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

Abstract

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 𝑞-Homotopy Analysis Method (𝑞-HAM)

Consider the following nonlinear partial differential equation: 𝑁[𝑢(𝑥,𝑡)]=0, (1) 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: 1−𝑛𝑞𝐿∅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, and 𝐻(𝑥,𝑡) denotes a non-zero auxiliary function. It is obvious that when 𝑞=0 and 𝑞=1/𝑛, (2) becomes ∅(𝑥,𝑡;0)=𝑢₀()1𝑥,𝑡,∅𝑥,𝑡;𝑛=𝑢(𝑥,𝑡), (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 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 ∅𝑥,𝑡;𝑞=𝑢₀(𝑥,𝑡)+^+∞_𝑚=1𝑢_𝑚(𝑥,𝑡)𝑞^𝑚, (4) where 𝑢_𝑚()=1𝑥,𝑡𝜕𝑚!^𝑚∅(𝑥,𝑡;𝑞)𝜕𝑞^𝑚_𝑞=0. (5) 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 𝑢()1𝑥,𝑡=∅𝑥,𝑡;𝑛=𝑢₀(𝑥,𝑡)+^+∞_𝑚=1𝑢_𝑚()1𝑥,𝑡𝑛^𝑚. (6) Let 𝑢_𝑟(𝑥,𝑡)=𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡),…,𝑢_𝑟(𝑥,𝑡). (7) 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: 𝐿𝑢_𝑚(𝑥,𝑡)−𝑘_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℎ𝐻(𝑥,𝑡)𝑅𝑢_𝑚^⇀_𝑚−1(𝑥,𝑡), (8) where 𝑅𝑢_𝑚^⇀_𝑚−1()=1𝑥,𝑡()!𝜕𝑚−1^𝑚−1𝑁∅𝑥,𝑡;𝑞−𝑓(𝑥,𝑡)𝜕𝑞^𝑚−1_𝑞=0,𝑘_𝑚=𝑛0𝑚≤1,otherwise. (9) It should be emphasized that 𝑢_𝑚(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: ()()()𝑢()𝐿𝑢𝑥,𝑡+𝐴𝑢𝑥,𝑡+𝐵𝑢𝑥,𝑡=0,𝑥,0=𝑓₀(𝑥),𝜕𝑢(𝑥,𝑡)𝜕𝑡_𝑡=0=𝑓₁(𝑥),⋮𝜕^𝑧−1𝑢(𝑥,𝑡)𝜕^𝑧−1_𝑡=0=𝑓_𝑧−1(𝑥), (10) where 𝐿=𝜕^𝑧/𝜕𝑡^𝑧, 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 1−𝑛𝑞𝐿∅𝑥,𝑡;𝑞−𝑢₀((()()(,𝑥,𝑡)=𝑞ℎ𝐻𝑥,𝑡)(𝐿𝑢𝑥,𝑡+𝐴𝑢𝑥,𝑡+𝐵𝑢𝑥,𝑡)) (11) we have the following 𝑚th order deformation equation: 𝐿𝑢_𝑚(𝑥,𝑡)−𝑘_𝑚𝑢_𝑚−1(()𝑥,𝑡)=ℎ𝐻𝑥,𝑡𝐿𝑢_𝑚−1(𝑥,𝑡)+𝐴𝑢_𝑚−1(𝑥,𝑡)+𝐵𝑢^⇀_𝑚−1(𝑥,𝑡). (12) Hence, 𝑢_𝑚(𝑥,𝑡)=𝑘_𝑚𝑢_𝑚−1(𝑥,𝑡)+ℎ𝐿⁻¹𝐻(𝑥,𝑡)𝐿𝑢_𝑚−1(𝑥,𝑡)+𝐴𝑢_𝑚−1(𝑥,𝑡)+𝐵𝑢^⇀_𝑚−1(𝑥,𝑡). (13) Now, the inverse operator 𝐿⁻¹ is an integral operator which is given by 𝐿⁻¹(⋅)=⋯(⋅)𝑑𝑡𝑑𝑡⋯𝑑𝑡_𝑧times+𝑐₁𝑡^𝑧−1+𝑐₂𝑡^𝑧−2+⋯+𝑐_𝑧, (14) where 𝑐₁,𝑐₂,…,𝑐_𝑧 are integral constants.

