A modified q-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.

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 the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M29"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Homotopy Analysis Method (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M30"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-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.

The q-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.

When z≥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.

For m>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 ExamplesExample 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.

h-curve for the nHAM (mq-HAM; n=1) approximation solution U6(x,t) of problem (28) at different values of x and t.

h-curve for the (mq-HAM; n=13) approximation solution U6(x,t;13) of problem (28) at different values of x and t.

Comparison between U6 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.

The absolute error of U6 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.

The comparison between the U3, U6, and U10 of nHAM (mq-HAM; n=1) and the exact solution of (28) at h=-1 and x=1.

The comparison between the U3, U6, and U10 of mq-HAM (n=75) and the exact solution of (28) at h=-49.25 and x=1.

The comparison between the absolute error of U3, U6, and U10 of nHAM (mq-HAM; n=1) of (28) at h=-1, x=1, and 0≤t≤1.5.

The comparison between the absolute error of U3, 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.

h-curve for the nHAM (mq-HAM; n=1) approximation solution U6(x,t) of problem (40) at different values of x and t.

h-curve for the mq-HAM (n=100) approximation solution U6(x,t;100) of problem (40) at different values of x and t.

Comparison between U6 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.

The absolute error of U6 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.

The comparison between the U3, U6 of nHAM (mq-HAM; n=1), and the exact solution of (40) at h=-1 and x=1.

The comparison between the U3, U6 of mq-HAM (n=100), and the exact solution of (40) at h=-79.5 and x=1.

The comparison between the absolute error of U3 and U6 of nHAM (mq-HAM; n=1) of (40) at h=-1, x=1, and 0≤t≤3.5.

The comparison between the absolute error of U3 and U6 of mq-HAM(n=100) of (40) at h=-79.5, x=1, and 0≤t≤3.5.

Figure 17 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.

The comparison between the U3 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 modified q-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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