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==Abstract==
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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.
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==Keywords==
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Laplace transform method; q-Homotopy analysis transform method; Fractional coupled Burgers’ equations; ℏℏ and nn-curves
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==1. Introduction==
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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]]].
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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.
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In this letter, we consider the following system of fractional coupled Burgers’ equations
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<span id='e0005'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|-
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|<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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
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|}
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subject to the initial conditions
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>u(x\mbox{,}0)=u_0=F(x)\mbox{,}\quad v(x\mbox{,}0)=</math><math>v_0=G(x)\mbox{,}</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
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|}
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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.
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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.
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==2. Basic idea of q-HATM==
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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:
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<span id='e0015'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
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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.
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By applying the Laplace transform operator on both sides of Eq. [[#e0015|(3)]], we get the following equation:
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>L\left[D_t^{\alpha }u\right]+L[Ru]+L[Nu]=L[g(x\mbox{,}t)]\mbox{.}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
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Making use of the differentiation property of the Laplace transform, it yields
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
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On simplifying, the above equation reduces to
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
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We define the nonlinear operator as
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
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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:
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<span id='e0040'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
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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
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
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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
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
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where
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
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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
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<span id='e0060'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
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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)]].
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Define the vectors in the following manner
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
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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:
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
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Finally applying the inverse Laplace transform, we have
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
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where the value of <math display="inline">R_m({\overset{\rightarrow}{u}}_{m-1})</math> is given as
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
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and <math display="inline">k_m</math> is defined as
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{| style="text-align: center; margin:auto;" 
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| <math>k_m=\begin{array}{ll}
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0\mbox{,} & m\leqslant 1\mbox{,}\\
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n\mbox{,} & m>1\mbox{.}
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\end{array}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
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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).
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==3. Application of the method==
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==Example 1.                     ==
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We consider the time-fractional coupled Burgers’ equation [[#b0170|[34]]] and [[#b0180|[36]]]
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <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>
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|<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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
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subject to the initial conditions
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
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Eqs. [[#e0090|(18)]] and [[#e0095|(19)]] advise that we define the nonlinear operator as
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
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and the Laplace operator as
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
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<span id='e0115'></span>
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{| style="text-align: center; margin:auto;" 
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
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where
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| <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>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
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|}
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Appling the inverse of Laplace transform on Eqs. [[#e0110|(22)]] and [[#e0115|(23)]], we get
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
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|}
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On solving the above equations, we have
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>u_0=sinx\mbox{,}\quad v_0=sinx\mbox{,}</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
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|}
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and so on.
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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
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<span id='e0160'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
416
|-
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|<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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
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|}
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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
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
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|}
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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.
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<span id='f0005'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[Image:draft_Content_928832423-1-s2.0-S1110016816300424-gr1.jpg|center|425px|(a)–(h) Represent six order approximations HAM (q-HATM, n=1) solution ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 1.
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(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)]].
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</span>
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|}
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[[#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]]).
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==Example 2.                     ==
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Finally, we consider the following space-fractional coupled Burgers’ equation [[#b0170|[34]]] and [[#b0180|[36]]]
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<span id='e0170'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|-
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|<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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
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|}
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subject to the initial conditions
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<span id='e0175'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
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|}
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<span id='t0005'></span>
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{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
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|+
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Table 1.
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Comparative study between HPM [[#b0180|[36]]] and q-HATM.
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|-
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! colspan="4" | Absolute error <math display="inline">E_4(u)=\vert u_{exa.}-u_{app.}\vert </math>
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|-
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! <math display="inline">x</math>
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! <math display="inline">t</math>
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! HPM [[#b0180|[36]]], <math display="inline">\vert u_{exa.}-u_{app.}\vert </math>
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! q-HATM, <math display="inline">\vert u_{exa.}-u_{app.}\vert \mbox{,}\quad \quad (\hslash \mbox{,}n)</math>
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|-
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| rowspan="5" | −10
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| rowspan="5" | 0.07
510
| rowspan="5" | 7.5 × 10<sup>−9</sup>
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| 7.5 × 10<sup>−9</sup>,    (−1, 1)
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|-
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| 4 × 10<sup>−10</sup>,    (−0.99, 1)
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|-
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| 2.0 × 10<sup>−9</sup>,    (−0.98, 1)
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|-
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| 1.5 × 10<sup>−9</sup>,    (−4.98, 5)
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|-
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| 1.0 × 10<sup>−9</sup>,    (−58.5, 60)
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|-
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| rowspan="4" | 15
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| rowspan="4" | 0.2
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| rowspan="4" | 0.0000016779
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| 0.0000016779,    (−1, 1)
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|-
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| 4.326 × 10<sup>−7</sup>,    (−0.99, 1)
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|-
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| 5.09 × 10<sup>−8</sup>,    (−9.8, 10)
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|-
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| 3.470 × 10<sup>−7</sup>,    (−44.5, 45)
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|}
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<span id='f0010'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 2.