To solve (10) by means of 𝑞-HAM, we choose the following initial approximation: 𝑢₀(𝑥,𝑡)=𝑓₀(𝑥)+𝑓₁(𝑥)𝑡+𝑓₂(𝑥)𝑡²2!+⋯+𝑓_𝑧−1(𝑥)𝑡^𝑧−1()!.𝑧−1 (15) Let 𝐻(𝑥,𝑡)=1, by means of (14) and (15); then (13) becomes 𝑢_𝑚(𝑥,𝑡)=𝑘_𝑚𝑢_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝜕^𝑧𝑢_𝑚−1(𝑥,𝜏)𝜕𝜏^𝑧+𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢^⇀_𝑚−1(𝑥,𝜏)𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times. (16) Now from ∫^𝑡₀∫^𝑡₀⋯∫^𝑡₀(𝜕^𝑧𝑢_𝑚−1(𝑥,𝜏)/𝜕𝜏^𝑧)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): 𝑢_𝑚(𝑥,𝑡)=𝑘_𝑚𝑢_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝜕^𝑧𝑢_𝑚−1(𝑥,𝜏)𝜕𝜏^𝑧𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times+ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢^⇀_𝑚−1(𝑥,𝜏)𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times=𝑘_𝑚𝑢_𝑚−1(𝑥,𝑡)+ℎ𝑢_𝑚−1()𝑢𝑥,𝑡−ℎ_𝑚−1(𝑥,0)+𝑡𝜕𝑢_𝑚−1(𝑥,0)𝑡𝜕𝑡+⋯+^𝑧−1()!𝜕𝑧−1^𝑧−1𝑢_𝑚−1(𝑥,0)𝜕𝑡^𝑧−1+ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢^⇀_𝑚−1(𝑥,𝜏)𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times. (17) Now, for 𝑚=1, 𝑘_𝑚=0, and 𝑢₀(𝑥,0)+𝑡𝜕𝑢₀(𝑥,0)+𝑡𝜕𝑡²𝜕2!²𝑢₀(𝑥,0)𝜕𝑡²𝑡+⋯+^𝑧−1()!𝜕𝑧−1^𝑧−1𝑢₀(𝑥,0)𝜕𝑡^𝑧−1=𝑓₀(𝑥)+𝑓₁(𝑥)𝑡+𝑓₂(𝑥)𝑡²2!+⋯+𝑓_𝑧−1(𝑥)𝑡^𝑧−1(𝑧−1)!=𝑢₀(𝑥,𝑡). (18) Substituting this equality into (17), we obtain𝑢₁(𝑥,𝑡)=ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝐴𝑢₀(𝑥,𝜏)+𝐵𝑢₀(𝑥,𝜏)𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times. (19) For 𝑚>1, 𝑘_𝑚=𝑛, and 𝑢_𝑚(𝑥,0)=0,𝜕𝑢_𝑚(𝑥,0)𝜕𝜕𝑡=0,₂𝑢_𝑚(𝑥,0)𝜕𝑡²𝜕=0,…,^𝑧−1𝑢_𝑚(𝑥,0)𝜕𝑡^𝑧−1=0. (20) Substituting this equality into (17), we obtain 𝑢_𝑚()=()𝑢𝑥,𝑡𝑛+ℎ_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀^𝑡₀⋯^𝑡₀𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢^⇀_𝑚−1(𝑥,𝜏)𝑑𝜏𝑑𝜏⋯𝑑𝜏_𝑧times. (21) The standard 𝑞-HAM is powerful when 𝑧=1, and the series solution expression by 𝑞-HAM can be written in the following form: 𝑢(𝑥,𝑡;𝑛;ℎ)≌𝑈_𝑀(𝑥,𝑡;𝑛;ℎ)=^𝑀_𝑖=0𝑢_𝑖()1𝑥,𝑡;𝑛;ℎ𝑛^𝑖. (22) 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 Modified 𝑞-Homotopy Analysis Method (m𝑞-HAM)

When 𝑧≥2, we rewrite (1) as in the following system of first-order differential equations: 𝑢_𝑡=𝑢1,𝑢1_𝑡⋮𝑢{}=𝑢2,𝑧−1_𝑡=−𝐴𝑢(𝑥,𝑡)−𝐵𝑢(𝑥,𝑡). (23) Set the initial approximation 𝑢₀(𝑥,𝑡)=𝑓₀(𝑥),𝑢1₀(𝑥,𝑡)=𝑓₁(𝑥),⋮𝑢{𝑧−1}₀(𝑥,𝑡)=𝑓_𝑧−1(𝑥). (24) Using the iteration formulas (19) and (21) as follows: 𝑢₁(𝑥,𝑡)=ℎ^𝑡₀−𝑢1₀(𝑥,𝜏)𝑑𝜏,𝑢1₁(𝑥,𝑡)=ℎ^𝑡₀−𝑢2₀()⋮𝑢{}𝑥,𝜏𝑑𝜏,𝑧−1₁(𝑥,𝑡)=ℎ^𝑡₀𝐴𝑢₀(𝑥,𝜏)+𝐵𝑢₀(𝑥,𝜏)𝑑𝜏. (25)

