An Analytical Approach to the New Solution of Family of Kuramoto Sivashinsky Equation by q-Homotopy Analysis Technique

Abstract

The current work aims to study a new analytical solution method for a family of Kuramoto–Sivashinsky equation (KSE). For this purpose, we implement the q-homotopy analysis method (q-HAM), a method extensively used due to its easy implementation, fast convergence, and presence of convergence-controlling parameters. This method does not require any transformation, linearization, or discretization. The applied method offers a well-convergent series solution. Results are obtained for some fixed values of physically significant parameters involved in the mathematical model. The computed results are compared with the travelling wave solutions and are illustrated graphically and in tabulated form for absolute error estimation. The comparison reveals the q-HAM provides better results than the existing methods.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, mathematicians and scientists have focused on finding solutions to differential equations (DE’s) that describe various important and complex physical phenomena. For this purpose, several solution techniques have been developed. Some well-known techniques are Variational Iteration Method (VIM) [1, 2], Differential Transform Method (DTM) [3, 4], Homotopy Analysis Method (HAM) [5–7], Adomian Decomposition Method (ADM) [8–11], and Homotopy Perturbation Method (HPM) [12, 13]. The q-HAM is the modified version of HAM, offering the flexibility to extend the convergence region. The q-HAM was initially developed by El-Tawil and Huseen [14] and has been extensively used to solve large classes of PDEs. The advancements in computer technology and hardware and the development of advanced Computer Algebra System (CAS) tools have made the implementation of these methods much easier. Literature review exposes that q-HAM is a powerful approximate analytical technique to deal with complicated and tedious mathematical models [15–17]. The studies also indicate that many researchers use numerous analytical and numerical techniques to solve KSE. We chose q-HAM due to its fast-converging property and flexibility to select the wide range of different values of the convergence-controlling parameter.

The generalized KSE is a mathematical model frequently used in exploring uninterrupted media exhibiting chaotic nature. This model was initially derived by Kuramoto and Sivashinsky [18, 19] for analysis of flame propagation and investigation of phase instability in reaction-diffusion systems. Literature review reveals that KSE is used to describe a large class of physical phenomena and models of various engineering problems of practical interest such a study of long waves on the interface of viscous fluids [20], instabilities in plasma, magnetized plasmas, and flame front propagation [21, 22].

Earlier studies have shown that various analytical, semianalytical, and numerical techniques have been implemented to explore solutions to the KSE. Lakestani and Dehghan [23] used B-spline functions to obtain the numerical solution, while Acan and Keskin [24] applied the reduced differential transformed method for the approximate solution of KSE. Khater and Temsah [25] used the Chebyshev spectral collocation method for the solution of KSE within a bounded domain, while Yousif et al. [26] explored the solution using the variational iteration method. Kurulay et al. [27] use HAM to approximate the solution of KSE. HPM is applied by Easif et al. to find the solution of KSE for small time scales [28]. Wazwaz proposed the new solitary wave solutions to the KSE using the homotopy analysis method [29]. However, to the best of our knowledge, the q-HAM has not explored for the solution of KSE. Therefore, we chose q-HAM due to its flexibility in solving this highly nonlinear and complex equation.

2. The KSE Equation

The general form of KSE is [26].$$\begin{array}{}\text{(1)}& {\Psi }_t+\Psi {\Psi }_x+\alpha {\Psi }_{xx}+\beta {\Psi }_{xxx}+\gamma {\Psi }_{xxxx}=0,\end{array}$$with the following initial and boundary conditions:$$\begin{array}{}\text{(2)}& \Psi x,0=fx,a\le x\le b,\text{(3)}& \Psi a,t=h_{1}t,\Psi b,t=h_{2}t,t>0,\end{array}$$where $\alpha ,\beta$, and $\gamma$ are constants. If $\beta =0$, equation (1) becomes KSE.

