American Journal of Computational Mathematics, 2012, 2, 282-286
http://dx.doi.org/10.4236/ajcm.2012.24038 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
Effective Solution of Riemann Problem for Fifth Order
Improperly Elliptic Equation on a Rectangle
Seyed Mohammadali Ali Raeisian
Institute of Mathematics, Armenian National Academy of Sciences, Yerevan, Armenia
Received August 12, 2012; revised October 17, 2012; accepted October 31, 2012
ABSTRACT
In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic
equation in complex plane. We reduce this problem to the boundary value problems for properly elliptic equations, and
then solve those problems by the grid methods.
Keywords: Polyharmonic Equation; Boundary Value Problem; Dirichlet Problem; Improperly Elliptic Equation;
Riemann Problem; Grid Method
1. Introduction
Let D be rectangle


,,Dxyaxbcyd

in a
complex plane with boundary D

 .
In this paper, we consider the elliptic equation:


32
5
32 5
1
,ii
2
,
uxy xy xy
zz
xy D


 

 
 

0,
(1.1)
The number of roots of the characteristic equation cor-
responding to Equation (1.1) in the upper half-plane and
lower half plane are not equal therefore it is an improp-
erly elliptic equation and general form of this kind of
equations was investigated in [1] in the form of: ***

 
1
,
1ii,0,,
2
n
pnp
pnp
n
uxy
zz
uxyxy D
xy xy





 
 

 

where


22
1,Dxyxy

1 is a unit disk in a com-
plex plane with boundary 1. As all classical boundary
conditions for improperly elliptic equations are not cor-
rect [1]; so we take following boundary conditions for
problem (1.1):
D
2
01
2
,,
uu
uf ff
NN







2
,
,

(1.2)
where






0
2, 1,
12
,,fCfC fC


(i.e. a function 0
f
with second order derivative
2
0
2
d
d
f
s
satisfy Holder condition on , 1 with first order de-
rivative satisfy Holder condition on and 2 satisfy
Holder condition on
ff

) are given functions on

. We
are seeking the solution of (1.1), (1.2) in the class of
functions . The problem (1.1),
(1.2) known as “Riemann Dirichlet type problem for fifth
order improperly elliptic equation”. The general form of
this kind of problem was considered in [1] and the solv-
ability of non-homogeneous problem and general solu-
tion of corresponding homogeneous problem was dis-
cussed there. The number of corresponding linearly in-
dependent solution of homogeneous problem in our case
is one (see [1]) therefore the problem is not uniquely
solvable and on the other hand we are dealing with com-
plex domain which makes more complicated obstacles in
realizing ideas. Although there was a few limited efforts
in the especial cases like first order improperly and sec-
ond order properly elliptic equations to present as effect-
tive finite difference method in [2] but generally speak-
ing; there is not too much practical suitable methods for
higher order improperly elliptic equations. So, in this
paper, our strategy based on converting boundary value
problem for elliptic equation with complex coefficients
into real coefficients cases; which could be solve by
many fast finite difference methods ([3,4]).



CD



,
D

52
C
2. Description of the Algorithm
The general solution of (1.1) can be represented in the
form:



 


2
01234
,uzzz zzzzz  

where


0,1 4
ii



 are arbitrary analytic functions
C
opyright © 2012 SciRes. AJCM

S. M. ALI RAEISIAN 283
in . D
So, by using Taylor’s expansion, we may replace:
 


1112 22
,zCzzzCzz    

,
Then, we will get:




 
2
011
234
,
uzz Czzzz Cz
zz zzz





2
or
  
  
2
012
34
,
uzz zzzz
zz
  
  (2.1)
On the other hand from (1.1) we get:
5


32 0,
u
zz
6
66
3 3333333
1 20,
2
uu
Uu
z zzzzzzz



 

 

 

6
u



u If we denote , we have a Dirichlet problem
for the determination of :
U U
 

3
01
2
2
2
0, ,,
,
U
UU ff
N
Uf
N






 



(2.2)
The solvability and smoothness of the solution of pro-
blem (2.2) follows from the general theory of elliptic
problems [5]. From the unique solution of Dirichlet
problem for third harmonic Equation (2.2) we have U on
all mesh points and a formula for representing unique
solution of above Dirichlet problem may be found in [6,
p. 149].
Applying bi-harmonic operator on real part of
in (2.1); we will get: ***
2
u
 

    
  
222
00
00
00 0
22
22
22 4,
Uzzzz
z zzzzzzz
zz z


 






