Iranian Journal of Optimization 3(2010) 469-476
Optimization
Adomian Decomposition Method On Nonlinear
Singular Cauchy Problem of
Euler-Poisson-Darbuox equation
Iyaya C. C. Wanjala
*School of Physical and Applied Science, Kenyatta University P.O BOX 43844-0010, Nairobi, Kenya
Abstract
In this paper, we apply Picard’s Iteration Method followed by Adomian Decomposition Method to solve a nonlinear Singular Cauchy Problem of Euler-Poisson- Darboux Equation. The solution of the problem is much simplified and shorter to arriving at the solution as compared to the technique applied by Carroll and Showalter (1976)in the solution of Singular Cauchy Problem.
Keywords:Adomian decomposition method, Singular Cauchy problem.
*Email address:
1. Introduction:
The Singular Cauchy Problem
2
2
;
, 0
, 0
;
, 0
0
0
m
m t
u
k u
u in
x
t
t
t
u x
f x
u x
on
t
(1)
has been studied since the time of Euler(1770).
Some classical results obtained are summarized as follows: (i). If k m 1, the solution as given by Arsgeirsson(1936) is
2
1
1
1
,
mi
m
u x t
f x
t d
(2)where
2 1 2
2 ;
2
m
m m
m
is the surface area of the
m
dimensional unit sphere.(ii). If k m 1, the solution obtained by Weinstein (1952) is
2
1
1 2 2
1
1 1
,
m1
i
k m
k m
k
u x t
f x
t
d
(3)where
d
d
1d
2
d
m©2010 Kenyatta University
1 2 1 2
, ,
n
k n
t k n
u x t t t u x t
t
(4)
where
n
is a positive integer chosen such that k2n m 1 andu
k2n is given by (2) or (3) with f replaced byf
k
1
k
2
k
2
n
1
.The solution of the singular Cauchy problem is unique for k 0 whereas for 0
k it is not unique as indicated in the work of Weinstein (1952).
Carroll and Showalter (1976) dealt primarily with the Cauchy problem for singular and degenerate equation of the form
tt
t
A t u
B t u
C t u
g
(5)where
u
is a function oft
, taking values in a separated locally convex space E, whileA t
,
B t
,
andC t
are families of linear or nonlinear differential type operators acting in E, some of which become zero or infinite at t0. They considered appropriate initial datau
0
andu
t
0
at t0, and ga suitable E -valued function.2. The Adomian Decomposition Method to Singular Cauchy Problem
The purpose of this paper is to apply Adomian Decomposition method to solve the problem (6).
2
2
;
, 0
, 0
;
, 0
0
0
p n
n t
u
k u
u
u
in
x
t
t
t
u x
f x
u x
on
t
(6)
where p0,ka parameter and
f x
is smooth with compact support.
2
2
2
;
, 0
, 0
;
, 0
0
0
p n
n t
d u
k du
u
in
dt
t dt
u
f
u
on
t
(7)
Then t1 t d tk du 2u up
dt dt
or
2 1
;
p t d k d
Lu u u L t t
dt dt
(8)
Then 1
2 1
1 1
0 0
;
t tp k k
u
L
u
L
u
L
t
t
dtdt
(9)Solving the homogeneous part of (8) by Picards iteration method with
u
0
f
;2 1 1 0 0
t k t k
n n
u
t
t
u
dtdt
So that we get
2 3
2 4 6
1 . ; 2 . ; 3 . ;
2! 1 3! 1 2
t t t
u f u f u f
k k k k k k
Therefore
2
4
2
6
2
2
, 1 . . 1 .
1 2! 1 2 3! 1 1 !
n n
n
t t t t
u t f
k k k k k k k k k n n
On takingk 2m1, we get
,
2
1
m m
m
2
0
1 1
1
! 2 1
n n
n
t
f m
n m n
(10)The nonlinear part is done by the Adomian Decomposition Method, Adomian(1994), where
0 0
1
;
0,1,
!
n
i
n n i
i
d
A
N
u
n
n d
(11)are Adomian polynomials with
a parameter introduced by convenience. Then Adomian suggests the inductive scheme:0 , 1 0, , n n 1
u f u A u A (12)
Thus from (9), we have
1 0 0
,
1,
,
,
0,1, 2,
t
n n n
u
t
A u
s u s
u
s
ds n
(13) So we have
0 0 1 1 0 1
2 2 1
2 0 1 0 2
3 3 2 1
2 0 1 0 1 2 0 3
1
1
2!
1
1
2
1
3!
p
p
p p
p p p
A
u
A
pu
u
A
p p
u
u
pu
u
A
p p
p
u
u
p p
u
u u
pu
u
0 1
2 2 1 2
3 3 1 2 3 2
3
4
2 4 3 2 4 1
4
2!
1
3!
2 1 1
4! ...
p
p
p
p p
p p
u t f
u t f t
t u t pf
t
u t p p f p f
t
u t p p f p p f
Thus for nonlinear part, we have
2 1
2
3 1
2 3 2
31
2!
3!
p p p
t
p pt
u t
f
f
t
pf
p p
f
p f
+
4
2 4 3 2 4 1
2
1
1
...
4!
p p
t
p
p
f
p
p
f
(14)
Using the equations (10) and (14); we write
,
l
,
n
,
u
t
u
t
u
t
(15)Therefore
,
l
,
n
,
u x t
f
u
t
f
u
t
(16)where
u
l
,
t
andu
n
,
t
are given in equations (10) and (14) respectively.
2 3 4
2 5
1 1
1
2
1
6
7
2
2!
3!
4!
1
1
1
pt
t
t
u t
t
p
p
p
p
p
p
O t
p
t
(17) Then
1 1
1 11
,
1
1
1
1
1
n
n p p
u
t
d
p
t
p
t
The first term in (15) then will be
2
2 0
1 1
, 1 2 1
! 2 1
n n m l m n t
u t m m t J t
n m n
and
,
1
2
n m i x l mt
u x t
e
J
t d
m
0 1cos sin 1
2
n
m
i x t
e
t
d d
m
(18)Now (18) is not easy to calculate directly since
u
l
,
t
is not an1
L
function. Here we shall consider the use of a cutoff functione
2 2 2;
0
(Folland1976) so that we have` ul
e 2 2 2u
l.l
u. Moreover, ul
L1
n , so we can calculate its inverse Fourier transform as an ordinary integral.3. Conclusion
On comparison of solution of homogeneous part the equation (2)-(3) with that used by Carroll (1951), this is far much simpler for we do not involve transformations which are not easy to identify. Thus Adomian decomposition method offers quite a simpler means of solving nonlinear differential equations.
References
[1] Adomian G.,” Solving Frontier Problems on physics, the decomposition method”.Kluwer Academic Publisher, Boston, 1994.
[2] Carroll R.,” Some Singular Cauchy Problems”, Ann.Mat.Pura Appl., 56, 1-31, 1951.
[3] Carroll R., and Showalter R.,” Singular and Degenerate Cauchy Problems”. Academic Press, New York, Vol.127-p 333, 1976.
[4] Euler L., ”Institiones Calculi Integralis ,III”. Petroli, 1770.
[5] Weinstein A., ”Sur le Proble’me de Cauchy pour1’ e’quation de poisson et 1’equationdes ondes.” Critical review, Paris, 234,2584-2585, 1952.