Numerical solution of time- and space-fractional coupled Burgers’ equations via homotopy algorithm

==Abstract==

In this paper, we constitute a homotopy algorithm basically extension of homotopy analysis method with Laplace transform, namely q-homotopy analysis transform method to solve time- and space-fractional coupled Burgers’ equations. The suggested technique produces many more opportunities by appropriate selection of auxiliary parameters <math display="inline">\hslash </math> and <math display="inline">n\quad (n\geqslant 1)</math> to solve strongly nonlinear differential equations. The proposed technique provides <math display="inline">\hslash </math> and <math display="inline">n</math>-curves, which describe that the convergence range is not a local point effects and finds elucidated series solution that makes it superior than HAM and other analytical techniques.

==Keywords==

Laplace transform method; q-Homotopy analysis transform method; Fractional coupled Burgers’ equations; ℏℏ and nn-curves

==1. Introduction==

Fractional calculus was utilized as an excellent instrument to discover the hidden aspects of various material and physical processes that deal with derivatives and integrals of arbitrary orders [[#b0005|[1]]], [[#b0010|[2]]], [[#b0015|[3]]], [[#b0020|[4]]] and [[#b0025|[5]]]. The theory of fractional differential equations translates the reality of nature excellently in a better and systematic manner [[#b0030|[6]]], [[#b0035|[7]]], [[#b0040|[8]]], [[#b0045|[9]]], [[#b0050|[10]]] and [[#b0055|[11]]]. In recent years, many authors have investigated partial differential equations of fractional order by various techniques such as homotopy analysis technique [[#b0060|[12]]], [[#b0065|[13]]] and [[#b0070|[14]]], operational matrix based method [[#b0075|[15]]], and tau method [[#b0080|[16]]].

This article considers the efficiency of q-homotopy analysis transform method (q-HATM) to solve time- and space- fractional coupled Burgers’ equations. The q-HATM is a graceful coupling of two powerful techniques namely q-HAM and Laplace transform algorithms and gives more refined convergent series solution. The q-HAM was initially introduced and nurtured by El-Tavil and Huseen [[#b0085|[17]]] and [[#b0090|[18]]]. The q-HAM is an extension of the embedding parameter <math display="inline">q\in [0\mbox{,}1]</math> arising in the study by Liao HAM [[#b0095|[19]]], [[#b0100|[20]]] and [[#b0105|[21]]] to <math display="inline">q\in \left[0\mbox{,}\frac{1}{n}\right]</math> that appears in q-HAM. The homotopy analysis method (HAM) is based on homotopy, a rudimentary concept in topology and differential geometry that has been notably applied for solving nonlinear problems occurring in different directions of scientific fields [[#b0110|[22]]], [[#b0115|[23]]], [[#b0120|[24]]], [[#b0125|[25]]], [[#b0130|[26]]], [[#b0135|[27]]] and [[#b0140|[28]]]. The HAM has also been united with Laplace transform to bringing out highly effective technique to investigate nonlinear problems of physical importance [[#b0145|[29]]], [[#b0150|[30]]] and [[#b0155|[31]]]. It is well-known fact the coupling of semi-analytical methods with Laplace transform giving time-consuming consequences and less C.P.U time to investigating nonlinear problems describing engineering applications.

In this letter, we consider the following system of fractional coupled Burgers’ equations

<span id='e0005'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\frac{{\partial }^{{\alpha }_1}u}{\partial t^{{\alpha }_1}}=</math><math>\frac{{\partial }^2u}{\partial x^2}+2u\frac{{\partial }^{{\alpha }_2}u}{\partial x^{{\alpha }_2}}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<{\alpha }_i\leqslant 1\mbox{,}</math>
|-
|<math>\frac{{\partial }^{{\beta }_1}v}{\partial t^{{\beta }_1}}=</math><math>\frac{{\partial }^2v}{\partial x^2}+2v\frac{{\partial }^{{\beta }_2}v}{\partial x^{{\beta }_2}}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<{\beta }_i\leqslant 1</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

subject to the initial conditions

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}0)=u_0=F(x)\mbox{,}\quad v(x\mbox{,}0)=</math><math>v_0=G(x)\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