 


 



Therefore



2
0
2U 

 ,z
(2.3)
where 0

 is analytic in , so is harmonic
function.
D 
z
0



From (2.3) we have:
     
00 0
1,
2Uzz z xy


 
 

2 (2.4)
Denoting




10
uz 

 and



10
v 



z, we
have Poincare problems for the determination of these
functions:

2
1
1
1
0, ,
2
u
uU
x


 


 (2.5)
and

2
1
1
1
0, ,
2
v
vU
y


 


 (2.6)
Solving these problems, we get: 2
and 4
0
111
uucyc 
0
11 3
vvcxc

 where , are uniquely de-
termined functions and are arbitrary
real constants.
0
1
u
cj
0
1
v

1, ,

4
j
We must mention that by Cauchy-Riemann equations
we have: 1
cc
3

 ,therefore we get a representation:
0
01
iCcz 
 0
0
1
Here 1 is complex and 0 c is real arbitrary con-
stants and is uniquely determined ana-
lytic function.
C
00
1
iuv 
By integration we have:




2
00 01
i,zwzCzCzC  
2
(2.7)
where 0 is real constant, 12
are arbitrary complex
constants, and is uniquely determined function.
C,CC
0
w
Now, replacing 0

in (2.1), we get:
  


  
2
2
001 2
34
i
i
uzwCzzzzzz
zz zz
 
 (2.8)
where
  
1112
zCz zz   

 
2
2234
zCz z z 
 
3112
zCz zz   

 


4
2
23 4
zCzzz
are arbitrary analytic functions.
Finally, we represent the solution in the form:
   
2
2
00
i,iuzwz CzzHzzhzz ,,
(2.9)
where




12
,

H
zz zzzz  and






34
,hzz zzzz. Here w0 is known func-
tion, and ,
H
h are real valued functions which satisfy
the condition 22
0Hh

 and constant is arbi-
trary real constant.
0
C
Now, we must determine real valued functions ,
H
h.
These functions satisfy bi-harmonic equation:
22
0Hh

 and from (2.9):

S. M. ALI RAEISIAN 284



2
0,,UzwzHxy
Hence, we have following Dirichlet conditions on the
boundary :

 

 

2
00
2
10
,
,
Hfzwz
Hfzwz
NN


 

 

Finally we get the “Dirichlet problem for bi-harmonic
equation”:



 

2
2
00
2
10
0,
,
,
H
Hfzwz
Hfzwz
NN



 

 
 
(2.10)
which has a unique solution.
Analogously, we get the same boundary problem:


 
2
2
00 ,,uzwzCzzhxy
i.e.:
 


2
2
00
,hxyuzw zCzz ,
(2.11)
for determination of function h. This problem includes
arbitrary constant , therefore must be modified.
0
So, first we find the function 0, by solving “Dirichlet
problem for bi-harmonic equation”:
Ch


2
0
2
0
2
0
0,
,
,
h
hzz
hzz
NN





 

 

 
(2.12)
and then ,we solve:
 







2
1
2
10
2
00
2 1
0
2
10
0,
,
,
h
hu zwz
fzwz
huzw
NN N
fzwz
N


z

 


 
 



 
 
 


 



(2.13)
These problems are uniquely solvable, and the solution
of (2.11) will be .
100
Replacing the function h into (2.9), we find:
hh Ch




   
2
0
2
100 0
,
i, ,i
uzwzHxy
hxy ChxyCzz

 ,
or:
 

2
2
010
,i,iuzwz HxyhxyC zzh 
0
,
So, during above argument we had proved that by us-
ing our algorithm; we can find the solution of the prob-
lems (1.1) and (1.2) with only one constant 0, and line-
arly independent solution for corresponding homogene-
ous problem is
C



2
00
i,uzzh
Here 0 is bi-harmonic function and uniquely deter-
mined from problem (2.12).
h
3. Numerical Solution
Let us divide the rectangle D by N − 1 straight lines, par-
allel to coordinate axes and for simplicity the nodes are
equidistant which denoted by:
,,0,
,,0,
k
j
ba ,.
,.
x
akhhk N
N
dc
ycjhhj N
N

 

 


Let’s consider Dirichlet problem for tri-harmonic
Equation (2.2). The tri-harmonic equation is a sixth order
elliptic partial differential equation:
6666
3
64224
3,
UUU
U6
U
x
xy xyy


 



which is encountered in viscous flow problems.
In the general case, poly-harmonic equation kUf

