Solving High-order Non-linear Partial Differential Equations by Modified q-Homotopy Analysis Method

www.ajms.com 25
ISSN 2581-3463
RESEARCH ARTICLE
Solving High-order Non-linear Partial Differential Equations by Modified
q-Homotopy Analysis Method
Shaheed N. Huseen1
, Magdy A. El-Tawil2
, Said R. Grace2
, Gamal A. F. Ismail3
1
Department of Mathematics, Faculty of Computer Science and Mathematics, University of Thi-Qar, Iraq,
2
Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza, Egypt, 3
Department
of Mathematics, Women’s Faculty, Ain Shams University, Egypt
Received: 25-04-2020; Revised: 25-05-2020; Accepted: 10-06-2020
ABSTRACT
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order
non-linear partial differential equations. This method improves the convergence of the series solution
and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM
which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases
are solved as illustrative examples of the proposed method.
Key words: Non-linear partial differential equations, q-homotopy analysis method, modified
q-homotopy analysis method
INTRODUCTION
Most phenomena in our world are essentially non-linear and are described by non-linear equations.
It is still difficult to obtain accurate solutions of non-linear problems and often more difficult to
get an analytic approximation than a numerical one of a given non-linear problem. In 1992, Liao[1]
employed the basic ideas of the homotopy in topology to propose a general analytic method for non-
linear problems, namely, homotopy analysis method (HAM). In recent years, this method has been
successfully employed to solve many types of non-linear problems in science and engineering.[2-11]
All of these successful applications verified the validity, effectiveness, and flexibility of the HAM.
The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and
control the convergence region and rate of convergence of the series solution. Moreover, by means
of the so-called h-curve, it is easy to determine the valid regions of h to gain a convergent series
solution. Hassan and El-Tawil[7]
presented a new technique of using HAM for solving high-order
non-linear initial value problems (nHAM) by transform the nth-order non-linear differential equation
to a system of n first-order equations. El-Tawil and Huseen[12]
established a method, namely, q-HAM
which is a more general method of HAM. The q-HAM contains an auxiliary parameter n as well
as h such that the case of n=1 (q-HAM; n=1) the standard HAM can be reached. The q-HAM has
been successfully applied to numerous problems in science and engineering.[12-22]
Huseen and Grace[23]
presented modifications of q-HAM (mq-HAM). They tested the scheme on two second-order non-
linear exactly solvable differential equations. The aim of this paper is to apply the mq-HAM to obtain
the approximate solutions of high-order non-linear problems by transform the nth-order non-linear
differential equation to a system of n first-order equations. We note that the case of n=1 in mq-HAM
(mq-HAM; n=1), the nHAM[7]
can be reached.
Address for correspondence:
Shaheed N. Huseen,
E-mail: shn_n2002@yahoo.com

Huseen, et al.: Solving high-order non-linear partial differential equations
AJMS/Apr-Jun-2020/Vol 4/Issue 2 26
ANALYSIS OF THE Q-HAM
Consider the following non-linear partial differential equation:
N[u(x, t)]=0(1)
Where, N is a non-linear operator, (x, t) denotes independent variables, and u(x, t) is an unknown function.
Let us construct the so-called zero-order deformation equation:
(1–nq)L[∅(x, t; q)–u0
(x, t)]=qhH(x, t)N[∅(x, t; q)](2)
where n≥1, q∈[0,
1
n
] denotes the so-called embedded parameter, L is an auxiliary linear operator with
the property L[f]=0 when f=0, h≠0 is an auxiliary parameter, H(x, t) denotes a non-zero auxiliary function.
It is obvious that when q=0 and q=
1
n
Equation (2) becomes
∅( ) = ( ) ∅





 =
x t u x t and x t
n
u x t
, ; , , ; ( , )
0
1
0 (3)
respectively. Thus, as q increases from 0 to
1
n
, the solution ∅(x, t; q) varies from the initial guess u0
(x, t)
to the solution (x, t). We may choose u0
(x, t), L, h, H (x, t) and assume that all of them can be properly
chosen so that the solution ∅(x, t; q) of Equation (2) exists for q∈[0,
1
n
].
Now, by expanding ∅(x, t; q) in Taylor series, we have
∅( ) = ( )+ =
+∞
∑
x t q u x t u x t q
m
m
m
, ; , ( , )
0 1
(4)
where
u x t
m
x t q
q
m
m
m q
,
!
( , ; )
|
( ) =
∂ ∅
∂
=
1
0 (5)
Next, we assume that h, H (x, t), u0
(x, t), L are properly chosen such that the series (4) converges at
q=
1
n
and:
u x t x t
n
u x t u x t
n
m m
m
, , ; , ,
( ) = ∅





 = ( )+ ( )





=
+∞
∑
1 1
0 1
(6)
We let
u x t u x t u x t u x t u x t
r r
, , , , , , , , ,
( ) = ( ) ( ) ( ) … ( )
{ }
0 1 2
Differentiating equation (2) m times with respect to q and then setting q=0 and dividing the resulting
equation by m! we have the so-called mth
order deformation equation
L u x t k u x t hH x t R u x t
m m m m m
, , , ( , )
( )− ( )

