order numerical method are described.
The numerical method has been applied to solve a practical numerical example and the results have been compared with exact
solution. The results were presented in tables using the MathCAD
12 software package when it is needed. The implicit finite difference
method appeared to be effective and reliable in solving the 3dimensional fractional diffusion equation. Keywords: Fractional
derivative, implicit Euler method, fractional diffusion equation,
stability, convergence.
The   u x  ,   u y  ,   u z  fractional derivatives by the
shifted Grunwald estimate formulae [ 2, 7]:
  u ( x, y , z , t )
1


x
(x) 
i 1
 g
k 0
  u ( x, y , z , t )
1

y 
(y ) 
  u ( x, y , z , t )
1

z 
(z ) 
1 INTRODUCTION
Various fields of science and engineering deal with dynamical
systems that can be described by fractional partial differential
equations, for example, systems biology, chemistry and biochemistry applications due to anomalous diffusion effects in constrained
environments. However, effective numerical methods and numerical analysis for fractional partial differential equations are still in
their infancy, [1, 2, 3, 4, 5, 6, 7, 11, 13,14,15].
In this work, we consider the 3-dimensional fractional Diffusion
equation:
u ( x, y, z, t )
  u ( x, y, z, t )
  u ( x, y, z, t )
 a ( x, y , z )
 b( x, y, z )


t
x
y 

c ( x, y , z )
where a , b , c and  are known functions of x, y and z,
and 1 is a known function of y, z and t, 2 is a known function of
x,z and t, 3 is a known function of x,y and t.  ,  and  are
given fractional number. q is a known function of x, y ,z and t.
 u ( x, y , z , t )
 q ( x, y , z , t )
z 
(1)
subject to the initial condition
u(x,y,z,0)=  (x,y,z), for x0 x  xR, y0 y yR and z0 zzR (2)
and the boundary conditions
u (x0,y,z,t)= 0, for y0 y yR , z0  z  zR and 0 t  Τ
u(x,y0,z,t)= 0, for x0x xR, z0  z  zR and 0 t  Τ
u(x,y,z0,t)= 0, for x0 x xR, y0 y yR and 0 t Τ
(3)
u(xR,y,z,t) = 1(y,z,t), for y0 y yR , z0 z  zR and 0 t  Τ
u(x,yR,z,t) =2(x,z,t), for x0 x  xR , z0 z  zR and 0 t  Τ
,k
u isk11, j , f  O(x)
j 1
 g
k 0
,k
f 1
 g
k 0
,k
u is,j1 k 1, f  O(y )
(4)
u is,j1, f  k 1  O(z )
In this paper, an implicit finite difference approximation for the
3-dimensional fractional diffusion equation is presented. The unconditional stability and convergence of the implicit finite difference approximation are analyzed and finally, we will present example to show the efficiency of our numerical method .
2 FINITE DIFFERENCE METHOD FOR SOLVING
THE 3-DIMENSIONAL FRACTIONAL DIFFUSION
EQUATION
In this section, we use the finite difference method to solve the
initial and boundary value problem 3-dimensional fractional diffusion equation (1)-(3).
The finite difference method starts by dividing the x-interval
[x0, xR] into n subintervals to get the grid points xi = x0 + ix,
where x  xR  x0  n and i = 0,1,…,n. and we divide the
y-interval [y0, yR] into m subintervals to get the grid points
yj = y0+jy, where y  y R  y 0 m and j=0,1,…,m. also we


divide the z-interval [z0, zR] into p subintervals to get the grid
points zg =z0+fz, where z  z R  z 0  p and f = 0,1,…,p.
Also, the t-interval [0,T ] is divided into M subintervals to get
the grid points ts = st,
s = 0,…, M, where t  T M .
Now, we evaluate eq. (1) at (x i, y j, ,zf ,t
implicit Euler method to get
s+1)
and we use the
1
I. I. Gorial
ui0, j , f  i , j , f , i=0,…, n , j=0,…, m and f = 0,1,…,p.
u ( xi , y j , z f , ts 1 )  u ( xi , y j , z f , ts )
t
  u ( xi , y j , z f , t s 1 )
b ( xi , y j , z f )
y