where <math display="inline">\left({\alpha }_i\mbox{,}{\beta }_i\mbox{,}\quad i=\right. </math><math>\left. 1\mbox{,}2\right)</math> are parameters describing the order of the time <math display="inline">\left(i=1\right)</math> and space <math display="inline">\left(i=2\right)</math> fractional derivatives, <math display="inline">x</math> is the space domain and <math display="inline">t</math> is time. When <math display="inline">{\alpha }_i={\beta }_i=1</math>, then the system of Eq. [[#e0005|(1)]] turns down to the classical coupled Burgers’ equations. The most important advantages of using fractional order derivative and integrals over the integer order derivatives and integrals are that they provide a powerful instrument for the description of memory and hereditary properties of different substances.

An excellent literature can be found to study the coupled Burgers’ equations, which are very significant for that the system of Eq. [[#e0005|(1)]] forming a simple model of sedimentations or evolution of scaled volume concentrations of two kinds of gravity [[#b0160|[32]]]. A logistic and remarkable study has been done by number of researchers pertaining to coupled Burgers’ equations [[#b0165|[33]]], [[#b0170|[34]]], [[#b0175|[35]]], [[#b0180|[36]]] and [[#b0185|[37]]]. Recently, Prakash et al. [[#b0190|[38]]] numerically solve the system of Eq. [[#e0005|(1)]] by making use of variational iteration method (VIM) and a systematic comparison has also been made with ADM, GDTM and HPM. In the present article, we observe a highly effective general approach say q-HATM to solve the system of fractional Eq. [[#e0005|(1)]] with concept of fractional Laplace transform of the Caputo derivative [[#b0195|[39]]] at large admissible domain.

==2. Basic idea of q-HATM==

In this section, we present the basic theory and solution procedure of proposed technique. We take a general fractional nonlinear non-homogeneous partial differential equation of the form:

<span id='e0015'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>D_t^{\alpha }u(x\mbox{,}t)+Ru(x\mbox{,}t)+Nu(x\mbox{,}t)=</math><math>g(x\mbox{,}t)\mbox{,}\quad n-1<\alpha \leqslant n</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

where is <math display="inline">D_t^{\alpha }u(x\mbox{,}t)</math> represents the fractional derivative of the function <math display="inline">u(x\mbox{,}t)</math> in terms of Caputo, <math display="inline">R</math> indicates the linear differential operator, <math display="inline">N</math> represents the general nonlinear differential operator and <math display="inline">g(x\mbox{,}t)</math> is the source term.

By applying the Laplace transform operator on both sides of Eq. [[#e0015|(3)]], we get the following equation:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L\left[D_t^{\alpha }u\right]+L[Ru]+L[Nu]=L[g(x\mbox{,}t)]\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

Making use of the differentiation property of the Laplace transform, it yields

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>s^{\alpha }L[u]-\sum_{k=0}^{n-1}s^{\alpha -k-1}u^{\left(k\right)}(x\mbox{,}0)+</math><math>L[Ru]+L[Nu]=L[g(x\mbox{,}t)]\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

On simplifying, the above equation reduces to

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[u]-\frac{1}{s^{\alpha }}\sum_{k=0}^{n-1}s^{\alpha -k-1}u^{\left(k\right)}(x\mbox{,}0)+</math><math>\frac{1}{s^{\alpha }}[L[Ru]+L[Nu]-L[g(x\mbox{,}t)]]=</math><math>0\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

We define the nonlinear operator as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>N[\phi (x\mbox{,}t\mbox{;}q)]=L[\phi (x\mbox{,}t\mbox{;}q)]-</math><math>\frac{1}{s^{\alpha }}\sum_{k=0}^{n-1}s^{\alpha -k-1}{\phi }^{\left(k\right)}(x\mbox{,}t\mbox{;}q)(0^+)+</math><math>\frac{1}{s^{\alpha }}[L[R\phi (x\mbox{,}t\mbox{;}q)]+</math><math>L[N\phi (x\mbox{,}t\mbox{;}q)]-L[g(x\mbox{,}t)]]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

where <math display="inline">q\in [0\mbox{,}1/n]</math> and <math display="inline">\phi (x\mbox{,}t\mbox{;}q)</math> are real functions of ''x'', ''t'' and ''q''. We construct a homotopy as follows:

