Journal of Applied Mathematics and Physics, 2013, 1, 58-64
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15009
Open Access JAMP
Hermite Matrix Polynomial Collocation Method for Linear
Complex Differential Equations and Some Comparisons
Mina Bagherpoorfard, Fahime Akhavan Ghassabzade*
Department of Mathematics, Islamic Azad University, Fasa Branch, Fasa, Iran
Email: mi.bagherpoorfard@um.ac.ir, *akhavan_gh@yahoo.com
Received August 20, 2013; revised September 25, 2013; accepted October 1, 2013
Copyright © 2013 Mina Bagherpoorfard, Fahime Akhavan Ghassabzade. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
ABSTRACT
In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex
differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of dif-
ferent polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equa-
tions leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient
matrix of final linear systems of equations. Some numerical examples will also be given.
Keywords: Approximate Solution; Collocation Methods; Complex Differential Equations; Hermite Polynomials;
Operational Matrix
1. Introduction
Complex differential equations and their solutions play
a major role in science and engineering. A physical
event can be modeled by complex differential equations.
Since a few of these equations cannot be solved explic-
itly, it is often necessary to resort to approximation and
numerical techniques. In recent years, the studies on
complex differential equations were developed very rap-
idly [1-6].
Since 1994, matrix polynomial collocation approaches
such as Taylor and Bessel matrix collocation methods
have been used by Sezer and colleagues [7-11] to solve
the complex linear differential equations.
The present work contains two main parts, in the first
part, we use Hermite matrix collocation method to find
the approximate solution of higher-order linear complex
differential equations of the following form.



0,
k
nk
k
p
zf gz
(1)
which is a generalized case of the complex differential
equations given in [5,6], with themixed conditions


1
00 ;0,1,,
Jm k
rkj r
kj
afr m


 1
(2)
in the following rectangular domain
,, ,,,,,DzxiyzcaxbcydabcdR ,
(3)
or elliptic domain
,, cos, sin,
02, ,,
Dzxiyzcxayb
axabybab R

 
  
(4)
In the second part, we will study the effect of using
different polynomial classes on the matrix polynomial
methods.
The outline of this paper is as follows. In Section 2, we
briefly introduce Hermite polynomial and describe de-
tails of using these polynomials in matrix polynomial
collocation method. Section 3 focuses on the comparison
of matrix collocation methods when different polynomi-
als are used. We present the results of numerical experi-
ments in Section 4. Finally, conclusions are drawn in
Section 5.
2. Hermite Matrix Polynomial Collocation
Method
In this section, we describe the matrix form of Hermite
polynomials and Hermite collocation Method for com-
plex differential equations. Our aim is to find an ap-
*Corresponding author.
M. BAGHERPOORFARD, F. A. GHASSABZADE 59
proximate solution of (1) defined by a truncated Hermite
series form
 
0,
Nnn
n
N
f
zaHzHz

A
(5)
where
01 N
HzHzH zHz 

such that
,0,1,,
n
H
zn N are the Hermite polynomials de-
fined by
 
0
2
2
2
!1
!2!
nj
jnj
n
N
j
Hz nz
jn j

,
where 2
n
N if n is even and 1
2
n
N
if n is odd
and . It is well known [12] that
the relation between the powers
T
12 N
Aaa a


2
2
20
2! ;0 1,
!2 !
2
n
n
nn
r
nHz
Zz
rn n'

(6)
and


21
21
21 0
21! ;0 1
!(21)!
2
n
r
n
nn
nHz
Zz
rn n'


(7)
By using the expression (6) and (7) and taking
0,1,,nN
we find the corresponding matrix relation
as follows
 

T
T
Z
zMHz,
and
N
z and Hermite poly-
nomials is

T
Z
zHzM, (8)
where
2
1N
Z
zzzz
, for odd N
1
2
11
24
31
48
!!
11
2!1!2 1!3!
22
100 0
000
00
00
00
NN
NN
NN
M

 
 
 















0
0
0
0
0
 
and for even N

10 0 00
000
00
00
1
2
11
24
31
48
!!
20!!
2!0!2 1!2!
22
00
N
NN
NN
NN N
M












 
 