For 𝑚>1, 𝑘_𝑚=𝑛, and 𝑢_𝑚(𝑥,0)=0,𝑢1_𝑚(𝑥,0)=0,𝑢2_𝑚(𝑥,0)=0,…,𝑢{𝑧−1}_𝑚(𝑥,0)=0. (26) Substituting in (17), we obtain 𝑢_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑢_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀−𝑢1_𝑚−1(𝑥,𝜏)𝑑𝜏,𝑢1_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑢1_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀−𝑢2_𝑚−1()⋮𝑢{}𝑥,𝜏𝑑𝜏,𝑧−1_𝑚()=()𝑥,𝑡𝑛+ℎ𝑢{𝑧−1}_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢_𝑚−1(𝑥,𝜏)𝑑𝜏. (27) 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: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+sin𝑢=0, (28) subject to the following initial conditions: 𝑢(𝑥,0)=0,𝑢_𝑡(𝑥,0)=4sech𝑥. (29) The exact solution is 𝑢(𝑥,𝑡)=4tan⁻¹(𝑡sech𝑥). (30) 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: 𝑢sin𝑢=𝑢−³6+𝑢⁵,120 (31) Then, (28) becomes 𝑢_𝑡𝑡−𝑢_𝑥𝑥𝑢+𝑢−³6+𝑢⁵120=0. (32) In order to solve (28) by m𝑞-HAM, we construct system of differential equations as follows: 𝑢_𝑡()(),𝑣𝑥,𝑡=𝑣𝑥,𝑡_𝑡()=𝜕𝑥,𝑡²𝑢(𝑥,𝑡)𝜕𝑥²𝑢−𝑢+³6−𝑢⁵,120 (33) with the following initial approximations: 𝑢₀(𝑥,𝑡)=0,𝑣₀(𝑥,𝑡)=4sech𝑥, (34) and the following auxiliary linear operators: ()=()𝐿𝑢𝑥,𝑡𝜕𝑢𝑥,𝑡()=()𝜕𝑡,𝐿𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡,𝜕𝑡𝐴𝑢_𝑚−1()𝜕𝑥,𝑡=−²𝑢_𝑚−1(𝑥,𝑡)𝜕𝑥²+𝑢_𝑚−1(),𝐵𝑥,𝑡_⇀𝑢_𝑚−1()1𝑥,𝑡=−6^𝑚−1_𝑗=0𝑢^𝑗_{𝑚−1−𝑗𝑖=0}𝑢_𝑖𝑢_𝑗−𝑖+1120^𝑚−1_𝑗=0𝑢^𝑗_{𝑚−1−𝑗𝑖=0}𝑢^𝑖_{𝑗−𝑖𝑘=0}𝑢^𝑘_{𝑖−𝑘𝑙=0}𝑢_𝑙𝑢_𝑘−𝑙. (35) From (25) and (27), we obtain 𝑢₁(𝑥,𝑡)=ℎ^𝑡₀−𝑣₀()𝑣𝑥,𝜏𝑑𝜏,₁(𝑥,𝑡)=ℎ^𝑡₀−𝜕²𝑢₀𝜕𝑥²+𝑢₀−𝑢³₀6+𝑢⁵₀120𝑑𝜏. (36) Now, for 𝑚≥2, we get 𝑢_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑢_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀−𝑣_𝑚−1()𝑣𝑥,𝜏𝑑𝜏,_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑣_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀𝐴𝑢_𝑚−1(𝑥,𝜏)+𝐵𝑢_𝑚−1(𝑥,𝜏)𝑑𝜏. (37) And the following results are obtained: 𝑢₁(𝑥,𝑡)=−4ℎ𝑡sech𝑣𝑥,₁()𝑢𝑥,𝑡=0,₂(𝑥,𝑡)=−4ℎ(ℎ+𝑛)𝑡sech𝑣𝑥,₂(𝑥,𝑡)=−4ℎ²𝑡²sech³𝑢𝑥,₃(𝑥,𝑡)=−4ℎ(ℎ+𝑛)²𝑡sech4𝑥+3ℎ³𝑡³sech³𝑥, (38)𝑢_𝑚(𝑥,𝑡), (𝑚=4,5,…) can be calculated similarly. Then, the series solution expression by m𝑞-HAM can be written in the following form: 𝑢(𝑥,𝑡;𝑛;ℎ)≌𝑈_𝑀(𝑥,𝑡;𝑛;ℎ)=^𝑀_𝑖=0𝑢_𝑖()1𝑥,𝑡;𝑛;ℎ𝑛^𝑖. (39) 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 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.