3. The q-Homotopy Analysis Method (q-HAM)

The q-HAM was initially developed by El-Tawil and Huseen [14] in 2012 after modification in HAM. They used q-HAM for several mathematical models for obtaining fast converging solutions. They have also shown that HAM is a special case of q-HAM with many additional advantages over HAM such as the large convergence region. It also provides a more appropriate way to control the convergence region. Later, this method was also used by many researchers in solving various problems of practical interest [15–17].

3.1. The Working Rule of q-HAM

This section is devoted to describe the basic concept of q-HAM. Consider the differential equation of the form,$$\begin{array}{}\text{(4)}& {\rm X}\Psi x,t-\xi x,t=0,\end{array}$$where ${\rm X}$ is the nonlinear operator, $\zeta x,t$ is known as a forcing function, and $\Psi x,t$ is an unknown function. The zero^th-order deformation equation can be written as$$\begin{array}{}\text{(5)}& 1-m\rho L\Psi x,t;\rho -{\Psi }_{0}x,t=\rho hHx,t{\rm X}\Psi x,t;\rho -\zeta x,t=0.\end{array}$$

In equation (5), $h$ is a nonzero auxiliary parameter, $L$ is a linear operator, $\rho \in 0,1/m$ is an inserting factor, $Hx,t$ is an auxiliary function, and ${\Psi }_{0}x,t$ is a primary solution of $\Psi x,t$. When $\zeta x,t=0$ and $\rho =0$ and $\rho =1/m$, equation (5) provides $\Psi x,t;0={\Psi }_{0}x,t$ and $\Psi x,t;1/m=\Psi x,t$, respectively.

By expanding $\Psi x,t;\rho$ using Taylor series, we have$$\begin{array}{}\text{(6)}& \Psi x,t;\rho ={\Psi }_{0}x,t+{\displaystyle \sum }_{n=1}^{\infty }{\Psi }_{n}x,t{\rho }^n,\end{array}$$where$$\begin{array}{}\text{(7)}& {{\Psi }_{n}x,t=\frac{1}{n!}\frac{{\partial }^n\Psi x,t}{\partial {\rho }^n}}_{\rho =0}.\end{array}$$

Equation (6) converges for a suitable choice of $L$, $h$, ${\rm H}x,t$, and ${\Psi }_{0}x,t$, then we get$$\begin{array}{}\text{(8)}& {\Psi }_{n}x,t={\frac{1}{n!}\frac{{\partial }^n\Psi x,t}{\partial {\rho }^n}}_{\rho =0}.\end{array}$$

By defining the vector,$$\begin{array}{}\text{(9)}& \stackrel{⟶}{\Psi }x,t={\Psi }_{0}x,t,{\Psi }_{1}x,t,{\Psi }_{2}x,t,\dots ,{\Psi }_{n}x,t.\end{array}$$

Using equation (5), the $n^{\text{th}}$-order deformation equation is given by,$$\begin{array}{}\text{(10)}& L{\Psi }_{n}x,t-{\lambda }_n{\Psi }_{n-1}x,t=hHx,t{R}_n{\stackrel{⟶}{\Psi }}_{n-1}x,t,\end{array}$$where,$$\begin{array}{}\text{(11)}& R_n{\stackrel{⟶}{\Psi }}_{n-1}x,t=\frac{1}{n-1!}\frac{{\partial }^{n-1}{\rm X}\Psi x,t;\rho }{\partial {\rho }^{n-1}},\end{array}$$and$$\begin{array}{}\text{(12)}& {\lambda }_n=\begin{array}{c}0,\text{\hspace{1em}when\hspace{1em}}n=1,\\ m,\text{\hspace{1em}when\hspace{1em}}n>1.\end{array}\end{array}$$

Simplifying equation (10), we get$$\begin{array}{}\text{(13)}& {\Psi }_{n}x,t=h{L}^{-1}{\rm H}x,t{R}_n{\stackrel{⟶}{\Psi }}_{n-1}x,t+{\lambda }_n{\Psi }_{n-1}x,t.\end{array}$$

The series solution $\Psi x,t$ is of the form$$\begin{array}{}\text{(14)}& \Psi x,t={\displaystyle \sum }_{n=0}^N{\Psi }_{n}x,t{\frac{1}{m}}^n.\end{array}$$

4. The Numerical Examples

In this section, we apply q-HAM on KSE for different values of parameters $\eta ,\tau$, and $\mu$. We also illustrate the computed results and its absolute error tables graphically and in the tabulated form, respectively.

