# Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

Authors: Shaheed N. Huseen; Said R. Grace
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting the nth-order nonlinear 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. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.


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## Body

## 1. Introduction

Homotopy analysis method (HAM) initially proposed by Liao in his Ph.D. thesis [1] is a powerful method to solve nonlinear problems. In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering [2–17]. 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, a valid region of h can be studied to gain a convergent series solution. More recently, a powerful modification of HAM was proposed [18–20]. Hassan and El-Tawil [21, 22] presented a new technique of using homotopy analysis method for solving nonlinear initial value problems (nHAM). El-Tawil and Huseen [23, 24] established a method, namely, q-homotopy analysis method, (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) and the standard homotopy analysis method (HAM) can be reached. In this paper, we present the modification of q-homotopy analysis method (mq-HAM) for solving nonlinear problems by transforming the nth-order nonlinear differential equation to a system of n first-order equations. we note that the nHAM is a special case of mq-HAM (mq-HAM; n=1).


## 2. Analysis of theq-Homotopy Analysis Method (q-HAM)

Consider the following nonlinear partial differential equation:(1)N[u(x,t)]=0,
where N is a nonlinear operator, (x,t) denotes independent variables, and u(x,t) is an unknown function. Let us construct the so-called zero-order deformation equation as follows:
(2)(1-nq)L[∅(x,t;q)-u0(x,t)]=qhH(x,t)N[∅(x,t;q)],
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, and H(x,t) denotes a non-zero auxiliary function. It is obvious that when q=0 and q=1/n, (2) becomes
(3)∅(x,t;0)=u0(x,t),∅(x,t;1n)=u(x,t),
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 u(x,t). We may choose u0(x,t),L,h, and H(x,t) and assume that all of them can be properly chosen so that the solution ∅(x,t;q) of (2) exists for q∈[0,1/n].Now, by expanding∅(x,t;q) in Taylor series, we have
(4)∅(x,t;q)=u0(x,t)+∑m=1+∞um(x,t)qm,
where
(5)um(x,t)=1m!∂m∅(x,t;q)∂qm|q=0.
Next, we assume that h,H(x,t),u0(x,t), and L are properly chosen such that the series (4) converges at q=1/n and that
(6)u(x,t)=∅(x,t;1n)=u0(x,t)+∑m=1+∞um(x,t)(1n)m.
Let
(7)ur(x,t)={u0(x,t),u1(x,t),u2(x,t),…,ur(x,t)}.
Differentiating equation (2) for 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 as follows:
(8)L[um(x,t)-kmum-1(x,t)]=hH(x,t)Rm(um-1⃑(x,t)),
where
(9)Rm(um-1⃑(x,t))=1(m-1)!∂m-1(N[∅(x,t;q)]-f(x,t))∂qm-1|q=0,km={0m≤1,notherwise.
It should be emphasized that um(x,t) for m≥1 is governed by the linear equation (8) 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 (2), standard HAM, can be reached.Theq-homotopy analysis method (q-HAM) can be reformatted as follows.We rewrite the nonlinear partial differential equation (1) in the following form:
(10)Lu(x,t)+Au(x,t)+Bu(x,t)=0,u(x,0)=f0(x),∂u(x,t)∂t|t=0=f1(x),⋮∂z-1u(x,t)∂z-1|t=0=fz-1(x),
where L=∂z/∂tz, z=1,2,… is the highest partial derivative with respect to t,A is a linear term, and B is a nonlinear term. The so-called zero-order deformation equation (2) becomes
(11)(1-nq)L[∅(x,t;q)-u0(x,t)]=qhH(x,t)(Lu(x,t)+Au(x,t)+Bu(x,t)),
we have the following mth order deformation equation:
(12)L[um(x,t)-kmum-1(x,t)]=hH(x,t)(Lum-1(x,t)+Aum-1(x,t)+B(um-1⃑(x,t))).
Hence,
(13)um(x,t)=kmum-1(x,t)+hL-1[H(x,t)(Lum-1(x,t)+Aum-1(x,t)+B(um-1⃑(x,t)))sssssssssssssssssszzzz+B(um-1⃑(x,t)))].
Now, the inverse operator L-1 is an integral operator which is given by
(14)L-1(·)=∫∫⋯∫(·)dtdt⋯dt︸ztimes+c1tz-1+c2tz-2+⋯+cz,
where c1,c2,…,cz are integral constants.To solve (10) by means of q-HAM, we choose the following initial approximation:
(15)u0(x,t)=f0(x)+f1(x)t+f2(x)t22!+⋯+fz-1(x)tz-1(z-1)!.
Let H(x,t)=1, by means of (14) and (15); then (13) becomes
(16)um(x,t)=kmum-1(x,t)+h∫0t∫0t⋯∫0t(∂zum-1(x,τ)∂τz+Aum-1(x,τ)ssssssssssssssssssszzzz+B(um-1⃑(x,τ)))dτdτ⋯dτ︸ztimes.
Now from ∫0t∫0t⋯∫0t(∂zum-1(x,τ)/∂τz)dτdτ⋯dτ︸ztimes, 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):
(17)um(x,t)=kmum-1(x,t)+h∫0t∫0t⋯∫0t∂zum-1(x,τ)∂τzdτdτ⋯dτ︸ztimes+h∫0t∫0t⋯∫0t(Aum-1(x,τ)B(um-1⃑(x,τ))sssssssssssssssssssszzz+B(um-1⃑(x,τ)))dτdτ⋯dτ︸ztimes=kmum-1(x,t)+hum-1(x,t)-h(um-1(x,0)+t∂um-1(x,0)∂tssssssssszzzs+⋯+tz-1(z-1)!∂z-1um-1(x,0)∂tz-1)+h∫0t∫0t⋯∫0t(Aum-1(x,τ)+B(um-1⃑(x,τ))ssssssssssszzzsssssssssssss+B(um-1⃑(x,τ)))dτdτ⋯dτ︸ztimes.
Now, for m=1, km=0, and
(18)u0(x,0)+t∂u0(x,0)∂t+t22!∂2u0(x,0)∂t2+⋯+tz-1(z-1)!∂z-1u0(x,0)∂tz-1=f0(x)+f1(x)t+f2(x)t22!+⋯+fz-1(x)tz-1(z-1)!=u0(x,t).
Substituting this equality into (17), we obtain(19)u1(x,t)=h∫0t∫0t⋯∫0t(Au0(x,τ)+B(u0(x,τ)))dτdτ⋯dτ︸ztimes.
For m>1, km=n, and
(20)um(x,0)=0,∂um(x,0)∂t=0,∂2um(x,0)∂t2=0,…,∂z-1um(x,0)∂tz-1=0.
Substituting this equality into (17), we obtain
(21)um(x,t)=(n+h)um-1(x,t)+h∫0t∫0t⋯∫0t(Aum-1(x,τ)+B(um-1⃑(x,τ))ssssssssssssssssssssssss+B(um-1⃑(x,τ)))dτdτ⋯dτ︸ztimes.
The standard q-HAM is powerful when z=1, and the series solution expression by q-HAM can be written in the following form:
(22)u(x,t;n;h)≌UM(x,t;n;h)=∑i=0Mui(x,t;n;h)(1n)i.
But when z≥2, there are too many additional terms where harder and more time consuming computations are performed. So, the closed form solution needs more numbers of iteration.