 a( xi , y j , z f )
 c ( xi , y j , z f )
 u ( xi , y j , z f , ts 1 )

u0s, j , f  0, j=0,…, m , f =0,1,…,p and s=1,…,M

uis,0, f  0,
i=0, … , n , f =0,1,…,p and s=1,…,M
(5)
uis, j ,0  0,
i=0, …, n , j=0,…, m and s=1,…,M
x
  u ( xi , y j , z f , t s 1 )
 z

q( xi , y j , z f , t s 1 )
Use fractional derivative of the shifted Grunwald estimate to the
,  ,  - the fractional derivative eq.(4), to reduce eq.(5) as in the
following form:
u is,j1, f  u is, j , f
i 1
1
 a ( xi , y j , z f )
x 
t
 g
k 0
,k
j 1
1
b ( xi , y j , z f )
y 
 g
1
z 
 g
c ( xi , y j , z f )
u isk11, j , f 
k 0
f 1
k 0
u
,k
,k
s 1
i , j  k 1, f

u is,j1, f  k 1 
q( xi , y j , z f , t s 1 )
uis,j1,f  uis, j , f

t
ai , j , f
x
v

i 1
 g ,k uisk11, j , f 
k 0
ci , j , f
z 
f 1
 g
k 0
,k
bi , j , f
y

j 1
 g
k 0
,k
uis,j1k 1, f 
u Rs , j , f   sj , f , j=0, …, m, f =0,1,…,p and s=1,…,M
uis, R, f   is, f , i=0, …, n, f =0,1,…,p and s=1,…,M
uis, j ,R   is, j , i=0, …, n, j=0,…, m and s=1,…,M
where
i , j , f   ( xi , y j , z f ),  sj , f   ( y j , z f , ts ) and
 is, j   ( xi , y j , ts )
3 STABILITY OF THE FINITE DIFFERENCE 3DIMENSIONAL
FRACTIONAL
DIFFUSION
EQUATION
We consider the following fractional partial difference operator:
 , x uis,j1, f  ai , j , f
uis,j1, f k 1  qis,j1,f
i  1,..., n  1 , j  1,..., m  1, f  1,..., p  1 , s  0,..., M
(6)
u
  , y uis,j1, f  bi , j , f
 u( xi , y j , z f , t s ), ai , j , f  a( xi , y j , z f ), bi , j , f  b( xi , y j , z f ),
 , z uis,j1, f  ci , j , f
ci , j , f  c( xi , y j , z f ), qis,j1,f  q( xi , y j , z f , t s 1 ),
 (  1)  (  k  1)
g , k  (1) k
k!
g  ,k  ( 1) k
g  , k  (1) k
, k=0,1,2,…
, k=0,1,2,…
The resulting equation can be implicitly solved for
uis,j1, f  ai , j , f
ci , j , f
t
z 
t
x
i 1
 g
k 0
f 1
 g
k 0
,k
,k
uisk11, j , f  bi , j , f
t
y 
s 1
i, j, f
u
j 1
 g
k 0
,k
uisk11, j , f
t
y 
t
z 
j 1
 g
k 0
,k
f 1
 g
k 0
,k
u is,j1 k 1, f
uis,j1, f  k 1
(1  t , x  t , y  t , z )uis,j1,f  uis, j , f  qis,j1,f t
to give

