Application of Homotopy and Variational Iteration Methods to the Atmospheric Internal Waves Model

ISSN: 2319-765X

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Journal of

Mathematics

Volume 10. Issue 5. Version 5

September-October 2014

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M. Ali Akbar,

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Kuldeep Narain Mathur,

l!niversiti Utam Malaysia, Malaysia

Mahana Sundaram Muthuvalu,

Universiti Tekno/ngi Petronas, Malaysia

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Harcourt Butler Technological Institute, Kanpur, India

Basant Kumar Jha,

Ahmadu Bello University, Nigeria

Jitendra Kumar Soni,

Bundelkhand University, Jhansi,

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M.K. Mishra,

EGS PEC Nagapattnam, Anna University,

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Manoj Kumar Patel,

NIT, Nagaland, Dimapw-, India

Sunil Kumar,

NIT, Jamshedpur, India

Suresh Kumar Sharma,

Punjab University, Chandigarh,

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Pradeep J. Jha,

L.J. Institute of Engineering and Technology,

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Xueyong Zhou,

Xinyang Normal University, China

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IOSR Journal.of Mathematics f/OSR-.IM;

e-ISSN: 2278-5728. p-ISSN: 2319- 765X Volume Ill, issue 5 Ver. I' !Sep-Oct. 2111-11. pp -16-53

H'll'H'.

iosrjourna/s_ org

Application of Homotopy and Variational Iteration Methods to

the Atmospheric Internal Waves Model

Zulfiqar Busrah

1,

Jaharuddin

2,

Toni Bakhtiar

3

J.J.J (Department o/A1athenuuics, Bogur Agricultural Unb·ersil\'.

JI. A4eranti, Kan1pus /PB Darmaga, Bogor 16680, Indonesia)

Abstract: Atmospheric intl!rna/ ll'tn•es c:un be represented by a nonline(lr ＺＬｾ｜ﾷｳｴ･ｩｮ＠ o(partial clUJ'i!renrial equation

(PDE) under shalloH':fluid ass111np1ion. Jn thi.\· pafh'r. \I'(.' (.'Xploitecl the hon101opy una(i·si.\' 1111.•thod tl!A.\IJ and

variational iteration n1ethud (VJ,\/) ro vb1ui11,111 approxin1ate ana(rth:ul solution.\ t!lthe .\r.-;11.•111. 7J1,· rc:.,11/1., 1!f

both methods are then con1paretl u·ith n11n1erical n1etht)(/. It is shou·n that hoth HAJJ and VIA'1 an..' ｾﾷＯｽｨﾷｴｬＧｮｴ＠ in

approxin1ating the numerical solutions. ·

Keywords: Atmospheric internal u·aves. ho11101opy ｡ｮ｡ｾｴＧＮ｜ﾷｩｳ＠ 111e1hod. nonlinear Pf)E .\ysfl•111. rariational iteration n1ethod

I.

Introduction

Internal \vaves. also called internal gravity \vaves. is a nalural phenon1enon \\'hose existence cannol he seen because it propagates in the interior of a nuid. rather than on its surf.1ce. Other than belO\\. of the ocean surface, there are also internal \vaves in the upper atmosphere. The propagation of internal \Vaves can he recognized by changes in temperature which occurs through the fluid in three dimensions. So. the wavelength consists of three spatial components. Internal \Vaves may also transfer their momentum and energy to the tlo\v with which they interact. such as winds or other \vaves [I]. Atmospheric internal \vaves can be visualized by wave clouds. In Australia, internal waves in the ｡ｴｭｯｾｰｨ･ｲ･＠ result in Morning Glory clouds. Many researchers have used propagation and interaction of internal \vaves in the atmosphere to 111athen1atical models and numerical simulations [1,2.3.4].

Internal waves in the atmosphere can be represented by a mathematical equation under the shallow fluid assumption. The name "shallow fluid" refers to the fact that the depth of the fluid layer is small compared to the horizootal scale of the perturbation [4]. Shallow fluid assumption has been applied to climate modeling [5), Kelvin waves [6), Rossby waves [7] and tsunami models [8]. Basically, the mathematical model of internal waves in the atmosphere is represented by a system of partial differential equations system (PDE). In many cases, these equations cannot be solved analytically. So. they must be converted to a tbrm that can provide an approximate solution. Liao in 1992 was introduced Homotopy Analysis Method (HAM) as an anal)tical approach [9]. The HAM is rather general and valid tbr nonlinear ordinary and partial ditl'erential equations in many types [10). It has been applied in a variety of nonlinear problems such as Klein-Gordon equation

I

11 ]. generalized Huxley equation [12), Zakharov-Kuznetsov equation [13]. and a single species population model (14). The advantage of HAM method is presence of auxiliary parameter that allows us to fine-tune the region and rate of convergence of a solution [15].