4.1. Example 1

Consider the KSE of the form after taking $\alpha =\gamma =1,\beta =0$ [24]$$\begin{array}{}\text{(15)}& {\Psi }_t+\Psi {\Psi }_x+{\Psi }_{xx}+{\Psi }_{xxxx}=0,\end{array}$$subject to$$\begin{array}{}\text{(16)}& \Psi x,0=c+\frac{5}{19}\sqrt{\frac{11}{19}}11{\tan h}^{3}kx-x_0-9\tan hkx-x_0.\end{array}$$

For $n=1$, equation (13) becomes$$\begin{array}{}\text{(17)}& {\Psi }_{1}x,t=h{L}^{-1}{\rm H}x,t{R}_1{\Psi }_{0}x,t+{\lambda }_1{\Psi }_{0}x,t.\end{array}$$

Taking ${\rm H}x,t=1$ and form equation (12) ${\lambda }_1=0$, equation (17) reduces to$$\begin{array}{}\text{(18)}& {\Psi }_{1}x,t=h{L}^{-1}{R}_1{\Psi }_{0}x,t.\end{array}$$

For $n=2$, equation (13) becomes$$\begin{array}{}\text{(19)}& {\Psi }_{2}x,t=h{L}^{-1}{\rm H}x,t{R}_2{\Psi }_{1}x,t+{\lambda }_2{\Psi }_{1}x,t.\end{array}$$

Taking ${\rm H}x,t=1$ and form equation (12) ${\lambda }_2=m$, equation (19) reduces to$$\begin{array}{}\text{(20)}& {\Psi }_{2}x,t=h{L}^{-1}{R}_2{\Psi }_{1}x,t+m{\Psi }_{1}x,t,\end{array}$$where ${\Psi }_{0}x,t$ is an initial guess and $L^{-1}$ is an integral operator. Taking IC as an initial guess$$\begin{array}{}\text{(21)}& \Psi x,0={\Psi }_{0}x,t=c+\frac{5}{19}\sqrt{\frac{11}{19}}11{\tan h}^{3}kx-x_0-9\tan hkx-x_0.\end{array}$$

Using equation (21) in (18) and after integration, we get$$\begin{array}{}\text{(22)}& {\Psi }_1=\frac{120kht}{361\cos h{kx-x_0}^7}\begin{array}{c}\cos h{kx-x_0}^5\sqrt{209}c-8\sin hkx-x_0\\ -\frac{55}{76}+k^3+\frac{1}{4}k\sqrt{209}\cos h{kx-x_0}^4-\frac{11\cos h{kx-x_0}^3\sqrt{209}c}{8}\\ +112\sin hkx-x_0-\frac{3025}{8512}+k^3+\frac{11}{224}k\sqrt{209}\\ \cos h{kx-x_0}^2+\sin hkx-x_0-165\sqrt{209}{k}^3+\frac{6655}{152}\end{array}.\end{array}$$

Next, using equation (22) in equation (20), we get$$\begin{array}{}\text{(23)}& {\Psi }_2=-\frac{120kht}{361\cos h{kx-x_0}^{11}}\begin{array}{c}-16k^{4}t+\frac{1}{4}{k}^{2}t+\frac{1}{16}h+\frac{n}{16}\sqrt{209}-\frac{55htk}{76}c\cos h{kx-x_0}^9+\dots \\ +415800htk{k}^6\sqrt{209}-\frac{1573k^3}{4788}+\frac{6655\sqrt{209}}{82966464}\sin hkx-x_0\end{array}.\end{array}$$

Similarly, $u_3,u_4,\dots$ can be calculated for $n=3,4,5,\dots$.

