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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\quad (n\geqslant 1)}
to solve strongly nonlinear differential equations. The proposed technique provides Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

-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 [1], [2], [3], [4] and [5]. The theory of fractional differential equations translates the reality of nature excellently in a better and systematic manner [6], [7], [8], [9], [10] and [11]. In recent years, many authors have investigated partial differential equations of fractional order by various techniques such as homotopy analysis technique [12], [13] and [14], operational matrix based method [15], and tau method [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 [17] and [18]. The q-HAM is an extension of the embedding parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q\in [0\mbox{,}1]}

arising in the study by Liao HAM [19], [20] and [21] to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q\in \left[0\mbox{,}\frac{1}{n}\right]}
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 [22], [23], [24], [25], [26], [27] and [28]. The HAM has also been united with Laplace transform to bringing out highly effective technique to investigate nonlinear problems of physical importance [29], [30] and [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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^{{\alpha }_1}u}{\partial t^{{\alpha }_1}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^2u}{\partial x^2}+2u\frac{{\partial }^{{\alpha }_2}u}{\partial x^{{\alpha }_2}}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<{\alpha }_i\leqslant 1\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^{{\beta }_1}v}{\partial t^{{\beta }_1}}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^2v}{\partial x^2}+2v\frac{{\partial }^{{\beta }_2}v}{\partial x^{{\beta }_2}}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<{\beta }_i\leqslant 1

(1)

subject to the initial conditions

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(2)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left({\alpha }_i\mbox{,}{\beta }_i\mbox{,}\quad i=\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 1\mbox{,}2\right)

are parameters describing the order of the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(i=1\right)}
and space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(i=2\right)}
fractional derivatives, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}
is the space domain and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}
is time. When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_i={\beta }_i=1}

, then the system of Eq. (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. (1) forming a simple model of sedimentations or evolution of scaled volume concentrations of two kinds of gravity [32]. A logistic and remarkable study has been done by number of researchers pertaining to coupled Burgers’ equations [33], [34], [35], [36] and [37]. Recently, Prakash et al. [38] numerically solve the system of Eq. (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. (1) with concept of fractional Laplace transform of the Caputo derivative [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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D_t^{\alpha }u(x\mbox{,}t)+Ru(x\mbox{,}t)+Nu(x\mbox{,}t)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g(x\mbox{,}t)\mbox{,}\quad n-1<\alpha \leqslant n

(3)

where is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_t^{\alpha }u(x\mbox{,}t)}

represents the fractional derivative of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)}
in terms of Caputo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R}
indicates the linear differential operator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
represents the general nonlinear differential operator and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g(x\mbox{,}t)}
is the source term.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\left[D_t^{\alpha }u\right]+L[Ru]+L[Nu]=L[g(x\mbox{,}t)]\mbox{.}

(4)

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s^{\alpha }L[u]-\sum_{k=0}^{n-1}s^{\alpha -k-1}u^{\left(k\right)}(x\mbox{,}0)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[Ru]+L[Nu]=L[g(x\mbox{,}t)]\mbox{.}

(5)

On simplifying, the above equation reduces to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[u]-\frac{1}{s^{\alpha }}\sum_{k=0}^{n-1}s^{\alpha -k-1}u^{\left(k\right)}(x\mbox{,}0)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s^{\alpha }}[L[Ru]+L[Nu]-L[g(x\mbox{,}t)]]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0\mbox{.}

(6)

We define the nonlinear operator as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N[\phi (x\mbox{,}t\mbox{;}q)]=L[\phi (x\mbox{,}t\mbox{;}q)]- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s^{\alpha }}\sum_{k=0}^{n-1}s^{\alpha -k-1}{\phi }^{\left(k\right)}(x\mbox{,}t\mbox{;}q)(0^+)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s^{\alpha }}[L[R\phi (x\mbox{,}t\mbox{;}q)]+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[N\phi (x\mbox{,}t\mbox{;}q)]-L[g(x\mbox{,}t)]]\mbox{,}

(7)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q\in [0\mbox{,}1/n]}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi (x\mbox{,}t\mbox{;}q)}
are real functions of x, t and q. We construct a homotopy as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (1-nq)L[\phi (x\mbox{,}t\mbox{;}q)-u_0(x\mbox{,}t)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hslash qH(x\mbox{,}t)N[\phi (x\mbox{,}t\mbox{;}q)]\mbox{,}

(8)

where L   denotes the Laplace transform, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\geqslant 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q\in \left[0\mbox{,}\frac{1}{n}\right]}

is the embedding parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(x\mbox{,}t)}
denotes a nonzero auxiliary function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash \not =0}
is an auxiliary parameter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_0(x\mbox{,}t)}
is an initial guess of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi (x\mbox{,}t\mbox{;}q)}
is an unknown function. It is obvious that, when the embedding parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=\frac{1}{n}}

, it holds the result

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \phi (x\mbox{,}t\mbox{;}0)=u_0(x\mbox{,}t)\mbox{,}\quad \phi \left(x\mbox{,}t\mbox{;}\frac{1}{n}\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(x\mbox{,}t)\mbox{,}