Another analytical approach that can be used in solving the nonlinear system of PDE is Variational Iteration Method (VIM). It was first introduced by He in 1997 [16]. VIM is a powerful tool which is capable of solving linear/nonlinear partial differential equation. The VIM has been successfully applied to nonlinear thermoelasticity (17]. Sawada-Kotera equation [18]. nonlinear Whitham-Breer-Kaup equation [ 19]. KdV-Burgers-Kuramoto equation [20]. and reaction-diffi.1ssion-convection prohlerns [::!:!). The \1_{1\f has 111any}

｡､ｶ｡ｮｴ｡ｧ･ｳｾ＠ such as it avoids linerization and perturbation in order to find solutions of a given nonlinear equatioos. In addition, the VIM provides explicit and numerical solutions with high accuracy [22].

In this paper, we focused on solving the mathematical model of atmospheric internal waves by implementing HAM and VIM as analytical approaches. Solutions by both methods were then compared with the solution by numerical method (NUM) of mathematical models of internal waves in the atmospheric problem.

II.

Atmospheric Internal WaHs Models

Models of internal waves in the atmosphere are represented b) a sys1en1 of nonlinear PDE. This 111odel was developed from the basic equations of fluid under shallo\v-tluid assun1ption. The fundamental equations of fluid motion in differential from are derived by conservation of mass and momentum. Shallo\v fluid refers to the fact that the depth of the fluid layer is the small compared with the wavelength. Here. the atmosphere is assumed to be fluid homogeneous (condition of fluid means that density does not vary in space). autobarotropic. and hydrostatic. The system is

.4pplication uj.Ho111utop)' anti Variational lter"tion .\letluuls tu till· .·lt1nos1J/Jeric Internal .

iJu iJu i)h

-+u--fv+g-=0

ilt iJx iJx ·

av

av

-+u-+fu+gll = 0

at ax ·

ah i)h au ll)

-+u-+vH +h-= 0

at ax a x ·

f_ H

=

--U.

g

\\'here xis a space coordinat, tis time. the independent variables u and v are the cartesian velocity. h is depth of a_ fluid, f represent the Coriolis parameter. g is acceleration of gra\'it).

H

represen1 111ean depth 0f a fluid. and U is the specified, constant mean geostrophic speed on \\·hich the u perturbation is superi1npused (..t].

Ill. Analysis of Method

3.1 Homotopy Analysis Method (HA:vt)

The principles of the Homotopy Analysis \1ethod (H . .\7\·1) an! gh-en in [9]. In this section

''e

ilu::itrate of the concept of HAM based on [ 10-14). V./e considered the follo,ving nonlinear equation in a general tbrm

N[u(x. t)J = 0, (2)

\vhere N is a nonlinear operator. u(x, t) is an unkno\vn function. x and t denote independent variables. Furthermore, \Ve defined a linear operator l \\'hich sati3tit!3

L[f(x. t)) = 0. when f(x. t) = 0. (:l)

Let u0(x, t) denotes an initial guess of the exact solution u(x, t). \'v'e constructed ho111otop) tJ>(x. t; q) ::::

n

x [0,1] __, Ill!. which satisfies

Jf[<l>(x, t; q), Uo(X, t), h .. q]

=

(1 - q)L[<l>(x, t; q) - Uo(X, t)) - qhN(<l>(x, t; q)J. (4) where q E [0,1] is the embedding parameter, h

*

0 is an auxiliary parameter. By the means of generalizing the traditional HAM. Liao constructed the zero-detbrmation equation

(1-q)L[<l>(x. t; q) - Uo(X, t))

=

qhN[<l>(x. t; q)]. (5)

Setting q

=

O. the zero-order deforn1ation equation (5) beco1ncs

• L[<l>(x,t; O) - u₀(x, t)) = 0, (6)

which gives. using the equation (2)

<l>(x, t; O) = u₀(x, t), (7)

when q

=

1, since h

*

O. the zero-order deformation equation (5) becomes

ｎｾｾｴｬＩ｝］ｾ＠ Ｈｾ＠

which is exactly the same as the original equation (2). provided

<l>(x. t; 1)

=

u(x. t). !9)

Thus according to (7) and (9). as q increase from 0 to I. the solution <l>(x, t; q) deforms continousl)' from the initial approximation u0(x, t) to the exact solution u(x, t) of the original equati1Jn ＨｾＩＮ＠ Liao 161 expanded <l>(x, t; q) in term of a power series of q as follows:

+oo

<l>(x, t; q)

=

u0(x, t)

+

L

um (x, t)qm, ( 10)

m=I with

( ) _ 1 amq,cx. t; q)I

Um X,t - I

a

m .

m. q q=O

( 11)

Assume that auxiliary linear operator £. the initial aproximation u0(x, t). the auxiliary paran1eter h. and the auxiliary function 1l(x, t) are properly chosen such that the series ( 10) is converges at q = 1.