4.1.1. The Error Analysis

In this subsection, comparison of the previous results obtained in [21] with the current results for fixed values of parameters $c=0.1,k=1/4\sqrt{11/19},x_0=-30$ is presented. The absolute error estimation shown in Table 1 uncovers the accuracy and efficacy of q-HAM over RDTM.

Table 1

Comparison of q-HAM and exact solution [21] by absolute error estimation for $n=10,h=-18$.

$x$	$t$	Absolute error (q-HAM)	Absolute error (RTDM)
0	0	0	$0.542\times {10}^{-9}$
0	0.5	$0.738\times {10}^{-7}$	$0.1057\times {10}^{-4}$
0	1	$0.217\times {10}^{-7}$	$0.202\times {10}^{-4}$
0.5	0	0	$0.604\times {10}^{-9}$
0.5	0.5	$0.585\times {10}^{-7}$	$0.865\times {10}^{-4}$
0.5	1	$0.176\times {10}^{-6}$	$0.167\times {10}^{-4}$
1	0	0	$0.595\times {10}^{-9}$
1	0.5	$0.490\times {10}^{-7}$	$0.716\times {10}^{-5}$
1	1	$0.147\times {10}^{-6}$	$0.138\times {10}^{-4}$

4.1.2. Graphical Comparison of q-HAM and Exact Solution for Example 1

In this subsection, comparison of q-HAM solution and exact solution for example 1 is presented graphically for fixed values of parameters at different time scales. Figures 1(a) and 1(b) show the accuracy and efficiency of the q-HAM.

[figure(s) omitted; refer to PDF]

4.1.3. The Convergence Analysis

In this subsection, h-curves have been plotted for different values of n in Figures 2(a), 2(b), 2(c), and 2(d). Increasing behavior of convergence region is observed with the increasing values of n which is shown in Table 2. Figure 2 shows the h-curve of equation (15) for different values of the time scale against different values of n and fixed values $x=0.2$, $x_0=-30$, $\alpha =1$, $\gamma =1$, and $k=1/4\sqrt{11/14}$. Here, h is known as a convergence controlling parameter. From Figures 2(a), 2(b), 2(c), and 2(d), one can observe the increase in convergence region.

[figure(s) omitted; refer to PDF]

Table 2

Region of convergence for increasing values of n in the case of example 1.

n	h
1 (HAM)	$-10\le h\le 10$
20	$-70\le h\le 30$
50	$-145\le h\le 45$
100	$-210\le h\le 48$

4.2. Example 2

Here, we consider KSE taking periodic boundary conditions [29].$$\begin{array}{}\text{(24)}& {\Psi }_t+\Psi {\Psi }_x+{\Psi }_{xx}+{\Psi }_{xxxx}=0,x\in 0,32\pi ,t\in 0,0.001,\end{array}$$subject to IC,$$\begin{array}{}\text{(25)}& \Psi x,0=\cos \frac{x}{16}1+\sin \frac{x}{16}.\end{array}$$

The true solution of equation (24) is$$\begin{array}{}\text{(26)}& \Psi x,t=\cos \frac{x}{16}-t1+\sin \frac{x}{16}-t.\end{array}$$

Taking IC as an initial guess, we have$$\begin{array}{}\text{(27)}& \Psi x,0={\Psi }_{0}x,t=\cos \frac{x}{16}1+\sin \frac{x}{16}.\end{array}$$

Using equation (27) in equation (18) and after integration, we get$$\begin{array}{}\text{(28)}& {\Psi }_1=\frac{ht}{65536}\begin{array}{c}8192\sin \frac{x}{16}{\cos }^2\frac{x}{16}+12288{\cos }^2\frac{x}{16}\\ -9200\sin \frac{x}{16}-8447\end{array}\cos \frac{x}{16}.\end{array}$$