<span id='e0040'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(1-nq)L[\phi (x\mbox{,}t\mbox{;}q)-u_0(x\mbox{,}t)]=</math><math>\hslash qH(x\mbox{,}t)N[\phi (x\mbox{,}t\mbox{;}q)]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

where ''L  '' denotes the Laplace transform, <math display="inline">n\geqslant 1</math>, <math display="inline">q\in \left[0\mbox{,}\frac{1}{n}\right]</math> is the embedding parameter, <math display="inline">H(x\mbox{,}t)</math> denotes a nonzero auxiliary function, <math display="inline">\hslash \not =0</math> is an auxiliary parameter, <math display="inline">u_0(x\mbox{,}t)</math> is an initial guess of <math display="inline">u(x\mbox{,}t)</math> and <math display="inline">\phi (x\mbox{,}t\mbox{;}q)</math> is an unknown function. It is obvious that, when the embedding parameter <math display="inline">q=0</math> and <math display="inline">q=\frac{1}{n}</math>, it holds the result

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\phi (x\mbox{,}t\mbox{;}0)=u_0(x\mbox{,}t)\mbox{,}\quad \phi \left(x\mbox{,}t\mbox{;}\frac{1}{n}\right)=</math><math>u(x\mbox{,}t)\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

respectively. Thus, as ''q  '' increases from 0 to <math display="inline">\frac{1}{n}</math>, the solution <math display="inline">\phi (x\mbox{,}t\mbox{;}q)</math> varies from the initial guess <math display="inline">u_0(x\mbox{,}t)</math> to the solution <math display="inline">u(x\mbox{,}t)</math>. Expanding the function <math display="inline">\phi (x\mbox{,}t\mbox{;}q)</math> in series form by employing Taylor theorem about ''q'', we have

<span id='e0050'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\phi (x\mbox{,}t\mbox{;}q)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t)q^m\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_m(x\mbox{,}t)=\frac{1}{m!}{\frac{{\partial }^m\phi (x\mbox{,}t\mbox{;}q)}{\partial q^m}}_{q=0}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

If the auxiliary linear operator, the initial guess, the auxiliary parameter <math display="inline">n\mbox{,}\hslash </math> and the auxiliary function are properly chosen, the series [[#e0050|(10)]] converges at <math display="inline">q=\frac{1}{n}\mbox{,}</math> and then we have

<span id='e0060'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}t)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

which must be one of the solutions of the original nonlinear equations. According to the definition [[#e0060|(12)]], the governing equation can be deduced from the zero-order deformation [[#e0040|(8)]].

Define the vectors in the following manner

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\overset{\rightarrow}{u}}_m=\lbrace u_0(x\mbox{,}t)\mbox{,}u_1(x\mbox{,}t)\mbox{,}\ldots \mbox{,}u_m(x\mbox{,}t)\rbrace \mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

Now, differentiating the zeroth-order deformation Eq. [[#e0040|(8)]]''m''-times with respect to ''q'' and then dividing them by ''m  ''! and finally setting <math display="inline">q=0\mbox{,}</math> we get the following ''m''th-order deformation equation:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[u_m(x\mbox{,}t)-k_mu_{m-1}(x\mbox{,}t)]=\hslash H(x\mbox{,}t)R_m({\overset{\rightarrow}{u}}_{m-1})\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

Finally applying the inverse Laplace transform, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_m(x\mbox{,}t)=k_mu_{m-1}(x\mbox{,}t)+\hslash L^{-1}[H(x\mbox{,}t)R_m({\overset{\rightarrow}{u}}_{m-1})]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

where the value of <math display="inline">R_m({\overset{\rightarrow}{u}}_{m-1})</math> is given as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R_m({\overset{\rightarrow}{u}}_{m-1})=\frac{1}{(m-1)!}{\frac{{\partial }^{m-1}N[\phi (x\mbox{,}t\mbox{;}q)]}{\partial q^{m-1}}}_{q=0}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

and <math display="inline">k_m</math> is defined as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>k_m=\begin{array}{ll}
0\mbox{,} & m\leqslant 1\mbox{,}\\
n\mbox{,} & m>1\mbox{.}
\end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

From study it should be analyzed that in special case <math display="inline">n=1</math>, q-HATM reduces to the homotopy analysis transform method (HATM).