Then we have the aproximate solution of equation (2), i.e.

+oo

u(x,t)

=

u0(x,t)

+

L

um(x,t). (12)

m=l

Define the vectors,

Um-I = ( Uo(X, t), u1 (x, t), u, (x, t), .... Um (x, t) ), ( 13)

Differentiating the zeroth-order deforn1ation equation (5) ni tin1es \Vith respect to the embedding paran1eter q. then setting q :::: 0 and finally dividing them by m!. \\"C ha,·e the ni-order detOnnation equation.

L[um(X,t)-xmum-1(x,t)) = h:Rrn[Urn_iJ, (14)

\vhere,

( 15)

and

Application of Ho111otop_v a1u.I Variational /terution AlethoJs to the .·ltnUJSfJheric: /ntl'rnal ...

(0, m S I

Xm=

l,m>l' (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 <I>) ｵｾｩｮｧ＠ ｾｹｲｮ｢ｯｬｩｾ＠ computation sofhvare such as Maple. tv1atlab or \<1athen1atica.

3.2. Variational Iteration Method (YIM)

Jn this section we ilustrate the basic ideas of variational iteration 1nethod (\'l\t) based on [ 15-20]. \\"c

consider the following partial deferential equation.

l[u(x, t)]

+

N[u(x, t)] = :J'(x, t), ( 17)

\vhere L is a linear differential operator, N is a nonlinear operator and 'F is

an

inho1nogencous tern1s. A.ccording to the VIM. \Ve constructed a correction functional

as

tbllo\\S [ 16-20].

'

uk+, (x, t)

=

uk(x, t)

+

J

ａＨｾＩ｛ｬ｛ｵｫＨｸＬ＠

OJ

+

N[uk (x.

OJ -

:J'(x. <J)

､ｾＮ＠

( 18!

0

where A is a general Lagrange n1ultiplier. \vhose optin1al value can be identitie<l b) using the ｾｈｮｩｯｮ｡ｲＩ＠

conditions of the variational theory. The second terms

on

the the right-hand side t 18) is called ｣ｯｲｲｾ｣ｴｩｯｮ＠ and Uk

is considered as a restricted variation, i.e 5i.ik = 0. The subscript k for k

=

0,1,2 ... , indicates the k'11-order approximation. As k tends to infinity. then iteration leads to the exact solution of ( 17). Consi:quently. the solution

u(x, t)

=

Jim uk(x, t).

ｫＭｾ＠ ( 19)

IV.

Aplication of HAM and VIM Methods

4. I Aplication of Homotopy Analysis Method

In this section. the homotopy analysis method is applied to solve the problem of internal \\'aves in the atmospheric. where system (I) will be solved by generalizing the described homotopy analysis me1hod. By

means of the homotopy analysis rnethocl. the linear operator can be defined as bclo\I,.,

l ; 'I'; X, ["' ( t )] ,q acp,(x,t,q) . at ,I

=

123 ' ' . (20)

According to system (I), nonlinear operators N₁ ,N₂ and

N

3 can be defined as follows:

acp,

acp,

acp,

.Nj[cp,,cp,.cp,] =--at+cp,

ax

-rcp,+g

ax'

[

l

acp,

acp,

r

H (21 >

. N,

cp,,cp,.cp3 =--at+cp,ai(+ cp, +g ·

acp,

acp,

acp,

N,[cp,,cp,.cp,] =--at+cp,

iix

+<p,H+<h

clx.

We construct the zeroth-order deformation equation

(1 -

_{q)£1 [ cjJ1} (x, t; _{q) - u0}(x, t)]

=

qh1 .Nj

[cp,, cp,. cp,J,

(1 - q)£2 [ cjJ2 (x, t; _{q) - v0}(x, t)] = _{qh 2}

N, [

cjJ1, cp,. cjJ3], (22) (1 - q)£₃[cjJ3(x, t; q) - h0(x, t)]

=

qh3N3[cjJ,,cjJ,. cjJ3 ].

According to equation (22). when q

=

0 we can write and when q

=

1,

we have

cjJ₁ (x, t, 0)

=

u₀(x, t)

=

u(x, 0), cjJ₂(x,t,O) = v₀(x,t) = v(x,O), cjJ₃(x,t,O) = h₀(x,t) = h(x,O),

cjJ1 (x, t, 0)

=

u(x, t), cjJ 2 (x, t, 0)

= v(x, t),

cjJ3 (x, t, O)

=

h(x, t). Thus, we obtain the mth _order deformation equation

£1 [um (x, t) - Xm Um-1 (x, t)]

=

h1 .'.R1,m (Um-I• vm-1 • hm-1 ).