,
in the discrete setting, are usually solved by applying a
Laplacian discretization repeatedly on a mesh, with the
cotangent formula being the most popular and the one
used in [7]. Although this formulation is efficient and
particularly useful in interactive applications, it sup-
ports only one particular way of specifying boundary
conditions and the results often have significant mesh
dependence. The convergent discretization based on a
quadratic fitting scheme could be found in [8]. Anyway,
Dang Q. A. [9] studied an iterative method for solving a
BVP for a tri-harmonic type equation based on using
boundary domain operator defined on pairs of boundary
and domain functions in combination with parametric
extrapolation technique and reduces the BVP for sixth
order equation to a sequence of BVPs for Poisson equa-
tion.
Next, For finding numerical solution of Poincare prob-
lems (2.5) and (2.6); we solve it in domain with the
boundary


, whereas




,:, ,
,1,
ij i
j
xyxa hbh
ychdh ijN

 
1,


where the values of are known on the boundary
2
hh
U

S. M. ALI RAEISIAN 285
points . Because of simplicity and getting rid of com-
plexity of the system , we may assume two points in (2.5)
on the top left and down right corner of the bound-
ary are fixed, i.e.:
 




00
,,
x
yahdh 

,
and

,,
11
x
yb

hc h

10111 11
,0,,uuxy u 
1
u
 
11
,,, 0,,kjkj
so:

100
uxy 0,
N
By denote the values of function in the mesh
points approximately:

j
k
uu
and from the grid boundary conditions:


2
11
1,,1,,1
4
jj j
iihh
i
uu UijN
h
  

1
we will find the values in the grid points on the sides of
parallel to OX axis.


Finally, the values of

1
j
i
u inside and on the
sides parallel to OY axis will be found from the sys-
tem of linear equation, main matrix of this system may
be reduced to the tri-diagonal form :




0... ...0
0...0
0..
0000...
AB
BAB
BAB
T.0
A












,
where:
222
22
22
11 00
141 0
14
00
,
41
11
000
000...00
0 10 ... 0 0
001...00
,
... ... ...... ...
000 ... 10
000...00
hh
hh h
hh
A
hh
hh
B

































 
 
 





the matrix T is diagonally dominant, therefore this linear
system is uniquely solvable; so considered modified
Poincare problem is uniquely solvable too and then from
maximum principle [10], we have unique solvability of
this system.
Next, we consider the problem (2.10). A fast solver for
compact discretizations of the bi-harmonic problem was
presented in [11-13], also Dang Q. A. in [14,15] with the
help of boundary or mixed boundary-domain operators
introduced appropriately, constructed iterative methods
for bi-harmonic and bi-harmonic type equations associ-
ated with the Dirichlet, Neumann or simple type of
mixed boundary conditions.
By finite difference method, we find the values of the
function
H
in the mesh points approximately:


,,1, ,,1, ,.
j
i
H
Hijim jn
we use the discrete analogue of the Laplace operator:

21
11
2
14,
jjjjj
hhiiiii
HHHHHH
h



1
and may discretisize biharmonic equation in (2.10) as




   
11
4
11
4
1111
1111
4
22
22
4
01
20 8
4
8
2
1,
,.
jjj
hh hiii
jj
ii
jjjj
iiii
jjj j
iii i
hh h
h
HHHH
h
h
HH
h
HHHH
h
HHH H
h
Hf Hf








 



 
(3)
Here forward divided differences for the operator N


in (2.10) noted by h

,and

h h
 
01
,
f
fre values of
functions
a


,
01
f
f
n ob
 the boundary points of mesh.
The
in
ditainn [16], tends
to
(3.1) approximates the problem
(2
scretizatioed here and also i
symmetric matrices so the convergence analysis is
simpler, but there now exist standard fast numerical al-
gorithms for this kind of boundary value problems too. In
[17-19] for the two dimensional case, problem (2.10) has
been reduced to second kind integral equations and a fast
numerical algorithm is developed based on the con-
structed SKIE in [20].
The discrete problem
.10) (see [21]), therefore, from the stability of the
problem (3.1), we have the convergence of the grid func-
tion to




,
ij
H
xy [3, p. 30, Theorem 2.5]. From the last
ation of the two equ(3.1); we find the values of the func-
tion h
H
in the points



,,0,1,,
ij
x
yij N and if
aluthe ves
j
i
H
are on im from nterior nodes, we find the

S. M. ALI RAEISIAN 286
duced to six
un
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