==3. Application of the method==

==Example 1.                     ==

We consider the time-fractional coupled Burgers’ equation [[#b0170|[34]]] and [[#b0180|[36]]]

<span id='e0090'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}=</math><math>\frac{{\partial }^2u}{\partial x^2}+2u\frac{\partial u}{\partial x}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\alpha \leqslant 1\mbox{,}</math>
|-
|<math>\frac{{\partial }^{\beta }v}{\partial t^{\beta }}=</math><math>\frac{{\partial }^2v}{\partial x^2}+2v\frac{\partial v}{\partial x}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\beta \leqslant 1\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|}

subject to the initial conditions

<span id='e0095'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}0)=u_0=F(x)=sinx\mbox{,}\quad v(x\mbox{,}0)=</math><math>v_0=G(x)=sinx\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|}

Eqs. [[#e0090|(18)]] and [[#e0095|(19)]] advise that we define the nonlinear operator as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>N^1[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]=</math><math>L[{\psi }_1(x\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}F(x)-</math><math>\frac{1}{S^{\alpha }}L\left[\frac{{\partial }^2{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. </math><math>\left. 2{\psi }_1(x\mbox{,}t\mbox{;}q)\frac{\partial {\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x}-\right. </math><math>\left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>N^2[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]=</math><math>L[{\psi }_2(X\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}G(x)-</math><math>\frac{1}{S^{\beta }}L\left[\frac{{\partial }^2{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. </math><math>\left. 2{\psi }_2(x\mbox{,}t\mbox{;}q)\frac{\partial {\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x}-\right. </math><math>\left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}

and the Laplace operator as

<span id='e0110'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[u_m(x\mbox{,}t)-k_mu_{m-1}(x\mbox{,}t)]=\hslash R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|}

<span id='e0115'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[v_m(x\mbox{,}t)-k_mv_{m-1}(x\mbox{,}t)]=\hslash R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]=</math><math>L\lbrace u_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}sinx-</math><math>\frac{1}{s^{\alpha }}L\left\{\frac{{\partial }^2u}{\partial x^2}+\right. </math><math>\left. 2\sum_{i=0}^{m-1}u_i\frac{\partial u_{m-i}}{\partial x}-\right. </math><math>\left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]=</math><math>L\lbrace v_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}sinx-</math><math>\frac{1}{s^{\beta }}L\left\{\frac{{\partial }^2v}{\partial x^2}+\right. </math><math>\left. 2\sum_{i=0}^{m-1}v_i\frac{\partial v_{m-i}}{\partial x}-\right. </math><math>\left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}

Appling the inverse of Laplace transform on Eqs. [[#e0110|(22)]] and [[#e0115|(23)]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_m(x\mbox{,}t)=k_mu_{m-1}(x\mbox{,}t)+\hslash L^{-1}\left\{R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\right\}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_m(x\mbox{,}t)=k_mv_{m-1}(x\mbox{,}t)+\hslash L^{-1}\left\{R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\right\}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|}

On solving the above equations, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_0=sinx\mbox{,}\quad v_0=sinx\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_1=\frac{\hslash sin{xt}^{\alpha }}{\Gamma (\alpha +1)}\mbox{,}\quad v_1=</math><math>\frac{\hslash sin{xt}^{\beta }}{\Gamma (\beta +1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_2=\frac{\hslash (\hslash +n)sin{xt}^{\alpha }}{\Gamma (\alpha +1)}+</math><math>{\hslash }^2sinx(1-2cosx)\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}+</math><math>2{\hslash }^2sinxcosx\frac{t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_2=\frac{\hslash (\hslash +n)sin{xt}^{\beta }}{\Gamma (\beta +1)}+</math><math>{\hslash }^2sinx(1-2cosx)\frac{t^{2\beta }}{\Gamma (2\beta +1)}+</math><math>2{\hslash }^2sinxcosx\frac{t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
|}

and so on.