£2[Vm (x, t) - Xm Vm-1 (x, t)] ;::;::. hi.'.R2.m

(rrm-1 •

v:n-1 • h:n-1 ).

£,[hm (x, t) - Xmhm-1 (x, t)]

=

h3'.R3.m (iim-1 • Ym-1 • hm-1 ). Now, the solution of the m'h-order deformation equation (25) form? I becomes.

'

Um (X, t)

=

Xm Um-I

(x,

t)

+

h1

f

'.R1.m (iim-1• Ym-1• h"'_,) ds

0

\\'\\" .iosrjournalS.llrg

(24)

rl.pplicutiun oj.J-/unzotup,r anti J ·ariutional li!!ru1iun .\h!tho<I ... to thL' .·ltnu1s1>l1L·ric /ntt'rnal

Where,

'

vm (x. t) ;;;; Xm vm-1 (x, t)

+

h2

f

'R.2.m (um-I• vm-1 • hm-1) ds

0

'

hm(x,t) = Xmh111 _1(x.t)

+

h:i

f

.'.R:i.:11(Um-J•V:u-i•hm-l)ds

u

m-1

ｾ＠

(- - -h ) aum-1 "\' aum-i-..

AJ,m Um-1• Vm-1• m-1

="at+

L

Un

ax

n=O

m-1

- - ... avm-1

I

avm-1 am-1

'R.2m(um-t•Ym-1•hm-1)

=-,-+

un-a-+fu0

+-

; : : i

-1gH,

· vt X

vqm-n=O m-1

(-

_ -

i

ahm-1

I

ahm-1-n ｡ｵｭＭＱＭｾＬ＠

.'.R3.m Um-I• Ym-1• hm-1 = -iJ-

+

Lin iJ

+

hn J

+

V:11-1 II.

t

x

c.x

n=O

According to equation ( 12), the results of system (I) can be obtained by sol\'ing the following series: +oo

u(x, t)

=

u0 (x, t) +

L

Um (x, t),

m=I

+oo

v(x. t) = v0 (x, t)

+

L

v,,, (x, t),

m= I

h(x, t) = h0(x, t) +

L

hm(x, t).

m=I

4.2 Aplication of Variational Iteration Method ,

(27)

t28)

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

'

uk+l (x, t)

=

u, +

J

'-1 co ( (u,J( + ii,(ii,), -

rv,

+ gii,(ii,)J

､ｾＮ＠

0

t

Vk+I (x, t)

=

v,

+

J

Az(O(

(v,)( +ii, (v,), +

ru,

+ gH) d<. (29)

0 t

h,+ 1 (x, t)

=

h, +

J

A.3(0 ( (h,J< + ii, (fi,J_ + Hv, + fi, (u, J,) d<.

0

where A.₁,A.₂ and A.₃are general Lagrange muhipliers which their optimal values can be found by

variational theory. Now, taking variation with respect to indepent variables ilk, Vk and ilk \Ve have

t .

ou,+l (x, t)

=

ou, + 0

J

A.,

co (

(u,)( + ii,(ii,J, -

rv,

+ gii,(ii,),) d<. 0

t

OVk+l (x, t)

=

ov,

+ 0

J

A.,(O( (v, )( +ii, (v,), +

ru,

+ gH) d<. 0

t

oh,+l(x.tl

=ov,

+o

J

A.,co(ch,)<+u,(fi,), +Hv, +ii,(u,J,)d<.

0

To find the optimal value ofA._1, A.2 and A.3 , we employ oii, = 0,

ov,

= 0, oii, = 0, from which we ha,·e

t

ou,+1 (x, t)

=

ou, (x, t) + o

J

A.1 ( (u,h) d<. 0

'

OVk+l (x, t)

=

ov,(x, t) + 6

J

A.,( (v,)d d<.

0

using

(30)

(31)

A{Jp!icutiun ＼ｾｬｬ＠ fu111oto/>Y Ulltl 1 ·ariutiu11u/ /1i.:ratio11 .\h,thud.' /u lhl' .-l11nu.\j>hl'ric /111ernul

'

oh•+• (x, t)

= liv• (x, t)

+

Ii

f

).

3 ( (h,

Jd

di;, 0

The considered stationary conditions are then obtained in tbllo\ving ti-0111:

1

+

;i.,

m

=

a.