 
 = ( ) ( )
− −
1 1

(7)
where,
R u x t
m
N x t q f x t
q
m m
m
m q
( , )
!
( , ; , )
|
−
−
− =
( ) =
−
( )
∂ ∅( )

 
 − ( )
∂
1
1
1 0
1
1

(8)

and
k
m
n otherwise
m =
≤



0 1
(9)
It should be emphasized that um
(x, t) for m≥1 is governed by the linear Equation (7) with linear boundary
conditions that come from the original problem. Due to the existence of the factor
1
n
m
, more chances for
convergence may occur or even much faster convergence can be obtained better than the standard HAM.
It should be noted that the case of n=1 in Equation (2), standard HAM can be reached. The q-HAM can
be reformatted as follows:
We rewrite the nonlinear partial differential equation (1) in the form
Lu x t Au x t Bu x t
, , ,
( )+ ( )+ ( ) = 0
u x f x
, ,
0 0
( ) = ( )

∂
∂
= ( )
=
u x t
t
f x
t
( , )
| ,
0 1 (10)
∂
∂
=
−
− = −
( )
( ) ( ) ( )
( , )
| ( ),
z
z t z
u x t
f x
1
1 0 1
Where, L
t
z
z
=
∂
∂
( )
, z=1,2,… is the highest partial derivative with respect to t, A is a linear term, and B is
non-linear term. The so-called zero-order deformation Equation (2) becomes:
1 0
−
( ) ∅( )−

 
 = ( ) ( )+ ( )+
nq L x t q u x t qhH x t Lu x t Au x t Bu x t
, ; ( , ) , ( , , ,
(
( ))(11)
we have the mth
order deformation equation
L u x t k u x t hH x t Lu x t Au x t B u
m m m m m
, , , ( , , (
( )− ( )

 
 = ( ) ( )+ ( )+
− − −
1 1 1 m
m x t
− ( )
1

, )) (12)
and hence
u x t k u x t hL H x t Lu x t Au x t B u
m m m m m m
, , [ , ( , , (
( ) = ( )+ ( ) ( )+ ( )+
−
−
− −
1
1
1 1 −
− ( )
1

x t
, ))](13)
Now, the inverse operator L–1
is an integral operator which is given by
L dt dt dt c t c t c
z times
z z
z
− − −
( ) = … ( ) … + + +…+
∫∫ ∫
1
1
1
2
2
. .

(14)
where c1
, c2
,…, cz
are integral constants.
To solve (10) by means of q-HAM, we choose the initial approximation:
u x t f x f x t f x
t
f x
t
z
z
z
0 0 1 2
2
1
1
2 1
,
! ( )!
( ) = ( )+ ( ) + ( ) +…+ ( )
−
−
−
(15)
Let (x, t)=1, by means of Equations (14) and (15) then Equation (13) becomes
u x t k u x t h
u x
Au x
m m m
t
t t
z
m
z m
, , (
,
,
( ) = ( )+ …
∂ ( )
∂
+ ( )+
−
−
−
∫
∫ ∫
1 0
0 0
1
1
τ
τ
τ B
B u x d d d
m
z times
( , ))
− ( ) …
1

τ τ τ τ (16)

Now from times 0
0 0
1
t
t t
z
m
z
z times
u x
d d d
∫
∫ ∫
…
∂ ( )
∂
…
−
(
,τ
τ
τ τ τ

, we observe that there are repeated computations
in each step which caused more consuming time. To cancel this, we use the following modification to
(16):
u x t k u x t h
u x
d d d
m m m
t
t t
z
m
z
z times
, ,
,
( ) = ( )+ …
∂ ( )
∂
…
−
−
∫
∫ ∫
1 0
0 0
1 τ
τ
τ τ τ

+
… ( )+ ( ) …
∫
∫ ∫ − −
h
Au x B u x d d d
t
t t
m m
z times
0
0 0 1 1
( , ( , ))
τ τ τ τ τ

= ( )+ ( )− ( )+
∂ ( )
∂
+…+
−
− − −
−
−
k u x t hu x t h u x t
u x
t
t
z
m m m m
m
z
1 1 1
1
1
0
0
1
, , ,
,
(
( )
∂ ( )
∂








+
+ … ( )+
−
−
−
−
∫
∫ ∫
!
,
( , (
z
m
z
t
t t
m
u x
t
h Au x B u
1
1
1
0
0 0 1
0
τ m
m
ztimes
x d d d
− ( ) …
1