 
0
0
0
!N



  
Then, by taking into account (5), we obtain
 

T
1
HzZz M
and we can replace series (6) in
the matrix form
 

T
1
N
f
zHzAZzM A
 . (9)
Furthermore, the relation between the matrix
Z
z
and its derivative


1
Z
z is
(1) T
Z
zZzB, (10)
where
T
010 0
002 0
000
000 0
B
N

And
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M. BAGHERPOORFARD, F. A. GHASSABZADE
60

0
T
100 0
010 0
000 1
B







From the matrix Equation (10), we get the following
relations:



 




 




 

2
21
TT
3
32
TT
1TT
,
,
.
k
kk
ZzZzBZzB
ZzZzBZzB
ZzZ zBZzB



(11)
By using relations (9) and (11), we have a recurrence
relation in what follows







T
1T
k
kk
N
T
1
f
zZzMAZzBM A

.
(12)
For the collocation points , the matrix relation
(12) becomes
pq
zz

 

T
T1
; ,0,1,, .
k
pq pq
k
N
f
zZzBMApq
N
N
(13)
For one can write the relation (13) in
the following form
0,1, ,p

 
 

 
 

 
 
T
T1
00
T
T1
11
T
T1
,
,
.
k
qq
k
qq
k
N
k
N
k
N
k
Nq Nq
f
zZzBMA
f
zZzBMA
f
zZzBM
A
or briefly






0
10,1,, ,
k
Nq
k
kNq
q
k
NNq
q
fz
fz
f
fz








N (14)
where



2
000 0
2
111 1
2
1
1
1
N
qqq q
N
qqq q
q
N
N
qNqNq
Nq
Zz zzz
Zz zz z
Z
zzz
Zz












 
Moreover, substituting the collocation points into Eq-
uation (3), we have

0;,0,1,, .
k
kpqN
m
pq pq
kPzfzgzpqN

(15)
By means of the expressions (13) and (14), we acquire
the fundamental matrix equation
 
T
T1
00 0
k
kq qq
k
m
qq
NN
PZ BMAG
 
 (16)
In which


0
1
00
00
00
kq
kq
kq
kNq
pz
pz
P
pz

and
 
,,,
qqqNq
Ggzgz gzT
01 . With the aid of
relation (12), we can obtain the corresponding matrix
form due to the condition (4) as follows


T
T1
00
0, 1;,,1,
k
rk j
k
m
j
J
r
aZB M
mAr



 

(17)
where
2
1N
jjj
Zj

.
Briefly, the system of the matrix Equation (17) can be
written in the matrix form
or ;
rr r
UAu r
, (18)
where
 

T
T1
00
01 0;,1,,
mk
rk j
kj
rN
J
rr
aZBM
uur mu



 1,


We can write Equation (16) in the form
WA G, (19)
such that

 
T
T1
00 ;
k
kq
m
q
q
st k
kPZ BMWw

 
1,,,,0,
s
tN
and
0
1
0q
q
N
N
g
g
GG
g







,
where Gq is defined in (16). The augmented matrix of
Equation (19) becomes
;;; ,0,1,
st s
WGw gstN,.
(20)
The augmented matrix of Equation (18) corresponds to
01
;
rrrrrN r
Uuuu;
, where Ur is dened in
(18).
Consequently, to find the unknown Hermite coeffi-
cients an, 0,1, ,nN
related to approximate solution
of the problem consisting of Equation (3) and condition
(4), we replace the matrices (20) by the last m rows of
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M. BAGHERPOORFARD, F. A. GHASSABZADE 61
the augmented matrix (19). Hence, we have a new aug-
mented matrix , where
*
WA G
**
00 01
10 11
20 21
01
00 01
10 11
20
10 11
21
Nm Nm
mm
ww
ww
ww
ww
uu
uu
uu
uu



*
020 0
1
22 2
2
020 0
121 1
2
12
12 1
2
22 2
11
;
N
NmNmN Nm
N
N
N
N
m Nm
N
m
WG
ww g
ww g
ww g
w wg
u u
u u
u u
u u
 