;i. ,

m

=

a

l+l.2(0=0. t.,(O =O

I

+

t.,

(0 = 0, !.', (\) = 0 Thus, the Lagrange multipliers are defined as follow:

ｴＮＬＨｾＩ］ＭＱＬ＠ t.,(0=-1, t.,(0=-1. Substituting Lagrange multipliers in (33) into the correction functional

ｦｯｲｭｵｬ｡ｾ＠

03)

in equation (28) givl!s the itl!ration

u,+1 (x, t) = u, - f ( (uk)( + u,(u,J., - fv, + g(h,)J

､ｾＮ＠

v,+,(x.t) =v• - fccv,J< +u,(v,J., +fu,

ＫｧｬｬＩ､ｾＮ＠

1.;41

h,+1 (x, t) = hk -

f

((hk)( + uk(h,), + Hv, + h,(u.JJ

､ｾＮ＠

Using the initial conditions u0 (x, t), v0(x, t) and h0(x, t) into (34) we obtain ihe following successive

approximations:

u(x, t)

=

lim "• (x, t), k-oo v(x, t) = lim vk (x, t),

k-oo

h(x. t)

=

lim

h,

(x. t).

ｫｾｶＺ＠

V.

Results

and

Discussions

In this section we compare solutions obtained by HAM and YIM with one obtained by numerical methods (NUM) toward system (I) through graphical representation. Suppose the following parameters are

given for numerical simulation: coriolis parameter f

=

2!lsinu. \vhere

n

=

7.29 x 10-s rad's and n:::::: :!. •

..

constant of gravity g = 9.8 ｭｩｳｾ＠ and constant of pressure gradient of desired nlagnitude H = - ; D. \\·here

U

= 2.5 mis is specified. To get solutions of HAM, VIM and NUM, we start the procedures with the given

initial aproximation:

u(x, 0) = e' sech2 _x.

v(x,O) = 2xsech2_{2x, (36)}

h(x, O) = x2 _sech2 _2x.

By means of the solution of the m"-order deformation equation (26) and the initial conditions (36) we obtain a number of terms as parts of series solution as follow:

u₀(x, t) = e' sech2 _x

Ut (x, t) = h(e' sech2 x (e' sech2 x - 2 · e' sech2 x tanhx)t + 19.5997474 xt sech2 2x - 39.2 x2_tsech2 _{2x tanh2x)}

v0(x, t)

=

2xsech2(2x)

v1 (x, t) = h(-0.0003156662596! + e' sech2 x (2 sech2 2x - 8x sech2 2x tanh2x)t + 0.0001266509 e'tsech2 _2x)

h0 (x, t) = x2 sech 2 2x

h1 (x, t) = h( e' sech2 x (2x sech2 2x - 4x2 sech2 2x tanh2x)t + x2 _sech2 _{2x (e' sech}2 _{x - 2 e' sech}2 _{x tanhx)t}

- 0.00006442168564xtsech2 _2x)

(37)

The rest of the components of the iteration formulas by HAM can easily be obtained by symbolic computation

software. Thus, we obtain the following aproximate solution in term of a series up to S1h -order:

u(x, t)"' u0(x, t) + u1 (x, t) + u2(x. t) +

u,(x.

t) + u.1(x, t) +

u,(x.

t},

v(x, t) • v0 (x, t) + v 1 (x, t) + v2 (x, t) + v3 (x, t) + v 4 (x, t) + v5 (x, t),

ｨｾＰｨｯｾＰＫｾｾＰＫｾｾＰＫｾｾＰＫｾｾＰＫｾｾｾ＠

'"'"''v.i

osrj ournals.org

138)

.-l.pp/icalion uj"//01110/ufJ)' un,J l'uriatiu11ul lt1.•ratio11 ;\/f!tluJ£l'i tu tht: ..t1111us1Jh1!ri1.· hlli!rnal .

Furthermore. by means of the iteration ti.lrn1ula equation (.l..J) 111 lht.• \ J\I and lht.• initial .... tlndititlfl l..'qu;111t1n ( .161. \Ve obtain the approxin1ations by symbolic computation sofl\,·arc.

In Fig. I, Fig. 2 and Fig. 3. we draw graphical solutions of u(x, t), v(x. t), h(x, t). where for I !.·\\I. we use auxiliary parameter h = -0.825 and a series up to S1

h-ordcr. and tOr YJ\t, \\"C use iteration until -tc11_-order.

By those figures, we can see that NL''.\1, HA'.\1 and ｖｉｾＡ＠ solutions are similar.

"II

:;J.c

ＭｾｦＧｬＧＭＭＮＮ｟ｉ＠

ｉｦＭｉＭｾＭＫＭＫＭ

- .[

.'::..>/ ... -.., •) ｾ＠ ＮｊｦＭＭｴＭＭＭＫＭＭｾ＠ r - --Ｍｾ＠ .

i

Ｍｾｾ＠

t·--·· :.-...

...

.,

I/

·-.

..

·-1--.

-.

