Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method

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 nHAMwhich 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.


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][3][4][5][6][7][8][9][10][11][12][13][14][15][16][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][19][20].Hassan and El-Tawil [21,22] presented a new technique of using homotopy analysis method for solving nonlinear initial value problems (HAM).El-Tawil and Huseen [23,24] established a method, namely, -homotopy analysis method, (-HAM) which is a more general method of HAM, The -HAM contains an auxiliary parameter  as well as ℎ such that the case of  = 1 (-HAM;  = 1) and the standard homotopy analysis method (HAM) can be reached.In this paper, we present the modification of -homotopy analysis method (m-HAM) for solving nonlinear problems by transforming the th-order nonlinear differential equation to a system of  first-order equations.we note that the HAM is a special case of m-HAM (m-HAM;  = 1).
To solve (10) by means of -HAM, we choose the following initial approximation: Let (, ) = 1, by means of ( 14) and (15); then (13) becomes , 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): Now, for  = 1,   = 0, and Substituting this equality into (17), we obtain For  > 1,   = , and Substituting this equality into (17), we obtain The standard -HAM is powerful when  = 1, and the series solution expression by -HAM can be written in the following form: But when  ≥ 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.

The Proposed Modified 𝑞-Homotopy Analysis Method (m𝑞-HAM)
When  ≥ 2, we rewrite (1) as in the following system of firstorder differential equations: Set the initial approximation Using the iteration formulas ( 19) and ( 21) as follows: For  > 1,   = , and Substituting in (17), we obtain It should be noted that the case of  = 1 in (27), the HAM, can be reached.
To illustrate the effectiveness of the proposed m-HAM, comparison between m-HAM and the HAM are illustrated by the following examples.

Illustrative Examples
Example 1.Consider the following nonlinear sine-Gordon equation: subject to the following initial conditions: The exact solution is In order to prevent suffering from the strongly nonlinear term sin  in the frame of -HAM, we can use Taylor series expansion of sin  as follows: Then, (28) becomes In order to solve (28) by m-HAM, we construct system of differential equations as follows: with the following initial approximations: and the following auxiliary linear operators: From ( 25) and ( 27), we obtain Now, for  ≥ 2, we get And the following results are obtained: (, ), ( = 4, 5, . ..) can be calculated similarly.Then, the series solution expression by m-HAM can be written in the following form: Equation ( 39) is a family of approximation solutions to the problem (28) in terms of the convergence parameters ℎ and .To find the valid region of ℎ, the ℎ-curves given by the 6th-order HAM (m-HAM;  = 1) approximation and the 6th-order m-HAM ( = 13) approximation at different values of ,  are drawn in Figures 1 and 2, respectively, and these figures show the interval of ℎ in which the value of  6 is constant at certain , , and ; we chose the horizontal line parallel to -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  6 of HAM and  6 of m-HAM using different values of  with the solution (30).The absolute errors of the 6th-order solutions HAM approximate and the 6th-order solutions m-HAM approximate using different values of  are shown in Figure 4.
The results obtained by m-HAM indicate that the speed of convergence for m-HAM with  > 1 is faster in comparison to  = 1 (HAM).The results show that the convergence region of series solutions obtained by m-HAM is increasing as  is decreased as shown in Figures 3 and 4.
By increasing the number of iterations by m-HAM, the series solution becomes more accurate, more efficient, and The exact solution is In order to solve (40) by m-HAM, we construct system of differential equations as follows: with the following initial approximations: and the following auxiliary linear operators: From ( 25) and (27), we obtain For  ≥ 2, we get The following results are obtained: Equation ( 49) is a family of approximation solutions to the problem (40) in terms of the convergence parameters ℎ and .To find the valid region of ℎ, the ℎ-curves given by the 6th-order HAM (m-HAM;  = 1) approximation and the 6th-order m-HAM ( = 100) approximation at different values of ,  are drawn in Figures 9 and 10; these figures show the interval of ℎ in which the value of  6 is constant at certain , , and ; we chose the horizontal line parallel to -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  6 of HAM and  6 of m-HAM using different values of  with the solution (42).The absolute errors of the 6th-order solutions HAM approximate and the 6thorder solutions m-HAM approximate using different values of  are shown in Figure 12.The results obtained by m-HAM indicate that the speed of convergence for m-HAM with  > 1 is faster in comparison to  = 1 (HAM).The results show that the convergence region of series solutions obtained by m-HAM is increasing as  is decreased as shown in Figures 11 and 12.
By increasing the number of iterations by m-HAM, the series solution becomes more accurate, more efficient, and the interval of  (convergent region) increases as shown in Figures 13,14,15,and 16. Figure 17 shows that the convergence of the series solutions obtained by the 3rd-order m-HAM ( = 100) is faster than that of the series solutions obtained by the 6th order HAM.This fact shows the importance of the convergence parameters  in the m-HAM.

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