(9)

respectively. Thus, as q   increases from 0 to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{1}{n}} , the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi (x\mbox{,}t\mbox{;}q)}

varies from the initial guess Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_0(x\mbox{,}t)}
to the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)}

. Expanding the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi (x\mbox{,}t\mbox{;}q)}

in series form by employing Taylor theorem about q, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \phi (x\mbox{,}t\mbox{;}q)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t)q^m\mbox{,}

(10)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_m(x\mbox{,}t)=\frac{1}{m!}{\frac{{\partial }^m\phi (x\mbox{,}t\mbox{;}q)}{\partial q^m}}_{q=0}\mbox{.}

(11)

If the auxiliary linear operator, the initial guess, the auxiliary parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\mbox{,}\hslash }

and the auxiliary function are properly chosen, the series (10) converges at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=\frac{1}{n}\mbox{,}}
and then we have

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(12)

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

Define the vectors in the following manner

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\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{.}

(13)

Now, differentiating the zeroth-order deformation Eq. (8)m-times with respect to q and then dividing them by m  ! and finally setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q=0\mbox{,}}

we get the following mth-order deformation equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{.}

(14)

Finally applying the inverse Laplace transform, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(15)

where the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_m({\overset{\rightarrow}{u}}_{m-1})}

is given as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(16)

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_m}

is defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k_m=\begin{array}{ll} 0\mbox{,} & m\leqslant 1\mbox{,}\\ n\mbox{,} & m>1\mbox{.} \end{array}

(17)

From study it should be analyzed that in special case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1} , q-HATM reduces to the homotopy analysis transform method (HATM).

Example 1.

We consider the time-fractional coupled Burgers’ equation [34] and [36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^2u}{\partial x^2}+2u\frac{\partial u}{\partial x}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\alpha \leqslant 1\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^{\beta }v}{\partial t^{\beta }}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\partial }^2v}{\partial x^2}+2v\frac{\partial v}{\partial x}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\beta \leqslant 1\mbox{,}

(18)

subject to the initial conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(x\mbox{,}0)=u_0=F(x)=sinx\mbox{,}\quad v(x\mbox{,}0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v_0=G(x)=sinx\mbox{,}

(19)

Eqs. (18) and (19) advise that we define the nonlinear operator as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N^1[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[{\psi }_1(x\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}F(x)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{S^{\alpha }}L\left[\frac{{\partial }^2{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2{\psi }_1(x\mbox{,}t\mbox{;}q)\frac{\partial {\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}

(20)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N^2[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[{\psi }_2(X\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}G(x)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{S^{\beta }}L\left[\frac{{\partial }^2{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2{\psi }_2(x\mbox{,}t\mbox{;}q)\frac{\partial {\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}

(21)

and the Laplace operator as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(22)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(23)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\lbrace u_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}sinx- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s^{\alpha }}L\left\{\frac{{\partial }^2u}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2\sum_{i=0}^{m-1}u_i\frac{\partial u_{m-i}}{\partial x}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}

(24)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\lbrace v_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}sinx- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s^{\beta }}L\left\{\frac{{\partial }^2v}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2\sum_{i=0}^{m-1}v_i\frac{\partial v_{m-i}}{\partial x}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}

(25)

Appling the inverse of Laplace transform on Eqs. (22) and (23), we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(26)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{.}

(27)

On solving the above equations, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_0=sinx\mbox{,}\quad v_0=sinx\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_1=\frac{\hslash sin{xt}^{\alpha }}{\Gamma (\alpha +1)}\mbox{,}\quad v_1= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\hslash sin{xt}^{\beta }}{\Gamma (\beta +1)}\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_2=\frac{\hslash (\hslash +n)sin{xt}^{\alpha }}{\Gamma (\alpha +1)}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\hslash }^2sinx(1-2cosx)\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2{\hslash }^2sinxcosx\frac{t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v_2=\frac{\hslash (\hslash +n)sin{xt}^{\beta }}{\Gamma (\beta +1)}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\hslash }^2sinx(1-2cosx)\frac{t^{2\beta }}{\Gamma (2\beta +1)}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2{\hslash }^2sinxcosx\frac{t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\mbox{,}

(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. (18) is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{.}

(29)

If we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1}

in Eq. (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash =-1}

, then we arrive at the results found by using HPM [36], DTM [37] and VIM [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =\beta =1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash =-1\mbox{,}\quad n=1}
then clearly we can observe that the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\sum }_{m=0}^Nu_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m}
when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N\rightarrow \infty }
converges to the exact solution of classical coupled Burgers’ equations, which is the special case of the system of Eq. (18) and is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u=v=sinx\left(1-t+\frac{t^2}{\Gamma (2+1)}-\frac{t^3}{\Gamma (3+1)}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \cdots +{\left(-1\right)}^r\frac{t^r}{\Gamma (r+1)}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \cdots \right)=e^{-t}sinx\mbox{.}

(30)

For simplicity, here we consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u=u(x\mbox{,}t)=v(x\mbox{,}t)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =\beta }
for every case. The efficiency of purpose method is noticed through the absolute error between exact solution and second order approximation shown in Fig. 1c and e.