In this manner the rest of the iterative components can be obtained. Therefore, the family of q-HATM series solutions of the system of Eq. [[#e0090|(18)]] is given by

<span id='e0160'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}t)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m\mbox{,}</math>
|-
|<math>v(x\mbox{,}t)=v_0(x\mbox{,}t)+\sum_{m=1}^{\infty }v_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|}

If we set <math display="inline">n=1</math> in Eq. [[#e0160|(29)]], we have the solutions derived by making use of HAM as a special case of q-HATM solution. On the other hand, if we take <math display="inline">n=1</math> and <math display="inline">\hslash =-1</math>, then we arrive at the results found by using HPM [[#b0180|[36]]], DTM [[#b0185|[37]]] and VIM [[#b0190|[38]]] as a particular case of q-HATM solution. Thus, we can conclude that the results obtained by using q-HATM contain the results obtained with the help of HAM, HPM, DTM and VIM. If we set <math display="inline">\alpha =\beta =1</math> and <math display="inline">\hslash =-1\mbox{,}\quad n=1</math> then clearly we can observe that the solution <math display="inline">{\sum }_{m=0}^Nu_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m</math> when <math display="inline">N\rightarrow \infty </math> converges to the exact solution of classical coupled Burgers’ equations, which is the special case of the system of Eq. [[#e0090|(18)]] and is given by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u=v=sinx\left(1-t+\frac{t^2}{\Gamma (2+1)}-\frac{t^3}{\Gamma (3+1)}+\right. </math><math>\left. \cdots +{\left(-1\right)}^r\frac{t^r}{\Gamma (r+1)}+\right. </math><math>\left. \cdots \right)=e^{-t}sinx\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}

For simplicity, here we consider <math display="inline">u=u(x\mbox{,}t)=v(x\mbox{,}t)</math> and <math display="inline">\alpha =\beta </math> for every case. The efficiency of purpose method is noticed through the absolute error between exact solution and second order approximation shown in [[#f0005|Fig. 1]]c and e.

<span id='f0005'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr1.jpg|center|425px|(a)–(h) Represent six order approximations HAM (q-HATM, n=1) solution ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 1.

(a)–(h) Represent six order approximations HAM (q-HATM, <math display="inline">n=1</math>) solution <math display="inline">u(x\mbox{,}t)=v(x\mbox{,}t)</math> of system of Eq. [[#e0090|(18)]].

</span>
|}

[[#t0005|Table 1]] shows that q-HATM, can provide many more acceptable solutions compared to all other analytical techniques for same grid point and order of solution series. A proper selection of auxiliary parameters <math display="inline">\hslash </math> and <math display="inline">n</math> gives more correct approximate solution which is identical to exact solution. A horizontal line segment represents the absolute convergence range for q-HATM solution series in <math display="inline">\hslash </math>-curve corresponding to <math display="inline">n\quad (n\geqslant 1)</math> (see [[#f0010|Figure 2]], [[#f0015|Figure 3]], [[#f0020|Figure 4]], [[#f0025|Figure 5]], [[#f0030|Figure 6]] and [[#f0035|Figure 7]]).

==Example 2.                     ==

Finally, we consider the following space-fractional coupled Burgers’ equation [[#b0170|[34]]] and [[#b0180|[36]]]

<span id='e0170'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\frac{\partial u}{\partial t}=\frac{{\partial }^2u}{\partial x^2}+</math><math>2u\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\alpha \leqslant 1\mbox{,}</math>
|-
|<math>\frac{\partial v}{\partial t}=\frac{{\partial }^2v}{\partial x^2}+</math><math>2v\frac{{\partial }^{\beta }v}{\partial x^{\beta }}-</math><math>\frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\beta \leqslant 1</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|}

subject to the initial conditions

<span id='e0175'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}0)=u_0=F(x)=x^2\mbox{,}\quad v(x\mbox{,}0)=</math><math>v_0=G(x)=x^3\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}

<span id='t0005'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 1.