Figure 4 

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.

Figure 5 

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.

Figure 6 

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.

Figure 7 

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.

Figure 8 

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: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+343𝑢−2𝑢³=0, (40) subject to the following initial conditions: 𝑢(𝑥,0)=−sech𝑢𝑥,_𝑡()=1𝑥,02sech𝑥tanh𝑥. (41) The exact solution is 𝑢(𝑥,𝑡)=−sech𝑡𝑥+2. (42)
In order to solve (40) by m𝑞-HAM, we construct system of differential equations as follows: 𝑢_𝑡()(),𝑣𝑥,𝑡=𝑣𝑥,𝑡_𝑡()=𝜕𝑥,𝑡²𝑢(𝑥,𝑡)𝜕𝑥²−343𝑢+2𝑢³, (43) with the following initial approximations: 𝑢₀(𝑥,𝑡)=−sech𝑥,𝑣₀()=1𝑥,𝑡2sech𝑥tanh𝑥, (44) and the following auxiliary linear operators: ()=()𝐿𝑢𝑥,𝑡𝜕𝑢𝑥,𝑡()=()𝜕𝑡,𝐿𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡,𝜕𝑡𝐴𝑢_𝑚−1()𝜕𝑥,𝑡=−²𝑢_𝑚−1(𝑥,𝑡)𝜕𝑥²+34𝑢_𝑚−1(),𝐵𝑥,𝑡_⇀𝑢_𝑚−1()3𝑥,𝑡=−2^𝑚−1_𝑗=0𝑢^𝑗_{𝑚−1−𝑗𝑖=0}𝑢_𝑖𝑢_𝑗−𝑖. (45) From (25) and (27), we obtain 𝑢₁(𝑥,𝑡)=ℎ^𝑡₀−𝑣₀()𝑣𝑥,𝜏𝑑𝜏,₁(𝑥,𝑡)=ℎ^𝑡₀−𝜕²𝑢₀𝜕𝑥²+34𝑢₀−32𝑢³₀𝑑𝜏. (46) For 𝑚≥2, we get 𝑢_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑢_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀−𝑣_𝑚−1()𝑣𝑥,𝜏𝑑𝜏,_𝑚(𝑥,𝑡)=(𝑛+ℎ)𝑣_𝑚−1(𝑥,𝑡)+ℎ^𝑡₀−𝜕²𝑢_𝑚−1(𝑥,𝑡)𝜕𝑥²+34𝑢_𝑚−1()−3𝑥,𝑡2^𝑚−1_𝑗=0𝑢^𝑗_{𝑚−1−𝑗𝑖=0}𝑢_𝑖𝑢_𝑗−𝑖𝑑𝜏. (47)

The following results are obtained: 𝑢₁()1𝑥,𝑡=−2ℎ𝑡sech𝑣𝑥tanh𝑥,₁()−3𝑥,𝑡=ℎ𝑡sech𝑥4+sech³𝑥2+sech𝑥tanh²𝑥,𝑢₂()3𝑥,𝑡=ℎ16ℎ𝑡²sech³1𝑥−16ℎ𝑡²cosh(2𝑥)sech³𝑥−12ℎ(ℎ+𝑛)𝑡sech𝑥tanh𝑥, (48)𝑢_𝑚(𝑥,𝑡), (𝑚=3,4,…) can be calculated similarly. Then, the series solution expression by m𝑞-HAM can be written in the following form: 𝑢(𝑥,𝑡;𝑛;ℎ)≌𝑈_𝑀(𝑥,𝑡;𝑛;ℎ)=^𝑀_𝑖=0𝑢_𝑖()1𝑥,𝑡;𝑛;ℎ𝑛^𝑖. (49)

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 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.

Figure 12 

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.

Figure 13 

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

Figure 14 

The comparison between the 𝑈3, 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 of  𝑈3 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 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.

Figure 17 

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.