, ))
τ τ τ τ
(17)
Now, for m=1, km
=0 and
u x t
u x
t
t u x
t
t
z
u x
z z
0
0
2 2
0
2
1 1
0
0
0
2
0
1
0
,
,
!
,
!
,
( )+
∂ ( )
∂
+
∂ ( )
∂
…+
−
( )
∂ (
− −
)
)
∂
= ( )+ ( ) + ( ) +…+ ( )
−
= ( )
−
−
−
t
f x f x t f x
t
f x
t
z
u x t
z
z
z
1
0 1 2
2
1
1
0
2 1
! ( )!
,
Substituting this equality into Equation (17), we obtain
u x t h Au x B u x d d d
t
t t
z times
1 0
0 0 0 0
, ( , ( , ))
( ) = … ( )+ ( ) …
∫
∫ ∫ τ τ τ τ τ

(18)
For m1, km
=n and
u x
u x
t
u x
t
u x
t
m
m m
z
m
z
, ,
,
,
,
, ,
,
0 0
0
0
0
0
0
0
2
2
1
1
( ) =
∂ ( )
∂
=
∂ ( )
∂
= …
∂ ( )
∂
=
−
−
Substituting this equality into Equation (17), we obtain
u x t n h u x t h Au x B u x
m m
t
t t
m m
, , ( , ( ,
( ) = +
( ) ( )+ … ( )+ ( )
− − −
∫
∫ ∫
1 0
0 0 1 1
τ τ

)
))d d d
z times
τ τ τ
…

(19)
We observe that the iteration in Equation (19) does not yield repeated terms and is also better than the
iteration in Equation (16).
The standard q-HAM is powerful when z=1, and the series solution expression by q-HAM can be written
in the form
( ) ( ) ( )
0
1
, ; ; , ; ; , ; ;
=
 
≅ =  
 
∑
i
M
M i
i
u x t n h U x t n h u x t n h
n
(20)
However, when z≥2, there are too much additional terms where harder computations and more time
consuming are performed. Hence, the closed form solution needs more number of iterations.

THE PROPOSED MQ-HAM
When z≥2, we rewrite Equation (1) as the following system of the first-order differential equations
ut
=u1
u1t
=u2
⋮ (21)
u{z–1}t
=–Au(x, t)–Bu(x, t)
Set the initial approximation
u0
(x, t)=f0
(x),
u10
(x, t)=f1
(x),
⋮ (22)
u{z–1}0
(x, t)=f(z–1) (x)
Using the iteration formulas (18) and (19) as follows
u x t h u x d
t
1 0 0
1
( , ) , ,
= − ( )
( )
∫ τ τ
u x t h u x d
t
1 2
1 0 0
( , ) ,
= − ( )
( )
∫ τ τ (23)
⋮
u z x t h Au x B u x d
t
{ } ( , ) , ( , )
− = ( )+ ( )
( )
∫
1 1
0
0 0
τ τ τ
For m1, km
=n and
um
(x, 0)=0, u1m
(x, 0)=0, u2m
(x, 0)=0,…,u{z–1}m
(x, 0)=0
Substituting in Equation (17), we obtain
u x t n h u x t h u x d
m m
t
m
, , , ,
( ) = +
( ) ( )+ − ( )
( )
− −
∫
1 0 1
1 τ τ
u x t n h u x t h u x d
m m
t
m
1 1 2
1 0 1
, , ,
( ) = +
( ) ( )+ − ( )
( )
− −
∫ τ τ (24)
⋮
u z x t n h u z x t h Au x B u x
m m
t
m m
{ } , { } , , ( ,
− ( ) = +
( ) − ( )+ ( )+ (
− − −
∫
1 1 1
0
1 1
τ τ )
)
( )
) dτ
To illustrate the effectiveness of the proposed mq-HAM, comparison between mq-HAM and the standard
q-HAM is illustrated by the following examples.
ILLUSTRATIVE EXAMPLES[8,9]
We choose the following two cases when z=2 and z=3.
Case 1. z=2
Consider the modified Boussinesq equation
utt
–uxxxx
– (u3
)xx
=0(25)
subject to the initial conditions

u x x
, [ ]
0 2
( ) = sech
u x x x
t , [ ]
0 2
( ) = [ ]
sech tanh (26)
The exact solution is
u x t x t
, [ ]
( ) = −
2sech (27)
This problem solved by HAM (q-HAM [n=1]) and nHAM (mq-HAM [n=1]),[7]
so we will solve it by
q-HAM and mq-HAM and compare the results.
IMPLEMENTATION OF Q-HAM
We choose the initial approximation
u0
(x, t)=u(x, 0)+tut
(x, 0)
= [ ]+ [ ]
2 2
sech sech tanh
x t x x
[ ](28)
and the linear operator:
L x t q
x t q
t
[ , ; ]
( , ; )
,
∅( ) =
∂ ∅
∂
2
2
(29)
with the property:
L[c0
+c1t]=0,(30)
where c0
and c1
are real constants.
We define the nonlinear operator by
N x t q
x t q
t
x t q
x x
x t q
∅( )

 
 =
∂ ∅
∂
−
∂ ∅
∂
−
∂
∂
∅( )
, ;
( , ; ) ( , ; )
[ , ; ]
2
2
4
4
2
2
3
(31)
According to the zero-order deformation Equation (2) and the mth-order deformation equation (7) with
R u
u
t
u
x x
u u u
m
m m
i
m
m i j
i
j i
−
− −
=
−
− − = −
( )=
∂
∂
−
∂
∂
−
∂
∂
∑ ∑
1
2
1
2
4
1
4
2
2 0
1
1 0