0

,

,;

0

.,

••

.,

OS I J 1 4 1 6 1 8

L

_

｟ｩＮ］ｓｴｍｾ＠

-

ｾ］ｾｾＺＭ

ﾷｾＺＡＡｾｾＭＭＺ｟ｾｾＺＭｾｾｾｍｍ＠

I

Figure I NUM. HAM and VIM solutions u(x, t), v(x. t), h(x, t). fort= 0.1 and 0 $ x $ 2.

'

·-

.

.

ｾＫＭＫＧ＠

.

ｲＭＭｾＺ＠ .

.

. _. : _.

1 ﾷﾷﾷﾷﾷﾷﾷﾷｾﾷﾷﾷ＠

.•

i

OS .

••

j I ' I

0.02 0.04 0 06 0 08 0.1

-

.

url.t!,!

'·.11,1

·'IUt.l

ＮＭＺＮｾＮＱ＠

1-.·:.1: "'Ill'.'

..,.:.·.·

;

I - ;

[image:11.595.66.549.66.786.2]

012 014 0.16

Figure 2 NUM, HAM and VIM solutions u(x, t), v(x, t), h(x, t), for x

=

1 and 0 $ t $ 0.15.

a.

ＺｓｾｴｵｵｯＺＱｾｾｾｵﾷ＠

ｨｾｴｬｔＡｍ＠

Solution uNuM

ＡＮｾ｜ﾷ＠ .... ｾｾ＠ ·f•u.• 1 . .,IJAM

b. Solution uHAM

\V\.\·\1,·. i osrj ourn als. org

c.

ｾｮｾＺｬＮＬ｣Ｎ＠ "!'·•<."

ﾷＭｾＺＮＺ＠

Solution UviM

[image:11.595.88.536.466.773.2]

.4pplication oj.Ho111oto/JY and Variational Iteration AletluJ£1s tu the ..J.111uJst>he1·ic Internal ...

d,

g,

Solution VNuM

ｬＬｾＢＢＢＧＧ＠ ,(liU•

\y:•UM

So_tution hNuM

e.

h.

,.,.

Solution vHAM "'·b ... ,_.., • ＮｦＱＬｾﾷ＠

ｾｹＱＱ＠ ... 1.1

Solution h11AM

'••

f.

i.

...

ｾ＠

Solution vv1M

l Ｎｾ＠ ... 1. ,.,;

ｾｬｾＧｩＡＮｬ＠

[image:12.608.80.524.102.422.2]

Solution hviM

Figure 3 NUM, HAM and VIM solutions u(x, t), v(x, t), h(x, t). for 0 $ x $ 2 and O $ t $ 0.1.

VI.

Conclusion

Homotopy analysis method and variational iteration method has been succesfully applied in finding the approximate solution of the atmospheric internal waves model, Solutions by those methods are then compared with one of numerical method. It is shown that all solutions are in excellent agreement. It means that both HAM and VIM are efficient in approximating the numerical solution.

References

{I) J.W. Rottman. D. Brottman and S.D. Eckermann. A fort."Cas1 model for <Hmosphcru: mtcrnal \\il\'CS produced l\y a mnunt:un lf•'

Australian Fluid Mechanics Confrence Cron11 Pla1.a. Goal Coast. All. 2007

[2) R.S. Lindzen, Internal gravity waves in a1mosphcrcs wilh realisuc d1ss1pa1ion and ti.:mpera1urc Pan I mathema11cal dc,·dopmcnt and propagation of waves into the thermosphcre. Geophysical Fluid Dynam1i.:s. I. 1 ')70. 303-355

[3) B.P. Cassady, Numerical simulations of aimospheric internal \\il\'CS \\llh ume-dcpcndcnt crit1i.:al levels and turning points.

Departinent of Mechanical Engineering. thesis. Brigham Young University. Ha\\nii. USA. 2010

[4) T.T. Warner, Numerical weather and climate predic11on. (New York. l !SA C:unhndgc ｕｭＬﾷ｣ｲｳＱＱｾ＠ l,rcss. 2011 l I ｾＭＱ＠ .J

[5] D.L. Williamson. J.B. Drake. J J llack. R Jacoh R. and P N A''anraubcr. A srnndard test for numerical appro\1mat1lm shallo\\· ''nter equations in spherical geometry. Journal of Computation Phys1c. 102. J 9'J2. 211-22.J

[6] C. Clancy and P.Lynch. Laplace transform 1ntcgra11on or lhc shallo\\ \\atcr co..tu<it1ons part 1 Eukrian fonnulatwn anJ Kchrn wave, Quarterly Journal of Royal Meteorological Society. 137 (656) . .'.!011. 79.'.!-799

[7] C.T. Duba and J.F. McKenzie. Propagation properties of Rosby,\nvcs for latitudinal ｾ＠ - plane varia1ions off and 7onal ,·anauons of the shallow water speed, Annales Geophysicae, 30. 2012, 849-855.