Next, using equation (28) in equation (20) and after integration, we get$$\begin{array}{}\text{(29)}& {\Psi }_2=\frac{3ht}{128}\cos \frac{x}{16}\begin{array}{c}ht\sin \frac{x}{16}+\frac{25}{12}{\cos }^4\frac{x}{16}+\begin{array}{c}-\frac{1975t}{768}+\frac{16}{3}h+\frac{16n}{3}\\ \sin \frac{x}{16}+-\frac{70765t}{24576}+8h+8n\end{array}{\cos }^2\frac{x}{16}\\ +\frac{646241t}{786432}-\frac{575}{96}h-\frac{575n}{96}\sin \frac{x}{16}+\frac{164855297t}{201326592}-\frac{8447}{1536}h-\frac{8447n}{1536}\end{array}.\end{array}$$

Likewise, one can evaluate the remaining terms of the series solution up to any required accuracy or a given tolerance.

4.2.1. The Error Analysis

In this subsection, comparison of the exact solution with the current results is presented. Table 3 uncovers the accuracy and applicability of q-HAM for solving highly nonlinear PDEs.

Table 3

Absolute error between approximate solution and exact solution when $t=0.0004$.

$x$	Approximate solution (q-HAM)	Exact solution (25)	Absolute error
0	0.9999777363	0.9995999200	$3.778\times {10}^{-4}$
6.4	1.279733159	1.279615936	$1.172\times {10}^{-4}$
12.8	1.196518667	1.196791917	$2.732\times {10}^{-4}$
19.2	0.7001196027	0.7007569806	$6.373\times {10}^{-4}$
25.6	−0.05838957728	−0.05758743503	$8.021\times {10}^{-4}$
32	−0.01866102249	−0.01872761446	$6.576\times {10}^{-4}$

4.2.2. Graphical Comparison of Exact and q-HAM Solution for Example 2

In this subsection, comparison of q-HAM solution and exact solution for example 2 are validated graphically for fixed values of parameters at different time scales. Figures 3(a) and 3(b) highlight the accuracy and applicability of q-HAM for periodic boundary conditions.

[figure(s) omitted; refer to PDF]

4.2.3. Convergence Analysis

In this section, form h-curves have been plotted for different values of n using Figures 4(a), 4(b), 4(c), and 4(d). Increase in the convergence region is observed with the increase of n as shown in Table 4. Figures 4(a), 4(b), 4(c), and 4(d) show the h-curve of equation (15) for different values of the time scale for different values of n and fixed values of $x=6.4$, $\alpha =1$, $\gamma =1$. From Figures 4(a), 4(b), 4(c), and 4(d), one can observe the increase in the convergence region for the increased values of n.

[figure(s) omitted; refer to PDF]

Table 4

Region of convergence for increasing values of “n” for example 2.

n	h
1 (HAM)	$-6\le h\le 6$
20	$-90\le h\le 50$
50	$-150\le h\le 48$
100	$-200\le h\le 45$

4.3. Example 3

Consider the KSE of the form for $\alpha =-1,\gamma =1,\beta =0$ [30],$$\begin{array}{}\text{(30)}& {\Psi }_t+\Psi {\Psi }_x-{\Psi }_{xx}+{\Psi }_{xxxx}=0,\end{array}$$subject to initial condition (IC),$$\begin{array}{}\text{(31)}& \Psi x,0=c+\frac{15}{19\sqrt{19}}{\tan h}^{3}kx-x_0-3\tan hkx-x_0.\end{array}$$

Using IC as an initial guess,$$\begin{array}{}\text{(32)}& \Psi x,0={\Psi }_{0}x,t=c+\frac{15}{19\sqrt{19}}{\tanh }^{3}kx-x_0-3\tanh kx-x_0.\end{array}$$

Using equation (32) in (18) and after integration, we get$$\begin{array}{}\text{(33)}& {\Psi }_1=-\frac{45kht}{361\cos h{kx-x_0}^7}\begin{array}{c}\cos h{kx-x_0}^3\sqrt{19}c-64\sin hkx-x_0\\ \frac{15}{608}+k^3-\frac{1}{16}k\sqrt{19}\cos h{kx-x_0}^2\\ +\sin hkx-x_{0}120\sqrt{19}{k}^3-\frac{15}{19}\end{array}.\end{array}$$