( j
j ) (32)
The solution of the mth-order deformation equation (7) for m≥1 takes the form
u x t k u x t h R u dt dt c c t
m m m m
, ,
( ) = ( )+ ( ) + +
− −
∫∫
1 1 0 1

(33)
where the coefficients c0
and c1
are determined by the initial conditions:
u x
u x
t
m
m
, ,
,
0 0
0
0
( ) =
∂ ( )
∂
= (34)
Obviously, we obtain
u x t ht x t x t
1
2 8 2 2
1
960 2
135 5 56 15 19 412
, [ ] ( ( ) [ ] ( )
( ) = − − + − +
Sech Cosh C
Cosh Cosh
Cosh Cosh Sinh
[ ] [ ]
[ ] [ ] [ ]
3 15 5
540 5 15 7 215
2
x x
t x x t x
− +
+ − +
6
6120 315 3 1836 3 95 5
108
3 3
t x t x t x t x
Sinh Sinh Sinh Sinh
[ ] [ ] [ ] [ ]
− − − +
t
t x t x
3
5 5 7
Sinh Sinh
[ ] [ ])
+


u x t h h n t x t x
2
2 8 2
1
960 2
135 5 56
15 19 4
, ( ) ( ( ) [ ]
(
( ) = − + [ ] − +
− +
Sech Cosh
1
12 3 15 5 540 5 15 7
215
2 2
t x x t x x
t
) [ ] [ ] [ ] [ ]
Cosh Cosh Cosh Cosh
Sin
− + +
− h
h Sinh Sinh Sinh
Sinh
[ ] [ ] [ ] [ ]
[
x t x t x t x
t x
+ − −
−
6120 315 3 1836 3
95 5
3 3
]
] [ ] [ ])
(
+ +
+ − [ ] + [ ]
108 5 5 7
1
160 2
1 2
3
10
t x t x
h ht x x
Sinh Sinh
Sech Cosh +
+ [ ]
( ) − +…
Sinh Cosh
2 1 6 2
3
x x
( [ ]
(34)
um
(x, t), (m=3,4,…) can be calculated similarly. Then, the series solution expression by q-HAM can be
written in the form:
n
M i
M
i
i
, ; ; , ; ; , ; ;
( ) ( ) = ( )





=
∑
≅ 0
1
(35)
Equation (35) is a family of approximation solutions to the problem (25) in terms of the convergence
parameters h and n. To find the valid region of h, the h curves given by the 3rd
order q-HAM approximation
at different values of x, t, and n are drawn in Figures 1-3. This figure shows the interval of h which the
value of U3
(x, t; n) is constant at certain x, t, and n, We choose the line segment nearly parallel to the
horizontal axis as a valid region of h which provides us with a simple way to adjust and control the
convergence region. Figures 4 and 5 show the comparison between U3
of q-HAM using different values
of n with the solution (27). The absolute errors of the 3rd
order solutions q-HAM approximate using
different values of n are shown in Figures 6 and 7.
IMPLEMENTATION OF MQ-HAM
To solve Equation (25) by mq-HAM, we construct system of differential equations as follows
ut
(x, t)=v(x, t),
v x t
u x t
x x
u x t
t ,
( , )
[ , ]
( ) =
∂
∂
+
∂
∂
( )
4
4
2
2
3
(36)
with initial approximations
u x t x v x t x x
0 0
2 2
, , , [ ]
( ) = [ ] ( ) = [ ]
sech sech tanh (37)
and the auxiliary linear operators
Figure 1: h curve for the (q-HAM; n=1) (HAM) approximation solution U3
(x, t; 1) of problem (25) at different values of x
and t

Lu x t
u x t
t
Lv x t
v x t
t
,
( , )
, ,
( , )
( ) =
∂
∂
( ) =
∂
∂
(38)
and
Au x t
u x t
x
m
m
−
−
( ) = −
∂
∂
1
4
1
4
,
( , )
Figure 2: h curve for the (q-HAM; n=50) approximation solution U3
(x, t; 50) of problem (25) at different values of x and t
and t
Figure 4: Comparison between U3
of q-HAM (n=1, 2, 5, 10, 20, 50, 100) with exact solution of Equation (25) at x=0 with
h=–1, h=–1.8, h=–4.5, (h=–8, h=–15.2, h=–37, h=–70), respectively

Bu x t
x
u x t u x t u x t
m i
m
m i j
i
j i j
− =
−
− − = −
( ) = −
∂
∂
( )
∑ ∑
1
2
2 0
1
1 0