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(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)]].
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</span>
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|}
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<span id='f0015'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 3.
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(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.
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</span>
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|}
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<span id='f0020'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 4.
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(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>.
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</span>
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|}
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<span id='f0025'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 5.
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(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>.
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</span>
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|}
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<span id='f0030'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 6.
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(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.
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</span>
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|}
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<span id='f0035'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[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 ...]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 7.
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(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.
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</span>
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|}
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Eqs. [[#e0170|(31)]] and [[#e0175|(32)]] advise that we define the nonlinear operator as
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
664
|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (33)
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <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>
679
|}
680
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
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|}
682
683
and the Laplace operator as
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<span id='e0190'></span>
686
{| class="formulaSCP" style="width: 100%; text-align: center;" 
687
|-
688
| 
689
{| style="text-align: center; margin:auto;" 
690
|-
691
| <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>
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|}
693
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
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|}
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<span id='e0195'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
700
{| style="text-align: center; margin:auto;" 
701
|-
702
| <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>
703
|}
704
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
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|}
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where
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
710
|-
711
| 
712
{| style="text-align: center; margin:auto;" 
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|-
714
| <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>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (37)
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
722
{| style="text-align: center; margin:auto;" 
723
|-
724
| <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>
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|}
726
| style="width: 5px;text-align: right;white-space: nowrap;" | 
727
|}
728
729
Obviously, the solution of the ''m''th-order deformation Eqs.  [[#e0190|(35)]] and [[#e0195|(36)]] for <math display="inline">m\geqslant 1</math> becomes
730
731
{| class="formulaSCP" style="width: 100%; text-align: center;" 
732
|-
733
| 
734
{| style="text-align: center; margin:auto;" 
735
|-
736
| <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>
737
|}
738
| style="width: 5px;text-align: right;white-space: nowrap;" | (38)
739
|}
740
741
{| class="formulaSCP" style="width: 100%; text-align: center;" 
742
|-
743
| 
744
{| style="text-align: center; margin:auto;" 
745
|-
746
| <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>
747
|}
748
| style="width: 5px;text-align: right;white-space: nowrap;" | (39)
749
|}
750
751
On solving the above equations, it gives the following results
752
753
{| class="formulaSCP" style="width: 100%; text-align: center;" 
754
|-
755
| 
756
{| style="text-align: center; margin:auto;" 
757
|-
758
| <math>u_0=x^2\mbox{,}\quad v_0=x^3\mbox{,}</math>
759
|}
760
| style="width: 5px;text-align: right;white-space: nowrap;" | 
761
|}
762
763
{| class="formulaSCP" style="width: 100%; text-align: center;" 
764
|-
765
| 
766
{| style="text-align: center; margin:auto;" 
767
|-
768
| <math>u_1=-\hslash \left(2-5x^4+4\frac{x^{4-\alpha }}{\Gamma (3-\alpha )}\right)t\mbox{,}</math>
769
|}
770
| style="width: 5px;text-align: right;white-space: nowrap;" | 
771
|}
772
773
{| class="formulaSCP" style="width: 100%; text-align: center;" 
774
|-
775
| 
776
{| style="text-align: center; margin:auto;" 
777
|-
778
| <math>v_1=-\hslash \left(6x-5x^4+12\frac{x^{6-\beta }}{\Gamma (4-\beta )}\right)t\mbox{,}</math>
779
|}
780
| style="width: 5px;text-align: right;white-space: nowrap;" | 
781
|}
782
783
{| class="formulaSCP" style="width: 100%; text-align: center;" 
784
|-
785
| 
786
{| style="text-align: center; margin:auto;" 
787
|-
788
| <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>
789
|}
790
| style="width: 5px;text-align: right;white-space: nowrap;" | 
791
|}
792
793
{| class="formulaSCP" style="width: 100%; text-align: center;" 
794
|-
795
| 
796
{| style="text-align: center; margin:auto;" 
797
|-
798
| <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>
799
|}
800
| style="width: 5px;text-align: right;white-space: nowrap;" | (40)
801
|}
802
803
and so on.
804
805
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
806
807
<span id='e0245'></span>
808
{| class="formulaSCP" style="width: 100%; text-align: center;" 
809
|-
810
| 
811
{| style="text-align: center; margin:auto;" 
812
|-
813
| <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>
814
|-
815
|<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>
816
|}
817
| style="width: 5px;text-align: right;white-space: nowrap;" | (41)
818
|}
819
820
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.
821
822
==4. Conclusions==
823
824
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.
825
826
==Acknowledgments==
827
828
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.
829
830
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831
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