Next, using equation (32) and (33) in equation (20), we get$$\begin{array}{}\text{(34)}& {\Psi }_2=\frac{90htk}{361\cos h{kx-x_0}^{11}}\begin{array}{c}-128k^{4}t-\frac{1}{16}{k}^{2}t+\frac{1}{256}h+\frac{n}{256}\sqrt{19}+\frac{15htk}{608}\\ c\cos h{kx-x_0}^7+\dots -151200h\\ t\sqrt{19}{k}^6-\frac{13k^3}{1596}+\frac{5\sqrt{19}}{9218496}\sin hkx-x_{0}k\end{array}.\end{array}$$

Similarly, ${\Psi }_3,{\Psi }_4,\dots$ can be calculated for $n=3,4,5,\dots$.

4.3.1. The Error Analysis

The comparison analysis is carried out in this subsection for example 3 in Table 5. For this purpose, some fixed values of involved physical parameters are taken into consideration. Table 5 validates the applicability and effectiveness of q-HAM. The table is constructed for fixed parameter values $c=5,k=1/2\sqrt{19},x_0=-25,n=10,h=-11$.

Table 5

Comparison between exact and q-HAM solutions for different values of x and t.

x	t	${\Psi }_{\text{approximate}}$	${\Psi }_{\text{exact}}$	Absolute error
−5	0.1	4.63803897006092	4.63803896999059	$7.033\times {10}^{-11}$
	0.5	4.63843960932536	4.63843955306175	$5.626\times {10}^{-8}$
	1	4.63980631834404	4.63981420314860	$7.885\times {10}^{-6}$
0	0.1	4.63779287238718	4.63779287237956	$7.620\times {10}^{-12}$
	0.5	4.63783494136028	4.63783493730662	$4.054\times {10}^{-9}$
	1	4.63798185888373	4.63798344165438	$1.583\times {10}^{-6}$
5	0.1	4.63776749758524	4.63776749758446	$7.800\times {10}^{-13}$
	0.5	4.63777179526045	4.63777179492130	$3.392\times {10}^{-10}$
	1	4.63778691685813	4.63778710621169	$1.893\times {10}^{-7}$

4.3.2. The Graphical Comparison

In this subsection, a comparison of q-HAM solution and exact solution for example 3 are presented graphically for some fixed values of involved factors at different time levels. Figures 5(a) and 5(b) highlight the accuracy and applicability of the q-HAM.

[figure(s) omitted; refer to PDF]

4.3.3. Convergence Analysis

In this subsection, the h-curves are constructed through plots by taking some fixed values of the involved parameters in the model equation $c=5,k=1/2\sqrt{19},x_0=-25,x=5,\alpha =-1,\gamma =1$ and $t=0.1,0.2,0.3.$ The convergence region is calculated and is illustrated in Figure 6. It is observed that the convergence region upsurges as we increase the values of n. Similar behavior is also predicted in Table 6.

[figure(s) omitted; refer to PDF]

Table 6

Region of convergence for increasing values of “n” for example 3.

n	h
1 (HAM)	$-15\le h\le 15$
20	$-60\le h\le 20$
50	$-115\le h\le 5$
100	$-225\le h\le 0$

5. Conclusion

In this study, we find the approximate analytical solutions for various forms of KS equation using the q-HAM. The results are compared with the exact solution through absolute error tables which reveals a good agreement between computed solutions and the exact solutions. In Table 1, the q-HAM results are compared with the RDTM results for $\alpha =\gamma =1,\beta =0$. The absolute error in Tables 1, 3, and 5 demonstrates that the q-HAM is more accurate than other methods like RDTM. The convergence analysis is performed using h-curves for each numerical example. The effectiveness and applicability of q-HAM are illustrated through h-curves in Figures 2(a), 2(b), 2(c), 2(d), 4(a), 4(b), 4(c), 4(d), 6(a), 6(b), 6(c), and 6(d). The convergence region for the embedded parameter h is calculated for some fixed values of independent variables x and t as well as other involved parameters. The accuracy of q-HAM is also explored through Figures 1(a), 1(b), 3(a), 3(b), 5(a), and 5(b) for examples 1, 2, and 3. A large convergence region is observed and tabulated in Tables 2, 4, and 6, highlighting the advantage of q-HAM over HAM. Given the fast convergence and ease of implementation of q-HAM, we plan to use this method for solving more complex and challenging problems in future.