, ( ( , ) , ( , ))
) (39)
From Equations (23) and (24) we obtain:
u x t h v x d
t
1 0 0
, ,
( ) = − ( )
( )
∫ τ τ (40)
(h=–1, h=–1.8, h=–4.5, h=–8, h=–15.2, h=–37, h=–70), respectively
Figure 6: The absolute error of U3
of q-HAM (n=1, 2, 5, 10, 20, 50, 100) for problem (25) at x=0 using (h=–1, h=–1.8,
h=–4.5, h=–8, h=–15.2, h=–37, h=–70), respectively
of q-HAM (n=1, 2, 5, 10, 20, 50, 100) for problem (25) at x=1 using (h=–1, h=–1.8,

v x t h
u x
x x
u x d
t
1 0
4
0
4
2
2 0
3
,
,
,
( ) = −
∂ ( )
∂
−
∂
∂
( )
( )








∫
τ
τ τ .
Now, form ≥2, we get
u x t n h u x t h v x d
m m
t
m
, , ,
( ) = +
( ) ( )+ − ( )
( )
− −
∫
1 0 1 τ τ (41)
v x t n h v x t h
u x
x x
u
m m
t
m
i
m
m
, ,
( , )
(
( ) = +
( ) ( )+ −
∂
∂
−
∂
∂
−
−
=
−
∫ ∑
1
0
4
1
4
2
2
0
1
τ
−
− −
=
−
∑ ( )








1
0
i
j
i
j i j
x u x u x d
( , ) , ( , ))
τ τ τ τ
And the following results are obtained
u x t ht x x
1 2
, [ ] [ ]
( ) = − Sech Tanh
v x t ht x x x
1
5 4
2 2
, ( [ ] [ ] [ ] )
( ) = −
Sech Sech Tanh
u x t
h t x x
h h n t x x
2
2 2 3
3 2
2 2
2
,
( [ ]) [ ]
( ) [ ] [ ]
( ) =
− +
− +
Cosh Sech
Sech Tanh
v x t
h t x x x
h h n t
2
2 2 3
11 2
2 2
2
,
( [ ]) [ ] [ ]
( ) ( [
( ) =
− +
+ +
Cosh Sech Tanh
Sech x
x x x
] [ ] [ ] )
5 4
2
− Sech Tanh
um
(x, t), (m=3, 4,…) can be calculated similarly. Then, the series solution expression by mq- HAM can
be written in the form:
n
M i
M
i
i
, ; ; , ; ; , ; ;
( ) ( ) = ( )





=
∑
≅ 0
1
(42)
parametershandn.Tofindthevalidregionofh,thehcurvesgivenbythe3rd
ordermq-HAMapproximation
at different values of x, t, and n are drawn in Figures 8-10. This figure shows the interval of h which
the value of U3
(x, t; n) is constant at certain x, t, and n. We choose the line segment nearly parallel to
the horizontal axis as a valid region of h which provides us with a simple way to adjust and control the
convergence region. Figure 11 shows the comparison between U3
of mq-HAM using different values
of n with the solution (27). The absolute errors of the 3th
order solutions mq-HAM approximate using
different values of n are shown in Figure 12. The results obtained by mq-HAM are more accurate than
Figure 8: h curve for the (mq-HAM; n=1) approximation solution U3

q-HAM at different values of x and n, so the results indicate that the speed of convergence for mq-HAM
with n1 is faster in comparison with n=1 (nHAM). The results show that the convergence region of
series solutions obtained by mq-HAM is increasing as q is decreased, as shown in Figures 11 and 12.
By increasing the number of iterations by mq-HAM, the series solution becomes more accurate, more
efficient and the interval of t (convergent region) increases, as shown in Figures 13-20.
Case 2. z=3
Consider the non-linear initial value problem:
u x t u x t x u x t u x t
ttt x
, , , ,
( )+ ( )− ( )
( ) + ( )
( ) =
2 6 0
2 4
(43)
and t
and t
(x, t) of mq-HAM (n=1, 2, 5, 10, 20, 50, 100) with exact solution of Equation (25) at
x=0 with (h=–1, h=–1.8, h=–4.5, h=–8, h=–15.2, h=–37, h=–70), respectively

Subject to the initial conditions
u x
x
u x
x
u x
x
t tt
, , , , ,
0
1
0
1
0
2
2 4 6
( ) = − ( ) = − ( ) = − (44)
The exact solution is
u x t
x t
,
( ) =
− +
1
2
(45)
of mq-HAM (n=1, 2, 5, 10, 20, 50, 100) for problem (25) at x=0 using (h=–1, h=–1.8,
Figure 13: The comparison between the U3
(x, t) of q-HAM (n=1), U3
(x, t) of mq-HAM (n=1), U5
(x, t) of mq-HAM (n=1),
and the exact solution of Equation (25) at h=–1 and x=0
and the exact solution of Equation (25) at h=–1 and x=1

