Application of Homotopy and Variational Iteration Methods to the Atmospheric Internal Waves Model
ISSN: 2319-765X
ｾＨ｝ﾧ＠
--------ｾﾷＭ＠
Ｍﾷｾ
Journal of
Mathematics
Volume 10. Issue 5. Version 5
September-October 2014
IOSR Journals
International Or9Jn1zat1on
of Sc1ent1f1c Reso.:arc!'!
IOSR Journal of Mathematics (IOSR-JM) is a douhll' blind pl'
(16)
The right hand sidt! of equation ( 1-t) depends onl) on the 1crrn"> li:u-: Thu'-. \\I..' t'a") h) nbtain th1..•
series of Um,nl = 1,2,3, ... by solving the linear high-order defonnation equation ) ｵｾｩｮｧ＠
ｾｹｲｮ｢ｯｬｩ＠
computation sofhvare such as Maple. tv1atlab or \l1L·ric /ntt'rnal
vm (x. t) ;;;; Xm vm-1 (x, t)
f'
f'
+ h2 'R.2.m (um-I• vm-1 •hm-1) ds
0
hm(x,t) = Xmh 111 _ 1(x.t) + h:i
.'.R:i.: 11 (Um-J•V:u-i•hm-l)ds
u
Where,
ｾ＠
AJ,m
(-
-
Um-1• Vm-1•
-h
m-1
)
m-1
"\'
aum-1
="at+ L
aum-i-..
ax
Un
n=O
I
=-,-+
i
I
+
m-1
.....
'R.2m(um-t•Ym-1•hm-1)
·
avm-1
- _ .'.R3.m (Um-I• Ym-1• hm-1
ahm-1
vt
avm-1
un-a-+fu 0
X
am-1
+gH,
vqm- 1
(27)
;::i-
n=O
m-1
= -iJt
ahm-1-n
iJ
Lin
n=O
x
+ hn
｡ｵｭＭＱｾＬ＠
J
c.x
+ V:11-1 II.
According to equation ( 12), the results of system (I) can be obtained by sol\'ing the following series:
+oo
L
+L
L
u(x, t) = u 0 (x, t) +
Um (x, t),
m=I
+oo
v(x. t) = v 0 (x, t)
v,,, (x, t),
t28)
m= I
h(x, t) = h 0 (x, t) +
hm(x, t).
m=I
4.2 Aplication of Variational Iteration Method
,
In this section, we implement VIM for obtaining the analytical approximate solution of system (I}. By
means of the variational iteration method refers to system (I), we construct correction functionals as follow
J'
uk+l (x, t) = u, +
'-1 co ( (u,J( + ii,(ii,), -
rv, + gii,(ii,)J ､ｾＮ＠
0
t
Vk+I (x, t)
= v, + JAz(O( (v,)( +ii, (v,), + ru, + gH) d
ｾＨ｝ﾧ＠
--------ｾﾷＭ＠
Ｍﾷｾ
Journal of
Mathematics
Volume 10. Issue 5. Version 5
September-October 2014
IOSR Journals
International Or9Jn1zat1on
of Sc1ent1f1c Reso.:arc!'!
IOSR Journal of Mathematics (IOSR-JM) is a douhll' blind pl'
(16)
The right hand sidt! of equation ( 1-t) depends onl) on the 1crrn"> li:u-: Thu'-. \\I..' t'a") h) nbtain th1..•
series of Um,nl = 1,2,3, ... by solving the linear high-order defonnation equation ) ｵｾｩｮｧ＠
ｾｹｲｮ｢ｯｬｩ＠
computation sofhvare such as Maple. tv1atlab or \l1L·ric /ntt'rnal
vm (x. t) ;;;; Xm vm-1 (x, t)
f'
f'
+ h2 'R.2.m (um-I• vm-1 •hm-1) ds
0
hm(x,t) = Xmh 111 _ 1(x.t) + h:i
.'.R:i.: 11 (Um-J•V:u-i•hm-l)ds
u
Where,
ｾ＠
AJ,m
(-
-
Um-1• Vm-1•
-h
m-1
)
m-1
"\'
aum-1
="at+ L
aum-i-..
ax
Un
n=O
I
=-,-+
i
I
+
m-1
.....
'R.2m(um-t•Ym-1•hm-1)
·
avm-1
- _ .'.R3.m (Um-I• Ym-1• hm-1
ahm-1
vt
avm-1
un-a-+fu 0
X
am-1
+gH,
vqm- 1
(27)
;::i-
n=O
m-1
= -iJt
ahm-1-n
iJ
Lin
n=O
x
+ hn
｡ｵｭＭＱｾＬ＠
J
c.x
+ V:11-1 II.
According to equation ( 12), the results of system (I) can be obtained by sol\'ing the following series:
+oo
L
+L
L
u(x, t) = u 0 (x, t) +
Um (x, t),
m=I
+oo
v(x. t) = v 0 (x, t)
v,,, (x, t),
t28)
m= I
h(x, t) = h 0 (x, t) +
hm(x, t).
m=I
4.2 Aplication of Variational Iteration Method
,
In this section, we implement VIM for obtaining the analytical approximate solution of system (I}. By
means of the variational iteration method refers to system (I), we construct correction functionals as follow
J'
uk+l (x, t) = u, +
'-1 co ( (u,J( + ii,(ii,), -
rv, + gii,(ii,)J ､ｾＮ＠
0
t
Vk+I (x, t)
= v, + JAz(O( (v,)( +ii, (v,), + ru, + gH) d