Authors’ Contributions

Safdar Hussain contributed to the formal analysis, methodology, and writing the original draft. Abdullah Shah supervised the overall activities, editing and modifying the revised draft.

References

[1] J. H. He, "Variational iteration method- a kind of non-linear analytical technique: some examples," International Journal of Non-Linear Mechanics, vol. 34 no. 4, pp. 699-708, DOI: 10.1016/s0020-7462(98)00048-1, 1999.

[2] S. Abbasbandy, "A new application of He’s variational iteration method for quadratic Riccati differential equation by using adomians polynomials," Journal of Computational and Applied Mathematics, 2006.

[3] A. Arikoglu, I. Ozkol, "Solution of fractional differential equations by using differential transform method," Chaos, Solitons and Fractals, vol. 34 no. 5, pp. 1473-1481, DOI: 10.1016/j.chaos.2006.09.004, 2007.

[4] I. H. Abdel-Halim Hassan, "Application to differential transformation method for solving systems of differential equations," Applied Mathematical Modelling, vol. 32 no. 12, pp. 2552-2559, DOI: 10.1016/j.apm.2007.09.025, 2008.

[5] S. J. Liao, "On the homotopy analysis method for nonlinear problems," Applied Mathematics and Computation, vol. 147 no. 2, pp. 499-513, DOI: 10.1016/s0096-3003(02)00790-7, 2004.

[6] M. Kumar, R. K. Jana, "Homotopy analysis method for forced KdV equation in unmagnetized superthermal plasmas," National Academy Science Letters, vol. 46 no. 3, pp. 229-232, DOI: 10.1007/s40009-023-01231-0, 2023.

[7] M. Sajid, T. Hayat, S. Asghar, "Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt," Nonlinear Dynamics, vol. 50 no. 1-2, pp. 27-35, DOI: 10.1007/s11071-006-9140-y, 2007.

[8] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, 1994.

[9] G. Adomian, G. Adomian, "A global method for solution of complex systems," Mathematical Modelling, vol. 5 no. 4, pp. 251-263, DOI: 10.1016/0270-0255(84)90004-6, 1984.

[10] M. Kumar, R. K. Jana, "Analysis of dust-ion acoustic soliton and shock waves of damped KdV burgers’ equation in superthermal plasmas: adomian decomposition approach," Brazilian Journal of Physics, vol. 54 no. 4, 2024.

[11] M. Kumar, R. K. Jana, P. Chatterjee, U. N. Ghosh, P. Chatterjee, "Regular and singular dust ion-acoustic soliton structures in superthermal plasmas: Adomian decomposition approach," Indian Journal of Physics, vol. 97 no. 13, pp. 4059-4068, DOI: 10.1007/s12648-023-02703-1, 2023.

[12] J. H. He, "Homotopy perturbation method: A new nonlinear analytical technique," Applied Mathematics and Computation, vol. 135 no. 1, pp. 73-79, DOI: 10.1016/s0096-3003(01)00312-5, 2003.

[13] J. Biazar, H. Ghazvini, "Convergence of the homotopy perturbation method for partial differential equations," Nonlinear Analysis: Real World Applications, vol. 10 no. 5, pp. 2633-2640, DOI: 10.1016/j.nonrwa.2008.07.002, 2009.

[14] M. A. El-Tawil, S. Huseen, "The q -homotopy analysis method ( q -HAM)," International Journal of Applied Mathematics and Mechanics, vol. 8, pp. 51-75, 2012.