[8] Blaise S, St-Cyr A. 2012. A dynamic hp-Adapth'e disconlinuous Galerkin method for shallow-water !lows on the sphere with application to a global tsunami simulation . .\mcni.:an ｾｫｬ｣ｯｲｯｬｯｧｩｩＮＺ｡ｬ＠ ｓｲＱ｣Ｑ｣ｴｾ＠ 1-10 978-996

{9) SJ.Liao. The proposed homotopy analysis technique for the solu11on nonlinear problem. PhD thesis. Shanghai Jiao Tong University. Sanghai. CN. 1992

{I

OJ

SJ.Liao, Beyond penurbation Introduction to the ｨｯｭｯｴｯｰｾ＠ analysis mc1hod. ｦｾｃ｜｜＠ York. l 'S A CRC Press C ｯｭｰ｡ｮｾＮ＠ 200.i 1 {I I J A.K . .-\lomari. N.S Nooram. and ｾｉ＠ '.".; RoslmJa :\ppro'l.Hnatc ｡ｮ｡ｬｾ＠ \lea! Ｚ＾ｯｬｵｴｵｭｾ＠ 11!' th..: Kk111-liurJ1111 ..:i.1uat11111 ｢ｾ＠ rn..:an:. \lt' lh\.'

homotopy analysis method. Journal of ｑｵ｡ｩｮｾ＠ ｾｫ｡ｳｵｲｩＮＺｭ･ｭ＠ anJ ａｮ｡ｬｾ＠ :.1:.. -1! l ) . .'.!llll8. -l:'-57

[12) M. Usman, T. Zubair. I Rashid. I Rashid. A Yildirim and S.T. Mohyud-Din. !v!od11icd Homotopy Analysis Mc1hod for lakharo'· Kuznetsov. Walailak. 10(5). 2013. 467-478

[13} K.H. Hemida and M.S. Mohamed. Numerical simulation oflhe generalized Hu ... lcy equa1ion hy homotopy ｡ｮ｡ｬｾ＠ sis mclhnd. Journal of Applied Functional Analysis. 5(4). 2010. 3-14-350

[14] Jaharuddin. A single species population model in pollult.'<i cn,·iromi.:nt soh·cd ｉ｜ｾ＠ homntop> ｡ｮ｡ｬｾｳＱｳ＠ method. Appl1ed :i.1:11hi:ma11o.: Sciences. 8(20). 201-1. 951-961.

[15) B. Froese. ｈｯｭｯＱｯｰｾ＠ analysis melhod lhr ｾｸＱｳｹｭｭ｣Ｑｲｊｃ＠ tlO\\ of a p1n\i:r la\\ 1h11d p;.1:>1 a s1ri:1d11ng ｾｨ･ＮＺＱＮ＠ ｔｲｭｮｾ＠ \\'c:.tcrn University. Kanada. CA. 2007

. I

I

i

!

I

I

I

.4pplication ＼ｾｬｈＰＱｮｯｴｯｰＩＧ＠ anll J,.ariationul iteration .\lctho1I.\· to the At11111sr'heric lntcrnu/

[ 16) J H lie, Vana11onal 1tcra1m11 method a J..111d of nonlmcar ｡ｮ｡ｬｾ＠ 11..;;1l 11:J...h111qu.: ｾｩＮｩｭＮＺ＠ ｃ｜Ｎ＼ｩｭｰｬ｣ｾ＠ Jni.:rn.itrun;il Jour11;1I 111 '0111 ＱＱＱ｣ｾｵ＠

Mechanics. 34(-1), i 999, b99-708

( 17] N H. Swe1lam and ｾＱ＠ｾｉ＠ Khadcr, Vana11onal 11crauon mc1hod !Or one d1mcns1on.it nonlrnc.ir ｴｨ｣ｲｭｯ､｡ｳｵＮＮＺＱＱｾＮ＠ Chao ... Sol1101h 1111J

Fracuils, 32(2007). 2007, 145-1..19

[18) H Jafan. A Yazdani, J Vah1d1 and DD GanJt. Appl1r.:.i11on of !k's Var1a11vnal licraln•n \1cthui.l ti•r -.11J,1ng \\.'\cnth vr1k1 Sawada-Kotera Equauons, Appl1.:d M.ithcma1u:al Sc1cm:c. 21 IU), 2UU8, -!71-177

(19] \·t Maunfar. A h:rc1doon. A Ｎﾷ｜ｨ｡ｳｧｨ｡ｮＱＱｾＱＮＧｨＮ＠ and \I Ghanhan, \'a11atwnal 1h:ra11on mctl111J lur ｾｵｨｭｧ＠ ｮｵｮｨｮｾＭＮｈ＠ \\Bh.