(21)
If det (W*) 0 then we can write

1
**
A
WG
,, ,
. The
unknown Hermite coefficients matrix A, is determined by
solving this linear system and 01
N
aa a

n
are sub-
stituted in Equation (3). Thus, we obtain the Hermite
polynomial solution

n
N
Nn
f
zaHz
.
3. Comparison of Matrix Polynomial
Collocation Methods
Theorem3.1. Let 0i be a base for vector space
S, then every member s S has a unique representation
in the form of linear combination of these vectors.

i
B
k
Proof. [13].
Based on the above theorem, if the bases of approxi-
mate space in collocation methods are chosen from com-
plex polynomials up to degree N, using different bases or
choosing of different complex polynomial classes as the
base has no effect on the approximate solution, theoretic-
cally. This means that if and


0
N
nn
z


0
N
nn
z
be
two different bases according to the uniqueness repre-
sentation, then approximate solution of (1) can be written
 
00
.
NN
n
nn
nn
Nn
f
za
0,1,,n
za





z
N
For this reason when we use different polynomials
(such as Taylor, Bessel, Hermite, etc.) in polynomial
Collocation methods one expects the equal results ob-
tained.
In the numerical implementation, to determine coef-
ficients an, in (5), we should solve a sys-
tem of equations in the form of WF = G and properties of
matrix W is directly depended on choosing the base. So
different bases result different matrix W with different
properties. Some of these properties such as condition
number has the direct influence on solution’s accuracy.
In addition CPU time for solving these systems differs
for different bases. Hence, different polynomial bases
can cause solutions with different accuracy.
Our experiences show that when we use different po-
lynomial classes in matrix polynomial collocation meth-
ods, there is negligible difference among approximated
solutions. In Section 4, we compare this matter for sev-
eral examples by using Taylor, Bessel and Hermite po-
lynomials.
4. Numerical Examples
Several numerical examples are studied in this section to
illustrate the accuracy and efficiently properties of Taylor,
Bessel and Hermite collocation method. In this paper,
collocation points in the rectangular domain (3) are de-
fined by
p
qp
zxiy
q
, such that
,,
;,0,1, ,
pq
q
ba dc,
p
x
apycqax
NN
cy dpqN
b


and in the elliptic domain (4) are defined by
,
pqpqpqpq pq
zxiyxy D
 
,
cos ,
pq a
x
pq
NN
sin ,
pq b
yp
NN
q
such that ,0,1,,pq N
; ,
02q
0
0, 0
00
zz
q
Examples show that the difference among collocation
methods based on these polynomials is negligible. All of
them are performed on a computer using programs writ-
ten in MATLAB 2011a. In this regard, we have reported
in the Tables the value of absolute error function
.
N
ezfNfz
N
at the selected points of the
domain.
4.1. Example 1
As the first example, [10], we consider the linear second
order complex differential equation

 
2
2
126
5cos2 sin,
zfz zfzfz
zzzz
 

 
with
 
1
0,0
2
ff
0
and exact solution
 
2
31
cos
22
f
zz z on elliptic domain with
1
1, 2
ab
and
. Absolute errors are listed in
able 1. T
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M. BAGHERPOORFARD, F. A. GHASSABZADE 63
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M. BAGHERPOORFARD, F. A. GHASSABZADE
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64
4.2. Example 2
In this example, [10], we consider the third order linear
complex differential equation

65 432
8e 12e30e 19e9e0,
zz zzz
fz fz
fz
 
 
with the condition , ,

2
0ef

2
03ef

2
014ef and exact solution
2
ee
z
z
fz in el-
liptic domain with a=1, 1
2
b
and
. Absolute
errors of the obtain solutions are given in Table 2.
4.3. Example 3
The last example, [11] is the second order complex dif-
ferential equation
ee
z
z
f
zzfz z
 ,
with the initial conditions
01f,

0f1
. The
exact solution is

e
z
fz on rectangular domain with
a = 1, b = 1, c = 1, d = 1. Absolute errors are listed in
Tables 3 and 4.
5. Conclusion
In this article, approximate solutions which can be ob-
tained by different polynomial collocation methods have
been compared. Our experiments show that using differ-
ent polynomials cannot significantly affect the numerical
solutions and the results are similar to each other.
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