-order
uis,j1k 1, f 
uis,j1, f k 1  uis, j , f  tqis,j1, f
(8)
eq.(8) may be written in form
(1  t , x )(1  t , y )(1  t , z )uis,j1,f  uis, j , f  tqis,j1,f (9)
Then
(7)
ETOU
Also form the initial condition and boundary conditions one can
get
2
,k
With these definitions, the implicit difference scheme (7) may be
written in the following compact form:
k!
k!
k 0
are O(y ) and O (z ) approximation of the  and
Grunwald shifted fractional derivatives term, respectively.
 (   1)(   k  1) k=0,1,2,…and
,
 (  1)  (  k  1)
i 1
 g
which is an O(x) approximation to the  th fractional derivative. Similarly, the following fractional partial difference operators
are defined.
Where
s
i, j , f
t
x 
s 1
U  R
s
s 1
Where
E  (1  t ,x ),
(10)
On the Numerical Solution of the 3-Dimensional Fractional
T  (1  t , y ),
U ks1,1k23  [u1s,k11,k32 , u 2s ,k11 ,3k2 ,  , u ns 11,k31 ,k2 ]T ,
O  (1  t ,z ),
U ks1 ,k2  t Qks1,1k2  [u1s,k1 ,k2  q1s,k11,k2 t , u 2s,k1 ,k2  q2s,k11 ,k2 t ,, u ns1,k1 ,k2  qns11,k1 ,k2 t ]T
and
Ci , j
Therefore the resulting matrix entries
U  [u1s,1,1 , u2s,1,1 , , uns1,1,1 , u1s, 2,1 , u2s, 2,1 , , uns1, 2,1 , , u1s,m1,1 , u2s,m1,1 , , uns1,m1,1
s
j  1,, n  1 are defined by
u1s,1, 2 , u2s,1,2 ,, uns1,1,2 , u1s, 2, 2 , u2s,2, 2 ,, uns1,2,2 ,, u1s,m1, 2 , u2s,m1, 2 ,, uns1,m1,2

Ci, j
u1s,1, p1 , u2s,1, p1 ,  , uns1,1, p1 , u1s, 2, p1 , u 2s, 2, p 1 ,  , uns1, 2, p 1 ,  , u1s,m1, p1 ,
u
s
2 ,m1, p 1
, , u
s
T
n 1,m1, p 1
and the vector R
]
s 1
is the forcing term .
Hence, we obtain the fractional implicit scheme at time
(1  t , x )uis,j1,f3  uis, j , f  tqis,j1,f
(1  t , y )uis,j ,2f 3  uis,j1,f3
1   i , k1 , k 2 g  ,1
 