Comparative study between HPM [[#b0180|[36]]] and q-HATM.

|-

! colspan="4" | Absolute error <math display="inline">E_4(u)=\vert u_{exa.}-u_{app.}\vert </math>
|-

! <math display="inline">x</math>
! <math display="inline">t</math>
! HPM [[#b0180|[36]]], <math display="inline">\vert u_{exa.}-u_{app.}\vert </math>
! q-HATM, <math display="inline">\vert u_{exa.}-u_{app.}\vert \mbox{,}\quad \quad (\hslash \mbox{,}n)</math>
|-

| rowspan="5" | −10
| rowspan="5" | 0.07
| rowspan="5" | 7.5 × 10<sup>−9</sup>
| 7.5 × 10<sup>−9</sup>,    (−1, 1)
|-

| 4 × 10<sup>−10</sup>,    (−0.99, 1)
|-

| 2.0 × 10<sup>−9</sup>,    (−0.98, 1)
|-

| 1.5 × 10<sup>−9</sup>,    (−4.98, 5)
|-

| 1.0 × 10<sup>−9</sup>,    (−58.5, 60)
|-

| rowspan="4" | 15
| rowspan="4" | 0.2
| rowspan="4" | 0.0000016779
| 0.0000016779,    (−1, 1)
|-

| 4.326 × 10<sup>−7</sup>,    (−0.99, 1)
|-

| 5.09 × 10<sup>−8</sup>,    (−9.8, 10)
|-

| 3.470 × 10<sup>−7</sup>,    (−44.5, 45)
|}

<span id='f0010'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr2.jpg|center|px|(a)–(f) Show the six order approximate q-HATM solution u(x,t)=v(x,t) of system ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 2.

(a)–(f) Show the six order approximate q-HATM solution <math display="inline">u(x\mbox{,}t)=v(x\mbox{,}t)</math> of system of Eq. [[#e0090|(18)]].

</span>
|}

<span id='f0015'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr3.jpg|center|px|(a)–(d) ℏ and n-curves at x=20,t=0.002 of system of Eq. (18) and show the valid ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 3.

(a)–(d) <math display="inline">\hslash </math> and <math display="inline">n</math>-curves at <math display="inline">x=20\mbox{,}\quad t=0.002</math> of system of Eq. [[#e0090|(18)]] and show the valid convergence range of <math display="inline">\hslash </math> and asymptotic behaviour of <math display="inline">u(x\mbox{,}t)=v(x\mbox{,}t)</math> respectively with different values of <math display="inline">\alpha =\beta </math>: (3a) for HAM, convergence range is <math display="inline">-1.99\leqslant \hslash <0</math>; (3c) for q-HATM, <math display="inline">n=20</math> convergence range is <math display="inline">-39.81\leqslant \hslash <0</math>; (3b) at <math display="inline">\hslash =-1</math> and (3d) at <math display="inline">\hslash =-19.8</math>, show the validity of corresponding <math display="inline">\hslash </math>-curves.

</span>
|}

<span id='f0020'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr4.jpg|center|508px|(a)–(f) Show the second order approximate q-HATM surface solution u(x,t) and ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 4.

(a)–(f) Show the second order approximate q-HATM surface solution <math display="inline">u(x\mbox{,}t)</math> and <math display="inline">v(x\mbox{,}t)</math> of system [[#e0170|(31)]] with different values of <math display="inline">\left(\hslash \mbox{,}n\mbox{,}\alpha \mbox{,}\beta \right)</math> versus time variable ''t  '' and space variable <math display="inline">x</math>.

</span>
|}

<span id='f0025'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr5.jpg|center|px|(a)–(d) ℏ-curves at x=t=0.05 for second order approximation of system of ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 5.

(a)–(d) <math display="inline">\hslash </math>-curves at <math display="inline">x=t=0.05</math> for second order approximation of system of fractional Eq. [[#e0170|(31)]] and show the valid convergence range of <math display="inline">u(x\mbox{,}t)</math> and <math display="inline">v(x\mbox{,}t)</math>: (5a) <math display="inline">-2.021\leqslant \hslash <0</math> and (5b) <math display="inline">-1.968\leqslant \hslash <0</math> for HAM; (5c) <math display="inline">-10.18\leqslant \hslash <0</math> and (5d) <math display="inline">-9.85\leqslant \hslash <0</math> for q-HATM, <math display="inline">n=5</math>.