## 3. The Proposed Modifiedq-Homotopy Analysis Method (mq-HAM)

Whenz≥2, we rewrite (1) as in the following system of first-order differential equations:
(23)ut=u1,u1t=u2,⋮u{z-1}t=-Au(x,t)-Bu(x,t).
Set the initial approximation
(24)u0(x,t)=f0(x),u10(x,t)=f1(x),⋮u{z-1}0(x,t)=fz-1(x).
Using the iteration formulas (19) and (21) as follows:
(25)u1(x,t)=h∫0t(-u10(x,τ))dτ,u11(x,t)=h∫0t(-u20(x,τ))dτ,⋮u{z-1}1(x,t)=h∫0t(Au0(x,τ)+B(u0(x,τ)))dτ.Form>1, km=n, and
(26)um(x,0)=0,u1m(x,0)=0,u2m(x,0)=0,…,u{z-1}m(x,0)=0.
Substituting in (17), we obtain
(27)um(x,t)=(n+h)um-1(x,t)+h∫0t(-u1m-1(x,τ))dτ,u1m(x,t)=(n+h)u1m-1(x,t)+h∫0t(-u2m-1(x,τ))dτ,⋮u{z-1}m(x,t)=(n+h)u{z-1}m-1(x,t)+h∫0t(Aum-1(x,τ)+B(um-1(x,τ)))dτ.
It should be noted that the case of n=1 in (27), the nHAM, can be reached.To illustrate the effectiveness of the proposed mq-HAM, comparison between mq-HAM and the nHAM are illustrated by the following examples.