i , k1 , k 2 g  , 2

   i , k1 , k 2 g  , 0

  i , k1 , k 2 g  , j 1
ji
for
for j  i  1
for j  i  1
for j  i  1
where the coefficients
t s 1 :
 i ,k
1 ,k 2
(11)
(12)
and
(1  t , z )uis,j1,f  uis,j ,2f 3
for i  1,2, , n  1 and
(13)
Thus, we require three steps to solve the third-dimensional fractional diffusion equation in one time step.
t
x
 ai ,k1 ,k2
To illustrate this matrix pattern, we list the corresponding equations for the rows i =1, 2 and n-1:
 1,k1 ,k2 g ,2 u0s,k11 /,k32  (1  1,k1 ,k2 g ,1 )u1s,k11,/k32  1,k1 ,k2 g ,0 u2s,k11 /,k32  u1s,k1 ,k2  tq1s,k11,k2
 2, k1 , k 2 g ,3u0s,k11 ,/k32  2, k1 , k 2 g , 2u1s,k11,/k32  (1  2, k1 , k 2 g ,1 )u2s,k11 ,/k32 
2, k , k g ,0u3s,k1 ,/k3  u2s, k , k  tq2s,k1 , k
1
2
1
2
1
2
1
2
Firstly: if (yj, zf ) is fixed we will obtain an intermediate solution
uis,j1,f3
from (11).
Second: if (xi, zf ) is fixed we will obtain an intermediate solution u is,j ,2f 3 from (12).
Third: if (xi, yj ) is fixed we will obtain an intermediate solution
from (13) using information compiled during Second step.
  n 1, k1 , k 2 g , nu0s,k11/,3k 2     n 1, k1 , k 2 g , 2uns 12/,3k1 , k 2  (1 
 n 1, k , k g ,1uns 11,/k3 , k   n 1, k , k g ,0uns,k1/,3k  uns 1, k , k  tqns 11, k , k
1
2
1
1
2
1
2
1
2
1
2
According to the Greshgorin theorem [9], the eigenvalues of the
matrix C lie in the union of the circles centered at
ri 
Now, we must prove each one-dimensional implicit system defined
by the linear difference eqs. (11), (12) and (13) is unconditionally
stable for all 1 <  ,  ,  < 2.
Theorem 3.1. The implicit system defined by the linear difference
eqs.(11),(12) and (13) is unconditionally stable for all
1 <  ,  ,  < 2.
2
ci ,i with radius
n
c
l 0
l i
i ,l
.
Here we have
ci ,i  1   i ,k1 ,k2 g  ,1  1   i ,k1 ,k2 
and
Proof:
At each grid point yk1, for k1  1,, m  1 , and zk2 , for
k 2  1,, p  1 , the system of equation defined by (11). This system
of
equations
may
be
written
as
n
i 1
l 0
l i
l 0
l i
ri   ci ,l  i ,k1 ,k2  g ,i l 1 i ,k1 ,k2   i ,k1 ,k2
Ck1 ,k2 U ks1,1k23  U ks1 ,k2  t Qks1,1k2
With strict inequality holding true when  is not an integer. This
implies that the eigenvalue of the matrix C k k has a real-part larger
where
than 1, and therefore a magnitude larger than 1. Hence the spectral
1, 2
3
I. I. Gorial
1
radius of the matrix C k ,k is less than 1.This proves that the meth1
2
od is stable [10].
Ai , j
And with the same method above; the results of equations system, defined by (11), can be defined as:
1   k1 , k 2 , f g  ,1
 

k1 , k 2 , f g  , 2

   k1 , k 2 , f g  , 0

  k1 , k 2 , f g  , j 1
ji
for
for j  i  1
for j  i  1
for j  i  1
S k1 ,k2 U ks1,k22/ 3  U ks1,1k2/ 3 ,
where the coefficients
where
 k ,k
U ks1,1k 2  [u ks1,11,k2 , u ks1,12,k 2 ,, u ks1,1m 1,k 2 ]T ,
1
U ks1,1k 2/ 3  [u ks1,11,/k32 , u ks1,12/, k32 ,, u ks1,1m/31,k 2 ]T ,
and S k ,k is the matrix of coefficients, and is the sum of a lower
1 2
triangular matrix and a super diagonal matrix at the grid point xk1
for k1  1,, n  1 and zk2 for k 2  1,, p  1 . Therefore the resulting matrix entries S i , j for i  1,2,  , m  1 and j  1,  , m  1 are
defined by
1   k1 , j , k 2 g  ,1
 