This problem solved by HAM (q-HAM (n=1)) and nHAM (mq-HAM (n=1)),[7]
so we will solve it by
q-HAM and mq-HAM and compare the results.
IMPLEMENTATION OF Q-HAM
We choose the initial approximation
u x t
x
t
x
t
x
0 2 4
2
6
1
,
( ) = − − − (46)
(x, t) of q-HAM (n=100), U3
(x, t) of mq-HAM (n=100), U5
(x, t) of mq-HAM
(n=100), and the exact solution of Equation (25) at h=–70 and x=0
(x, t) of q-HAM (n=100), U3
(x, t) of mq-HAM (n=100), U5
(x, t) of mq-HAM
(n=100), and the exact solution of Equation (25) at h=–70 and x=1
Figure 17: The comparison between the absolute error of U3
(x, t) of q-HAM (n=1) and U3
(x, t) of mq-HAM (n=1) of
Equation (25) at h=–1, x=0 and-1≤t≤1

and the linear operator:
L x t q
x t q
t
[ , ; ]
( , ; )
∅( ) =
∂ ∅
∂
3
3
(47)
with the property:
L c c t c t
0 1 2
2
0
+ +

 
 = (48)
where c0
, c1
, and c2
are real constants.
(x, t) of q-HAM (n=100) and U3
Equation (25) at h=–70, x=0 and –1≤t≤1
(x, t) of mq-HAM (n=1) and U5
Equation (25) at h=–1, x=1 and –1.5≤t≤1.5
(x, t) of mq-HAM (n=100) and U5
(x, t) of mq-HAM (n=100)
of Equation (25) at h=–70, x=1 and –1.5≤t≤1.5

Next, we define the nonlinear operator by
N x t q
x t q
t
x t q
x
x x t q x
∅( )

 
 =
∂ ∅
∂
+
∂∅
∂
− ∅( ) + ∅
, ;
( , ; ) ( , ; )
[ , ; ] [
3
3
2
2 6 ,
, ; ]
t q
( ) 4
(49)
According to the zero-order deformation Equation (2) and the mth
-order deformation equation (7) with
R u
u
t
u
x
x u u u
m
m m m
i m i
m
m
i i
−
− − −
− −
−
− −
= =
( )=
∂
∂
+
∂
∂
− +
∑ ∑
1
3
1
3
1 1
1
1
1
0 0
2 6

i
i i j k j k
j
i
k
j
u u u
= =
− −
∑ ∑
0 0
(50)
The solution of the mth
-order deformation equation (7) for m≥1 becomes:
u x t k u x t h R u dt dt dt c c t c t
m m m m
, ,
( ) = ( )+ ( ) + + +
− −
∫∫∫
1 1 0 1 2
2

(51)
where the coefficients c0
, c1
and c2
are determined by the initial conditions:
u x
u x
t
u x
t
m
m m
, ,
,
,
,
0 0
0
0
0
0
2
2
( ) =
∂ ( )
∂
=
∂ ( )
∂
= (52)
We now successively obtain:
u x t
x
ht t t x t x t x
t x
1 24
3 8 7 2 6 4 5 6
2 12
1
2310
14 77 275 660
2310
, (
( ) = + + +
+ + 2
2310 2310 22 57 77 24
14 16 4 8 5 3 10 5
tx x t x x t x x
+ − − + − − +
( ) ( ))
u x t
x
hnt t t x t x t x
t x
2 24
3 8 7 2 6 4 5 6
2 12
1
2310
14 77 275 660
2310
, (
( ) = + + + +
+
+ + − − + − − +
−
2310 2310 22 57 77 24
1
244432
14 16 4 8 5 3 10 5
tx x t x x t x x
( ) ( ))
1
18800
519792 5197920 30603300
1272889
42
2 3 17 16 2 15 4
x
h t t t x t x
( + + +
8
80 10475665200
14 6 5 24
t x t x
+ −…
um
(x, t), (m=3,4,…) can be calculated similarly. Then, the series solution expression by q- HAM can be
written in the form:
n
M i
M
i
i
, ; ; , ; ; , ; ;
( ) ( ) = ( )





=
∑
≅ 0
1
(53)
Equation(53)isafamilyofapproximationsolutionstotheproblem(43)intermsoftheconvergenceparameters
h and n. To find the valid region of h, the h curves given by the 5th
order q-HAM approximation at different
values of x, t, and n are drawn in Figures 21-23. This figure shows the interval of h which the value of U5
(x, t; n) is constant at certain x, t and n. We choose the line segment nearly parallel to the horizontal axis as a
valid region of h which provides us with a simple way to adjust and control the convergence region. Figure 24
shows the comparison between U5
of q-HAM using different values of n with the solution 45. The absolute
errors of the 5th
order solutions q-HAM approximate using different values of n are shown in Figure 25.
IMPLEMENTATION OF MQ-HAM
To solve Equation (43) by mq-HAM, we construct system of differential equations as follows
ut
(x, t)=v(x, t),
vt
(x, t)=w(x, t)(54)
With initial approximations
u x t
x
v x t
x
w x t
x
0 2 0 4 0 6
1 1 2
, , , , ,
( ) = − ( ) = − ( ) = − (55)