</span>
|}

<span id='f0030'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr6.jpg|center|px|(a)–(b) n-curves at x=t=0.05 for second order approximation of system (31) and ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 6.

(a)–(b) <math display="inline">n</math>-curves at <math display="inline">x=t=0.05</math> for second order approximation of system [[#e0170|(31)]] and show the asymptotic behavior of <math display="inline">u(x\mbox{,}t)</math> and <math display="inline">v(x\mbox{,}t)</math> also describe the validity of <math display="inline">\hslash </math>-curve.

</span>
|}

<span id='f0035'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr7.jpg|center|px|(a)–(b) Show the comparative behaviors of u(x,t) and v(x,t) at t=0.05 versus ...]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 7.

(a)–(b) Show the comparative behaviors of <math display="inline">u(x\mbox{,}t)</math> and <math display="inline">v(x\mbox{,}t)</math> at <math display="inline">t=0.05</math> versus space variable <math display="inline">x</math>. It’s clear to see that both the functions are continuously increasing functions.

</span>
|}

Eqs. [[#e0170|(31)]] and [[#e0175|(32)]] advise that we define the nonlinear operator as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>N^1[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]=</math><math>L[{\psi }_1(x\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}F(x)-</math><math>\frac{1}{s}L\left[\frac{{\partial }^2{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. </math><math>\left. 2{\psi }_1(x\mbox{,}t\mbox{;}q)\frac{{\partial }^{\alpha }{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^{\alpha }}-\right. </math><math>\left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>N^2[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]=</math><math>L[{\psi }_2(X\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}G(x)-</math><math>\frac{1}{s}L\left[\frac{{\partial }^2{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. </math><math>\left. 2{\psi }_2(x\mbox{,}t\mbox{;}q)\frac{{\partial }^{\beta }{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^{\beta }}-\right. </math><math>\left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
|}

and the Laplace operator as

<span id='e0190'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[u_m(x\mbox{,}t)-k_mu_{m-1}(x\mbox{,}t)]=\hslash R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
|}

<span id='e0195'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>L[v_m(x\mbox{,}t)-k_mv_{m-1}(x\mbox{,}t)]=\hslash R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]=</math><math>L\lbrace u_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}x^2-</math><math>\frac{1}{s}L\left\{\frac{{\partial }^2u}{\partial x^2}+\right. </math><math>\left. 2\sum_{i=0}^{m-1}u_i\frac{{\partial }^{\alpha }u_{m-i}}{\partial x^{\alpha }}-\right. </math><math>\left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (37)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]=</math><math>L\lbrace v_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}x^3-</math><math>\frac{1}{s}L\left\{\frac{{\partial }^2v}{\partial x^2}+\right. </math><math>\left. 2\sum_{i=0}^{m-1}v_i\frac{{\partial }^{\beta }v_{m-i}}{\partial x^{\beta }}-\right. </math><math>\left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Obviously, the solution of the ''m''th-order deformation Eqs.  [[#e0190|(35)]] and [[#e0195|(36)]] for <math display="inline">m\geqslant 1</math> becomes

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_m(x\mbox{,}t)=k_mu_{m-1}(x\mbox{,}t)+\hslash L^{-1}\left\{R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\right\}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (38)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_m(x\mbox{,}t)=k_mv_{m-1}(x\mbox{,}t)+\hslash L^{-1}\left\{R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]\right\}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (39)
|}