k1 , j , k 2 g  , 2

   k1 , j , k 2 g  , 0

  k1 , j , k 2 g  , j 1
S i, j
1
2
1
1
k
is less than 1, this proves that
s2 / 3
k1 , k 2
tral radius of the matrix E
Since
U
U
s2 / 3
k1 , k 2
Ak1 ,k2
 [u
s 1
k1 , k 2 ,1
 [u
,u
s 1
k1 , k 2 , 2
s2 / 3
k1 , k 2 ,1
,u
is xk1 for
,
is less than 1, and hence the spec-
is less than 1.
spectral radius of the matrix
and
O
are in the un-
1
T
and
in
U
n
O
1
are less than 1.
 0 in U 0 results in an error
given by
 n  ( ETO) n  0
s 1
T
k1 , k 2 , m 1
, , u
T
ion of the Greschgorin disks for the matrices S k and O . Again
the argument of Theorem 3.2 may be applied to show that the
 0 at time t n
] ,
When the matrices E ,
s2 / 3
T
k1 , k 2 , m 1
k1  1,, n  1
] ,
and
O
yk2 for
k 2  1,, m  1 . Therefore the resulting matrix entries Ai , j for i  1,2,, m  1 and j  1,, m  1 are defined by
1
T and O commute,
E
i.e.
1
,
T
1
and
commute, we have
 n  E n T n O n  0
As the spectral radius of each matrix E
than one, it follows that
n  ,
4
1
1
Note that eq.(9) implies that an error
, , u
s2 / 3
k1 , k 2 , 2
E
Similarly, the eigenvalues of the matrix
where
s 1
k1 , k 2
commute.
value of the inverse matrix
Now, resulting the system of equations defined by (12) is then
defined by:
U
O
E ,T
theorem 3.1, it follows that every eigenvalue of the matrix E has
a real-part larger than 1. Therefore, the magnitude of every eigen-
the method is stable [10].
Ak1 , k2 U
conditionally stable for 1<  ,  ,  < 2 if the matrices
E k are in the union of the
Greschgorin disks for the matrices C k . Applying the argument of
part larger than 1, and therefore a magnitude larger than 1. Hence
s 1 / 3
k1 , k 2
Theorem 3.2. The implicit-Euler method, defined by (9), is un-
The eigenvalues of the matrix
for j  i  1
So, and in the same way, eigenvalue of the matrix S k has a realthe spectral radius of the matrix S
therefore a magnitude larger than 1. Hence the spectral radius of
the matrix Ak1 is less than 1, and hence this system is also uncon-
Proof.
for j  i  1
t
y 
2
t
z 
 ck1 ,k2 , f
ji
for j  i  1
where the coefficients
 k , j ,k  bk , j ,k
f
So, eigenvalue of the matrix Ak has a real-part larger than 1, and
and
for
2,
E
n
1
 0, T
,
n
T
1
and
0,
O
O
n
1
is less
 0 , as
On the Numerical Solution of the 3-Dimensional Fractional
where 0 denotes the zero matrix . Therefore, the stability of the
implicit Euler method follows.
c ( x, y , z )
1..5u ( x, y, z, t )
 q( x, y, z, t )
z1..5
With the coefficient function:
4 CONSISTENCY AND CONVERGENT OF THE
FINITE
DIFFERENCE
3-DIMENSIONAL
FRACTIONAL DIFFUSION EQUATION
To obtain the consistency of the 3-dimensional fractional diffusion equation, note that the time difference operator in (8) has alocal truncation error of order O ( t ), and the three space difference
operators in (8) have local truncation errors of orders
O(x), O(y ) and O ( z ) respectively. Similar to Lemma 2.1 in
paper of Meerschaert et al., (2006), below we can in our paper, we
have the results:
  