And the auxiliary linear operators
Lu x t
u x t
t
Lv x t
v x t
t
Lw x t
w x t
t
,
( , )
, ,
( , )
, ,
( , )
( ) =
∂
∂
( ) =
∂
∂
( ) =
∂
∂
(56)
And
Au x t
u x t
x
m
m
−
−
( ) =
∂
∂
1
1
,
( , )
Bu x t x u u u u
m i
m
i m i i
m
m i j
i
i j k
j
− =
−
− − =
−
− − = − =
( ) = − +
∑ ∑ ∑
1 0
1
1 0
1
1 0 0
2 6

, ∑
∑ −
u u
k j k (57)
Figure 21: h curve for the (q-HAM; n=1) (HAM) approximation solution U5
(x, t; 1) of problem (43) at different values of
x and t
and t
and t

From Equations (23) and (24) we obtain
u x t h v x d
t
1 0 0
, ,
( ) = − ( )
( )
∫ τ τ
v x t h w x d
t
1 0 0
, ,
( ) = − ( )
( )
∫ τ τ (58)
w x t h
u x
x
x u x u x d
t
1
0
0
0
2
0
4
2 6
,
,
, ,
( ) =
∂ ( )
∂
− ( )
( ) + ( )
( )






∫
τ
τ τ τ
For m≥2,
u x t n h u x t h v x d
m m
t
m
, , ,
( ) = +
( ) ( )+ − ( )
( )
− −
∫
1 0 1 τ τ
v x t n h v x t h w x d
m m
t
m
, , ,
( ) = +
( ) ( )+ − ( )
( )
− −
∫
1 0 1 τ τ (59)
w x t n h w x t h
u x t
x
x u u
m m
t
m
i
m
i m i
, ,
( , )
( ) = +
( ) ( )+
∂
∂
− +
−
−
=
−
− −
∫
∑
1 0
1
0
1
1
2 6
6
0
1
1
0 0
i
m
m i
j
i
i j
k
j
k j k
u u u u d
=
−
− −
=
−
=
−
∑ ∑ ∑







 τ
The following results are obtained
(h=–1, h=–1.97, h=–4.83, h=–8.45, h=–18.3, h=–44.75, h=–86), respectively
of q-HAM (n=1, 2, 5, 10, 20, 50, 100) for problem (43) at x=4 using h=–1, h=–1.97,
h=–4.83, h=–8.45, h=–18.3, h=–44.75, h=–86), respectively

u x t
ht
x
1 4
,
( ) =
u x t
h t
x
h h n t
x
2
2 2
6 4
,
( )
( ) = − +
+
u x t h
h t
x
h t
x
hnt
x
h n
h t
x
h h n t
x
3
2 3
8
2 2
6
2
6
2 2
6 4
, ( ) ( )(
( )
)
( ) = − − + + − +
+
um
(x, t), (m=4, 5,…) can be calculated similarly. Then, the series solution expression by mq-HAM can
be written in the form:
n
M i
M
i
i
, ; ; , ; ; , ; ;
( ) ( ) = ( )





=
∑
≅ 0
1
(60)
parametershandn.Tofindthevalidregionofh,thehcurvesgivenbythe5th
ordermq-HAMapproximation
at different values of x, t, and n are drawn in Figures 26-28. This figure shows the interval of h which
the value of U5
(x, t; n) is constant at certain x, t, and n. We choose the line segment nearly parallel to
the horizontal axis as a valid region of h which provides us with a simple way to adjust and control the
convergence region. Figure 29 shows the comparison between U5
of mq-HAM using different values
of n with the solution (45). The absolute errors of the 5th
order solutions mq-HAM approximate using
different values of n are shown in Figure 30. The results obtained by mq-HAM are more accurate than
q-HAM at different values of x and n, so the results indicate that the speed of convergence for mq-HAM
with n1 is faster in comparison with n=1. (nHAM). The results show that the convergence region
of series solutions obtained by mq-HAM is increasing as q is decreased, as shown in Figures 29-36.
and t

By increasing the number of iterations by mq-HAM, the series solution becomes more accurate, more
efficient and the interval of t (convergent region) increases, as shown in Figures 31-36.
and t
of mq-HAM (n=1, 2, 5, 10, 20, 50, 100) with exact solution of Equation (43) at x=4
with (h=–1, h=–1.97, h=–4.83, h=–9.45, h=–18.3, h=–44.75, h=–86), respectively
of mq-HAM (n=1, 2, 5, 10, 20, 50, 100) for problem (43) at x=4, –20≤t≤5 using h=–1,
h=–1.97, h=–4.83, h=–9.45, h=–18.3, h=–44.75, h=–86), respectively

U7
(x, t) of mq-HAM (n=1), and the exact solution of Equation (43) at h=–1 and x=4
(x, t) of q-HAM (n=20), U3
(x, t) of mq-HAM
(n=20), U7
(x, t) of mq-HAM (n=20), and the exact solution of Equation (43) at h=–18.3 and x=4
(x, t) of q-HAM (n=100), U3
(x, t) of mq-HAM (n=100), U5
(x, t) of mq-HAM
(n=100), U7
(x, t) of mq-HAM (n=100), and the exact solution of (43) at h=–86 and x=4
(x, t) of
mq-HAM (n=1), and U7
(x, t) of mq-HAM (n=1) of Equation (43) at h=–1, x=4 and –15≤t≤2