On solving the above equations, it gives the following results

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_0=x^2\mbox{,}\quad v_0=x^3\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_1=-\hslash \left(2-5x^4+4\frac{x^{4-\alpha }}{\Gamma (3-\alpha )}\right)t\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_1=-\hslash \left(6x-5x^4+12\frac{x^{6-\beta }}{\Gamma (4-\beta )}\right)t\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u_2=(\hslash +n)u_1+{\hslash }^2(-84x^2+30x^5+35x^6+</math><math>4(4-\alpha )(3-\alpha )+8)\frac{x^{2-\alpha }}{\Gamma (3-\alpha )}-</math><math>\left(\frac{4(7-\alpha )}{\Gamma (3-\alpha )}+\frac{240}{\Gamma (5-\alpha )}+\right. </math><math>\left. \frac{20}{\Gamma (3-\alpha )}\right)x^{6-\alpha }+</math><math>\left(\frac{8\Gamma (5-\alpha )}{\Gamma (5-2\alpha )}+\right. </math><math>\left. \frac{16}{\Gamma (3-\alpha )}\right)\frac{x^{6-2\alpha }}{\Gamma (3-\alpha )}-</math><math>12(8-\beta )\frac{x^{7-\beta }}{\Gamma (4-\beta )})\frac{t^2}{\Gamma (3-1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>v_2=(\hslash +n)v_1+{\hslash }^2\left(-84x^2+30x^5+\right. </math><math>\left. 35x^6+\left(\frac{12(6-\beta )(5-\beta )}{\Gamma (4-\beta )}+\right. \right. </math><math>\left. \left. \frac{12}{\Gamma (2-\beta )}+\frac{72}{\Gamma (4-\beta )}\right)x^{4-\beta }-\right. </math><math>\left. \left(\frac{12(8-\beta )}{\Gamma (4-\beta )}+\right. \right. </math><math>\left. \left. \frac{240}{\Gamma (5-\beta )}+\frac{60}{\Gamma (4-\beta )}\right)x^{7-\beta }+\right. </math><math>\left. \left(\frac{24\Gamma (7-\beta )}{\Gamma (7-2\beta )}+\right. \right. </math><math>\left. \left. \frac{144}{\Gamma (4-\beta )}\right)\frac{x^{9-2\beta }}{\Gamma (4-\beta )}-\right. </math><math>\left. 4(7-\alpha )\frac{x^{6-\alpha }}{\Gamma (3-\alpha )}\right)\frac{t^2}{\Gamma (3-1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (40)
|}

and so on.

In this manner the rest of the iterative components can be found. Therefore, the q-HATM approximate series solutions of system of Eq. [[#e0170|(31)]] are presented as

<span id='e0245'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u(x\mbox{,}t)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m</math>
|-
|<math>v(x\mbox{,}t)=v_0(x\mbox{,}t)+\sum_{m=1}^{\infty }v_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41)
|}

Eq. [[#e0245|(41)]] represents the family of q-HATM solutions of system of Eq. [[#e0170|(31)]], which converges rapidly. Although exact solution of system of Eq. [[#e0170|(31)]] is not available so diagrammatical representations to elucidate proposed method. If we set <math display="inline">n=1</math> in Eq. [[#e0245|(41)]], we get the solutions obtained by using HAM as a special case of q-HATM solution. On the other hand, if we let <math display="inline">n=1</math> and <math display="inline">\hslash =-1</math>, then we arrive at the results obtained by HPM [[#b0180|[36]]], DTM [[#b0185|[37]]] and VIM [[#b0190|[38]]] as a particular case of q-HATM solution. Thus, we can conclude that the results obtained by using q-HATM contain the results obtained with the help of HAM, HPM, DTM and VIM.

==4. Conclusions==

In this paper, the q-homotopy analysis transform method (q-HATM) has been successfully employed to time- and space- fractional coupled Burgers’ equations with entice solution procedures. The validity of family of purposed solution in large admissible convergent region, is noticed by <math display="inline">\hslash </math> and <math display="inline">n</math>-curves. Positivisms of proposed method is that it provides nonlocal effect, promising large convergence region, straight forward solution procedure and free from any assumption, calculating complicated polynomials and integrations, small/large physical parameters. Thus, it can be winded up that the scheme is highly systematic and can be applied to investigate nonlinear mathematical models describing realistic problems.

==Acknowledgments==

The authors are highly grateful to the anonymous referee for carefully reading the paper and for his constructive comments and suggestions which have improved the paper.

==References==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='b0005'></span>
[[#b0005|[1]]] K.B. Oldham, J. Spanier; The Fractional Calculus; Acad. Press, New York, NY, USA (1974)</li>
<li><span id='b0010'></span>
[[#b0010|[2]]] K.S. Miller, B. Ross; An Introduction to the Fractional Calculus and Fractional Differential Equations; Willey, New York, NY, USA (1993)</li>
<li><span id='b0015'></span>
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