[15] S. Mohamed, Y. S. Hamed, "Solving the convection–diffusion equation by means of the optimal q-homotopy analysis method (Oq-HAM)," Results in Physics, vol. 6, pp. 20-25, DOI: 10.1016/j.rinp.2015.12.008, 2016.

[16] K. M. Saad, E. H. AL-Shareef, M. S. Mohamed, X. J. Yang, "Optimal q-homotopy analysis method for time-space fractional gas dynamics equation," European Physical Journal A: Hadrons and Nuclei, vol. 132 no. 1,DOI: 10.1140/epjp/i2017-11303-6, 2017.

[17] A. Prakash, M. Goyal, S. Gupta, "q-homotopy analysis method for fractional Bloch model arising in nuclear magnetic resonance via the Laplace transform," Indian Journal of Physics, vol. 94 no. 4, pp. 507-520, DOI: 10.1007/s12648-019-01487-7, 2020.

[18] Y. Kuramoto, "Diffusion-induced chaos in reaction systems," Progress of Theoretical Physics—Supplement, vol. 64, pp. 346-367, DOI: 10.1143/ptps.64.346, 1978.

[19] G. I. Sivashinsky, "Nonlinear analysis of hydrodynamic instability in laminar flames I: derivation of basic equations," Acta Astronautica, vol. 4 no. 11-12, pp. 1177-1206, DOI: 10.1016/0094-5765(77)90096-0, 1977.

[20] A. P. Hooper, R. Grimshaw, "Nonlinear instability at the interface between two viscous fluids," The Physics of Fluids, vol. 28 no. 1, pp. 37-45, DOI: 10.1063/1.865160, 1985.

[21] Y. Kuramoto, T. Tsuzuki, "Persistent propagation of concentration waves in dissipative media far from thermal equilibrium," Progress of Theoretical Physics, vol. 55 no. 2, pp. 356-369, DOI: 10.1143/ptp.55.356, 1976.

[22] G. I. Sivashinsky, "Instabilities, pattern-formation, and turbulence in flames," Annual Review of Fluid Mechanics, vol. 15 no. 1, pp. 179-199, DOI: 10.1146/annurev.fl.15.010183.001143, 1983.

[23] M. Lakestani, M. Dehghan, "Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions," Applied Mathematical Modelling, vol. 36 no. 2, pp. 605-617, DOI: 10.1016/j.apm.2011.07.028, 2012.

[24] O. Acan, Y. Keskin, "Approximate solution of kuramoto-sivashinsky equation using reduced differential transform method," AIP Conference Proceedings, vol. 1648, 2014.

[25] A. Khater, R. Temsah, "Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods," Computers and Mathematics with Applications, vol. 56 no. 6, pp. 1465-1472, DOI: 10.1016/j.camwa.2008.03.013, 2008.

[26] M. A. Yousif, S. A. Manaa, F. H. Easif, "Solving the kuramoto-sivashinsky equation via variational iteration method," International Journal of Applied Mathematical Research, vol. 3,DOI: 10.14419/ijamr.v3i3.2901, 2014.

[27] M. Kurulay, A. Secer, M. A. Akinlar, "A new approximate analytical solution of Kuramoto-Sivashinsky equation using homotopy analysis method," Applied Mathematics and Information Sciences, vol. 7 no. 1, pp. 267-271, DOI: 10.12785/amis/070133, 2013.

[28] F. H. Easif, M. Saad, E. Fadhil, "The homotopy perturbation method for solving the Kuramoto—Sivashinsky equation," IOSR Journal of Engineering, vol. 3 no. 12, pp. 23-27, DOI: 10.9790/3021-031232327, 2013.

[29] A. M. Wazwaz, "New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations," Applied Mathematics and Computation, vol. 182 no. 2, pp. 1642-1650, DOI: 10.1016/j.amc.2006.06.002, 2006.

[30] R. C. Mittal, G. Arora, "Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation," Communications in Nonlinear Science and Numerical Simulation, vol. 15 no. 10, pp. 2798-2808, DOI: 10.1016/j.cnsns.2009.11.012, 2010.