ｾＧ｜ｪｵ｡ｬＱｯｮｳＮ＠ Jn1crnmlllnal Journal ｵｦＧＮＧ｜ｯｮｬｭｾｾｪｲ＠ Snl.'ncl.', 81-l;, 2Ull'l. -119--123

[201 ｾ＠ Satan.DI> Gan.11. and \1 ｜ｨＩｾﾷｳｫｭｩＮ＠ Appl11:a11nn ot'lk's \'arwtwnal 1tcr;it1un ＱＱＱｾﾷＱｨＱＱｊ＠ anJ ·\Jom1a1f-. i.kn1mpos1110n rm:1h1iJ tu

1hc tfac.:11om1I Kd\. Burgl.'rs Kur;imolo l.'ljuauo11. t'ompu!l.'r' ,1nJ Ｎ｜ｬ［ｵｨｾﾷｭ｡ｬｲｬﾷ［､＠ n 1!h ·\pplrl'.Llwn .. 'Si ,'.'.1irN1 ,'.'.1irl'J 2'"' ! . .;'"'':'

(21] S DuangpuaJ.. and ｾｉ＠ ｔｯｮ｡ＱＱ｡ｮ｡ｾｮＮ＠ yar1a11onal 11cra1mn ma1huJ for ｾｯｬＬｲｮｧ＠ nonl1ncar ｲｾﾷ［ｩｲＮＺｵｵＱＱＮｊＱｲｬｩＮｊＺＮＱＰｮﾷＮＮＺｵｮ｜ｬＧ｜ＺｬＱｵｮ＠ prubkn1:>. Applh.'<l Ma1hema11cal Sc.:1ences, b(l7). 2012. 8-13-8-!9

[22) MU ｾＱＱ｡ｮｳ｡ｲｩＬ＠ ME !l-1iansao, A Baran and DD GanJ1. Apph:alion of lle's \'iHJUl1unal 111:1a1mn mc1hoJ lo nunlml.'at lklmhull.t.

and lill·order KdV equauons. Journal of 1he ａｰｰｨｾＧ＼ｬ＠ t..ta1hema1ics. 5( I). 2009, S-20

•

IOSR Journal of Mathematics

Volume

10,

Issue 5, Version 5

September-October 2014

ISSN: 2319-765X

Contents

1

Existence of Solution of Antiperiodic Boundal)· Value Problems of Fractional Order

1 -

9

0

<

a

<

3,

R.P. Pathak, P.S. Somvanshi

DOI:

10.9790/5728-10550109

2

Combinatorial Theo!)· of

a Cnmplete Graph KS.

N.K.

Geethll

10 -

IC!

DOI:

10.9790/5728-10551012

3

Analytical Solution of Dynamic Thermo Elasticity Problem for the FGM Thick-walled

13 -

ｾＺｬ＠

Sphere,

S.M. Bashusqeh

,

A.S. Miavaghi

,

M.A. Khameslou

DOI:

10.9790/5728-10551323

4

Pebbling Graphs of Various Diameter,

B. Tamilselvi, R.Sowentharyt1

24 -

;J8

DOI: 10.9790/5728-105524

5

Dynamics of Allelopathic Two Species Model Having Delay in Predation,

39 -

45

M.A.S. Srinivas, B.S.N. Murthy, A.Prasanthi

DOI: 10.9790/5728-10553945

6

Application of Homotopy and Variational Iteration Methods to the Atmospheric

46 - 53

Internal Waves Model,

Z.

Busrah, Jaharuddin, T. Bakhtiar

DOI: 10.9790/5728-10554653

7

Non-Static Plane Symmetric Massive String Magnetized Perfect Fluid Cosmological

54 - 59

Models in Bimetric Theory of Gra\itation,

R.C. Sahu, B. Behera, S.P. Misra

DOI: 10.9790/5728-10555459

8

Adomain Decomposition Method for Solving Non Linear Partial Differential

60 - 66

Equations,

F.H. Easif

DOI:

10.9790/5728-10556066

9

Wiener Index of Directed and Weighted Graphs by MATLAB Program,

K. Thilakam,

67 - 71

A.Sumathi

DOI: 10.9790/5728-10556771

10 Modification Adomian Decomposition Method for Sohing Seventh Order

Integro-differential Equations,

S.M. Yassien

DOI:

10.9790/5728-10557277

72-77

11

Occurrence of Naked Singularities in Higher Dimensional Dust Collapse,

K.D. Patil,

78 - 88

M.S.Patil

DOI:

10.9790/5728-10557888

12 Strong Equality of MAJORITY Domination Parameters,

J.J. Manora, B. John

DOI:

10.9790/5728-10558995

.