f ( x, y, z )   x  y f ( x, y, z )  (x  y )
x  y 
  
f ( x, y, z )   x  z f ( x, y, z )  (x  z )
x z 
  
f ( x, y, z )   y  z f ( x, y, z )  (y  z )
y  z 
then
    
f ( x, y, z )   x  y  z f ( x, y, z )  (x  y  z )
x y  z 
a ( x, y , z ) 
(1.4) x 8 (1  x) 7 ,
2
b( x, y, z )  (0.2) y 8 (1  y) 7 ,
c( x, y, z )  (2.5) z 7 (1  z ) 7 ,
and the source function:
q( x, y, z, t )  x 7.4 (1  x) 7 yz 3e 3t  x 2 y 7.2 (1  y) 7 z 3e 3t  x1.5 y(1  y) 7 z 7 e 3t
 x 7.4 (1  x) 7 yz 3 e 3t  x 2 y 7.2 (1  y) 7 z 3 e 3t  x1.5 y(1  y) 7 z 7 e 3t  3x 2 yz 3 e 3t
subject to the initial condition
u (x,y,z,0) = x2yz3, 0  x  1, 0 < y < 1, 0 < z < 1
and the boundary conditions
u (0,y,z,t) = 0, 0 < y < 1, 0 < z < 1, 0  t  0.025
u (x,0,z,t) = 0, 0 < x < 1, 0 < z < 1, 0  t  0.025
u (x,,y,0,t) = 0, 0 < x < 1, 0 < y < 1, 0  t  0.025
u (1,y,z,t) = yz3e-3t, 0 < y < 1, 0 < z < 1, 0  t  0.025
u (x,1,z,t) = x2z3e-3t, 0 < x < 1, 0 < z < 1, 0  t  0.025
u (x,,y,1,t) = x2ye-3t, 0 < x < 1, 0 < y < 1, 0  t  0.025
Note that the exact
u(x,y,z,t) = x2yz3e-3t.
solution
to
this
problem
is:
Table1 and 2 show the numerical solution using the implicit finite difference approximation. From table 1 and 2, it can be seen that that good
agreement between the numerical solution and exact solution.
Which leads to the 3-dimensional fractional diffusion equation
with order O(t )  O(x)  O(y )  O(z ) .
We show above that implicit Euler method is consistent and unconditionally stable, then by Laxs equivalence theorem, [12], it
convergence at the rate O(x  y  z  t ) .
Table 1: The numerical solution of example using the finite difference
method (x  0.2, y  0.2, z  0.2, t  0.0125)
x = y =z
t
Numerical Solution Exact Solution
|uex -uapprox.|
0.2
0.0125 6.169E-5
0.61644E -4
4.55573 E -8
0.4
0.0125 3.949 E-3
0.39452E -2
3.75567 E -6
0.6
0.0125 0.045
0.44939 E -1
6.12012 E -5
0.8
0.0125 0.253
0.25296
5.04363 E -4
0.2
0.0250 5.947 E-5
0.59376E-4
9.44171 E -8
0.4
0.0250 3.807 E-3
0.38000E -2
6.96268 E -6
0.6
0.0250 0.043
0.43285 E -1
2.84800 E -4
0.8
0.0250 0.244
0.24320
7.97612 E -4
5 NUMERICAL TEST EXAMPLE
In this section, we implement the proposed method to solve a 3dimensional the fractional diffusion equation (1). Also, a comparison with numerical solution and exact solution, which is based on
the implicit finite difference approximation of fractional derivative,
is given.
Example:
In
this
example,
we
with   1.6 ,   1.8 ,   1.5 , of the form:
consider
(1)
u ( x, y, z, t )
1.6u ( x, y, z, t )
1.8u ( x, y, z, t )
 a( x, y, z )
 b( x, y, z )

1.6
t
x
y1.8
5
I. I. Gorial
Table 2: The numerical solution of example using the finite difference
method. (x  0.25, y  0.25, z  0.25, t  0.0125)
x = y =z
t
NumericalSolution Exact solution
|uex -uapprox.|
0.25
0.0125
2.353 E – 4
0.23515 E -3
1.45113E-7
0.50
0.0125
0.015
0.15050 E -1
4.99128 E-5
0.75
0.0125
0.172
0.17143
5.72087 E-4
0.25
0.0250
2.269 E – 4
0.22650E -3
4.00125 E-7
0.50
0.0250
0.015
0.14496 E -1
5.04008 E-4
0.75
0.0250
0.165
0.16512
1.18409 E-4
Table 3: Maximum error for the numerical solution of example using
the finite difference method.
x = y = z
t
Maximum Error
0.20
0.0125
0.0002848
0.25
0.0125
0.0001184
6 CONCLUSIONS
In this paper
(1)
Numerical method for solving the 3-dimensional twosided fractional Diffusion equation has been described and
demonstrated.
(2)
The implicit Euler method is proved to be unconditionally
stable and converges.
(3)
Numerical examples indicate the convergence of the solution with exact results.
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