CONCLUSION
A mq-HAM was proposed. This method provides an approximate solution by rewriting the nth-order
non-linear differential equation in the form of n first-order differential equations. The solution of these n
differential equations is obtained as a power series solution. It was shown from the illustrative examples
that the mq-HAM improves the performance of q-HAM and nHAM.
REFERENCES
1. Liao SJ. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis. China:
Shanghai Jiao Tong University; 1992.
2. Abbasbandy S. The application of homotopy analysis method to nonlinear equations arisingin heat transfer. Phys Lett A
2006;360:109-13.
3. Abbasbandy S. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV
equation. Phys Lett A 2007;361:478-83.
4. Bataineh S, Noorani MS, Hashim I. Solutions of time-dependent emde-fowler type equations by homotopy analysis
method. Phys Lett A 2007;371:72-82.
5. Bataineh S, Noorani MS, Hashim I. The homotopy analysis method for Cauchy reaction-diffusion problems. Phys Lett
A 2008;372:613-8.
6. Bataineh S, Noorani MS, Hashim I. Approximate analytical solutions of systems of PDEs by homotopy analysis method.
Comput Math Appl 2008;55:2913-23.
7. Hassan HN, El-Tawil MA. A new technique of using homotopy analysis method for solving high-order nonlinear
differential equations. Math Methods Appl Sci 2010;34:728-42.
8. Hassan HN, El-Tawil MA. A new technique of using homotopy analysis method for second order nonlinear differential
equations. Appl Math Comput 2012;219:708-28.
9. Saberi-Nik H, Golchaman M. The Homotopy Analysis Method for Solvingdiscontinued Problems Arising in
Nanotechnology. Italy: World Academy of Science, Engineering and Technology; 2011. p. 76.
(x, t) of q-HAM (n=20), U3
(x, t)
of mq-HAM (n=20), and U7
(x, t) of mq-HAM (n=20) of Equation (43) at h=–18.3, x=4 and –15≤t≤2
(x, t) of q-HAM (n=100), U3
U5
(x, t) of mq-HAM (n=100), and U7
(x, t) of mq-HAM (n=100) of Equation (43) at h=–86, x=4 and –15≤t≤2

10. Hayat T, Khan M. Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dynam
2005;42:395-405.
11. Huseen SN, Mkharrib HA. On a new modification of homotopy analysis method for solving nonlinear nonhomogeneous
differential equations. Asian J Fuzzy Appl Math 2018;6:12-35.
12. El-Tawil MA, Huseen SN. The q-homotopy analysis method (q-HAM). Int J Appl Math Mech 2012;8:51-75.
13. El-Tawil MA, Huseen SN. On convergence of the q-homotopy analysis method. Int J Contemp Math Sci 2013;8:481-97.
14. Huseen SN, Grace SR, El-Tawil MA. The optimal q-homotopy analysis method (Oq-HAM). Int J Comput Technol
2013;11:2859-66.
15. Huseen SN. Solving the K(2,2) equation by means of the q-homotopy analysis method (q-HAM). Int J Innov Sci Eng
Technol 2015;2:805-17.
16. Huseen SN. Application of optimal q-homotopy analysis method to second order initial and boundary value problems.
Int J Sci Innov Math Res 2015;3:18-24.
17. Huseen SN. Series solutions of fractional initial-value problems by q-homotopy analysis method. Int J Innov Sci Eng
Technol 2016;3:27-41.
18. Huseen SN. A numerical study of one-dimensional hyperbolic telegraph equation. J Math Syst Sci 2017;7:62-72.
19. Huseen SN,Ayay NM.Anew technique of the q-homotopy analysis method for solving non-linear initial value problems.
J Prog Res Math 2018;14:2292-307.
20. Huseen SN, Shlaka RA. The regularization q-homotopy analysis method for (1 and 2)-dimensional non-linear first kind
Fredholm integral equations. J Prog Res Math 2019;15:2721-43.
21. Akinyemi L. q-homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg-deVries and
Sawada-Kotera equations. Comp Appl Math 2019;38:191.
22. Akinyemi L, Huseen SN. A powerful approach to study the new modified coupled Korteweg-de Vries system. Math
Comput Simul 2020;177:556-67.
23. Huseen SN, Grace SR.Approximate solutions of nonlinear partial differential equations by modified q homotopy analysis
method. J Appl Math 2013;2013:9.

Solving High-order Non-linear Partial Differential Equations by Modified q-Homotopy Analysis Method

More Related Content

What's hot (18)

Similar to Solving High-order Non-linear Partial Differential Equations by Modified q-Homotopy Analysis Method (20)

More from BRNSS Publication Hub (20)

Recently uploaded (20)

Solving High-order Non-linear Partial Differential Equations by Modified q-Homotopy Analysis Method