## 4. Illustrative Examples

Example 1.
Consider the following nonlinear sine-Gordon equation:(28)utt-uxx+sinu=0,
subject to the following initial conditions:
(29)u(x,0)=0,ut(x,0)=4sechx.
The exact solution is
(30)u(x,t)=4tan-1(tsechx).
In order to prevent suffering from the strongly nonlinear term sinu in the frame of q-HAM, we can use Taylor series expansion of sinu as follows:
(31)sinu=u-u36+u5120,
Then, (28) becomes
(32)utt-uxx+u-u36+u5120=0.
In order to solve (28) by mq-HAM, we construct system of differential equations as follows:
(33)ut(x,t)=v(x,t),vt(x,t)=∂2u(x,t)∂x2-u+u36-u5120,
with the following initial approximations:
(34)u0(x,t)=0,v0(x,t)=4sechx,
and the following auxiliary linear operators:
(35)Lu(x,t)=∂u(x,t)∂t,Lv(x,t)=∂v(x,t)∂t,Aum-1(x,t)=-∂2um-1(x,t)∂x2+um-1(x,t),Bum-1⃑(x,t)=-16∑j=0m-1um-1-j∑i=0juiuj-i+1120∑j=0m-1um-1-j∑i=0juj-i∑k=0iui-k∑l=0kuluk-l.
From (25) and (27), we obtain
(36)u1(x,t)=h∫0t(-v0(x,τ))dτ,v1(x,t)=h∫0t(-∂2u0∂x2+u0-u036+u05120)dτ.
Now, for m≥2, we get
(37)um(x,t)=(n+h)um-1(x,t)+h∫0t(-vm-1(x,τ))dτ,vm(x,t)=(n+h)vm-1(x,t)+h∫0t(Aum-1(x,τ)+B(um-1(x,τ)))dτ.
And the following results are obtained:
(38)u1(x,t)=-4htsechx,v1(x,t)=0,u2(x,t)=-4h(h+n)tsechx,v2(x,t)=-4h2t2sech3x,u3(x,t)=-4h(h+n)2tsechx+43h3t3sech3x,um(x,t), (m=4,5,…) can be calculated similarly. Then, the series solution expression by mq-HAM can be written in the following form:
(39)u(x,t;n;h)≌UM(x,t;n;h)=∑i=0Mui(x,t;n;h)(1n)i.
Equation (39) is a family of approximation solutions to the problem (28) in terms of the convergence parameters h and n. To find the valid region of h, the h-curves given by the 6th-order nHAM (mq-HAM; n=1) approximation and the 6th-order mq-HAM (n=13) approximation at different values of x,t are drawn in Figures 1 and 2, respectively, and these figures show the interval of h in which the value of U6 is constant at certain x,t, and n; we chose the horizontal line parallel to x- axis (h) as a valid region which provides us with a simple way to adjust and control the convergence region. Figure 3 shows the comparison between U6 of  nHAM and U6 of mq-HAM using different values of n with the solution (30). The absolute errors of the 6th-order solutions nHAM approximate and the 6th-order solutions mq-HAM approximate using different values of n are shown in Figure 4. The results obtained by mq-HAM indicate that the speed of convergence for mq-HAM with n>1 is faster in comparison to 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 3 and 4.Figure 1
h-curve for the nHAM (mq-HAM; n=1) approximation solution U6(x,t) of problem (28) at different values of x and t.Figure 2
h-curve for the (mq-HAM; n=13) approximation solution U6(x,t;13) of problem (28) at different values of x and t.Figure 3
Comparison betweenU6 of nHAM (mq-HAM; n=1) and U6mq-HAM (n=5.5,13,30,75) with exact solution of (28) at x=1 with (h=-1, h=-4.9, h=-10.8, h=-23.15, h=-49.25), respectively.Figure 4
The absolute error ofU6 of  nHAM (mq-HAM; n=1) and U6 mq-HAM (n=5.5,13,30,75) for problem (28) at x=1 using (h=-1, h=-4.9, h=-10.8, h=-23.15, h=-49.25), respectively.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 5, 6, 7, and 8.Figure 5
The comparison between theU3, U6, and U10 of nHAM (mq-HAM; n=1) and the exact solution of (28) at h=-1 and x=1.Figure 6
The comparison between theU3, U6, and U10 of  mq-HAM (n=75) and the exact solution of (28) at h=-49.25 and x=1.Figure 7
The comparison between the absolute error ofU3, U6, and U10 of nHAM (mq-HAM; n=1) of (28) at h=-1, x=1, and 0≤t≤1.5.Figure 8
The comparison between the absolute error ofU3, U6, and U10 of mq-HAM (n=75) of (28) at h=-49.25, x=1, and 0≤t≤1.5.Example 2.
Consider the following Klein-Gordon equation:(40)utt-uxx+34u-32u3=0,
subject to the following initial conditions:
(41)u(x,0)=-sechx,ut(x,0)=12sechxtanhx.
The exact solution is
(42)u(x,t)=-sech(x+t2).
In order to solve (40) by mq-HAM, we construct system of differential equations as follows:
(43)ut(x,t)=v(x,t),vt(x,t)=∂2u(x,t)∂x2-34u+32u3,
with the following initial approximations:
(44)u0(x,t)=-sechx,v0(x,t)=12sechxtanhx,
and the following auxiliary linear operators:
(45)Lu(x,t)=∂u(x,t)∂t,Lv(x,t)=∂v(x,t)∂t,Aum-1(x,t)=-∂2um-1(x,t)∂x2+34um-1(x,t),Bum-1⃑(x,t)=-32∑j=0m-1um-1-j∑i=0juiuj-i.
From (25) and (27), we obtain
(46)u1(x,t)=h∫0t(-v0(x,τ))dτ,v1(x,t)=h∫0t(-∂2u0∂x2+34u0-32u03)dτ.
For m≥2, we get
(47)um(x,t)=(n+h)um-1(x,t)+h∫0t(-vm-1(x,τ))dτ,vm(x,t)=(n+h)vm-1(x,t)+h∫0t(-∂2um-1(x,t)∂x2+34um-1(x,t)∑j=0m-1um-1-jsssssssssss-32∑j=0m-1um-1-j∑i=0juiuj-i)dτ.The following results are obtained:(48)u1(x,t)=-12htsechxtanhx,v1(x,t)=ht(-3sechx4+sech3x2+sechxtanh2x),u2(x,t)=h(316ht2sech3x-116ht2cosh(2x)sech3x)-12h(h+n)tsechxtanhx,um(x,t), (m=3,4,…) can be calculated similarly. Then, the series solution expression by mq-HAM can be written in the following form:
(49)u(x,t;n;h)≌UM(x,t;n;h)=∑i=0Mui(x,t;n;h)(1n)i.Equation (49) is a family of approximation solutions to the problem (40) in terms of the convergence parameters h and n. To find the valid region of h, the h-curves given by the 6th-order nHAM (mq-HAM;n=1) approximation and the 6th-order mq-HAM (n=100) approximation at different values of x,t are drawn in Figures 9 and 10; these figures show the interval of h in which the value of U6 is constant at certain x,t, and n; we chose the horizontal line parallel to x-axis(h) as a valid region which provides us with a simple way to adjust and control the convergence region. Figure 11 shows the comparison between U6 of nHAM and U6 of mq-HAM using different values of n with the solution (42). The absolute errors of the 6th-order solutions nHAM approximate and the 6th-order solutions mq-HAM approximate using different values of n are shown in Figure 12. The results obtained by mq-HAM indicate that the speed of convergence for mq-HAM with n>1 is faster in comparison to 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.Figure 9
h-curve for the nHAM (mq-HAM; n=1) approximation solution U6(x,t) of problem (40) at different values of x and t.Figure 10
h-curve for the mq-HAM (n=100) approximation solution U6(x,t;100) of problem (40) at different values of x and t.Figure 11
Comparison betweenU6 of  nHAM (mq-HAM; n=1) and U6 of  mq-HAM (n=5,20,50,100) with exact solution of (40) at x=1 with (h=-1,h=-4.85,h=-18.55,h=-43.11,h=-79.5), respectively.Figure 12
The absolute error ofU6 of nHAM (mq-HAM; n=1) and U6 of mq-HAM (n=5,20,50,100) for problem (40) at x=1 using (h=-1,h=-4.85,h=-18.55,h=-43.11,h=-79.5), respectively.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, 14, 15, and 16.Figure 13
The comparison between theU3, U6 of nHAM (mq-HAM; n=1), and the exact solution of (40) at h=-1 and x=1.Figure 14
The comparison between theU3, U6 of mq-HAM (n=100), and the exact solution of (40) at h=-79.5 and x=1.Figure 15
The comparison between the absolute error ofU3 and U6 of  nHAM (mq-HAM; n=1) of (40) at h=-1, x=1, and 0≤t≤3.5.Figure 16
The comparison between the absolute error ofU3 and U6 of mq-HAM(n=100) of (40) at h=-79.5, x=1, and 0≤t≤3.5.Figure17 shows that the convergence of the series solutions obtained by the 3rd-order mq-HAM (n=100) is faster than that of the series solutions obtained by the 6th order nHAM. This fact shows the importance of the convergence parameters n in the mq-HAM.Figure 17
The comparison between theU3 of mq-HAM (n=100), U6 of  nHAM (mq-HAM; n=1), and the exact solution of (40) at (h=-79.5,h=-1) and x=1.


## 5. Conclusion

In this paper, a modifiedq-homotopy analysis method was proposed (mq-HAM). This method provides an approximate solution by rewriting the nth-order nonlinear differential equations in the form of system of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution, which converges to a closed form solution. The mq-HAM contains two auxiliary parameters n and h such that the case of n=1 (mq-HAM; n=1); the nHAM which is proposed in [21, 22] can be reached. In general, it was noticed from the illustrative examples that the convergence of mq-HAM is faster than that of nHAM.



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