Figure 1. (a)–(h) Represent six order approximations HAM (q-HATM, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1} ) solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)=v(x\mbox{,}t)} of system of Eq. (18).

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash }

-curve corresponding to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n\quad (n\geqslant 1)}

(see Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7).

Example 2.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial u}{\partial t}=\frac{{\partial }^2u}{\partial x^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2u\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\alpha \leqslant 1\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial v}{\partial t}=\frac{{\partial }^2v}{\partial x^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2v\frac{{\partial }^{\beta }v}{\partial x^{\beta }}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial (uv)}{\partial x}\mbox{,}\quad 0<\beta \leqslant 1

(31)

subject to the initial conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(x\mbox{,}0)=u_0=F(x)=x^2\mbox{,}\quad v(x\mbox{,}0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v_0=G(x)=x^3\mbox{,}

(32)

Table 1. Comparative study between HPM [36] and q-HATM.

Absolute error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_4(u)=\vert u_{exa.}-u_{app.}\vert }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}	HPM [36], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vert u_{exa.}-u_{app.}\vert }	q-HATM, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vert u_{exa.}-u_{app.}\vert \mbox{,}\quad \quad (\hslash \mbox{,}n)}
−10	0.07	7.5 × 10−9	7.5 × 10−9,    (−1, 1)
			4 × 10−10,    (−0.99, 1)
			2.0 × 10−9,    (−0.98, 1)
			1.5 × 10−9,    (−4.98, 5)
			1.0 × 10−9,    (−58.5, 60)
15	0.2	0.0000016779	0.0000016779,    (−1, 1)
			4.326 × 10−7,    (−0.99, 1)
			5.09 × 10−8,    (−9.8, 10)
			3.470 × 10−7,    (−44.5, 45)

Figure 2. (a)–(f) Show the six order approximate q-HATM solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)=v(x\mbox{,}t)} of system of Eq. (18).

Figure 3. (a)–(d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} -curves at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=20\mbox{,}\quad t=0.002} of system of Eq. (18) and show the valid convergence range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash } and asymptotic behaviour of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)=v(x\mbox{,}t)} respectively with different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =\beta } (3a) for HAM, convergence range is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -1.99\leqslant \hslash <0} (3c) for q-HATM, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=20} convergence range is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -39.81\leqslant \hslash <0} (3b) at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash =-1} and (3d) at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash =-19.8} , show the validity of corresponding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash } -curves.

Figure 4. (a)–(f) Show the second order approximate q-HATM surface solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v(x\mbox{,}t)} of system (31) with different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\hslash \mbox{,}n\mbox{,}\alpha \mbox{,}\beta \right)} versus time variable t   and space variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x} .

Figure 5. (a)–(d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash } -curves at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=t=0.05} for second order approximation of system of fractional Eq. (31) and show the valid convergence range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v(x\mbox{,}t)} (5a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -2.021\leqslant \hslash <0} and (5b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -1.968\leqslant \hslash <0} for HAM; (5c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -10.18\leqslant \hslash <0} and (5d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -9.85\leqslant \hslash <0} for q-HATM, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=5} .

Figure 6. (a)–(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} -curves at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=t=0.05} for second order approximation of system (31) and show the asymptotic behavior of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v(x\mbox{,}t)} also describe the validity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash } -curve.

Figure 7. (a)–(b) Show the comparative behaviors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u(x\mbox{,}t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v(x\mbox{,}t)} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=0.05} versus space variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x} . It’s clear to see that both the functions are continuously increasing functions.

Eqs. (31) and (32) advise that we define the nonlinear operator as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N^1[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[{\psi }_1(x\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}F(x)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s}L\left[\frac{{\partial }^2{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2{\psi }_1(x\mbox{,}t\mbox{;}q)\frac{{\partial }^{\alpha }{\psi }_1(x\mbox{,}t\mbox{;}q)}{\partial x^{\alpha }}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}

(33)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N^2[{\psi }_1(x\mbox{,}t\mbox{;}q)\mbox{,}{\psi }_2(x\mbox{,}t\mbox{;}q)]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L[{\psi }_2(X\mbox{,}t\mbox{;}q)]-\left(1-\frac{k_m}{n}\right)\frac{1}{s}G(x)- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s}L\left[\frac{{\partial }^2{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2{\psi }_2(x\mbox{,}t\mbox{;}q)\frac{{\partial }^{\beta }{\psi }_2(x\mbox{,}t\mbox{;}q)}{\partial x^{\beta }}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial ({\psi }_1(x\mbox{,}t\mbox{;}q){\psi }_2(x\mbox{,}t\mbox{;}q))}{\partial x}\right]\mbox{,}

(34)

and the Laplace operator as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(35)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(36)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_{1\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\lbrace u_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}x^2- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s}L\left\{\frac{{\partial }^2u}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2\sum_{i=0}^{m-1}u_i\frac{{\partial }^{\alpha }u_{m-i}}{\partial x^{\alpha }}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}

(37)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_{2\mbox{,}m}\left[{\overset{\rightarrow}{u}}_{m-1}\mbox{,}{\overset{\rightarrow}{v}}_{m-1}\right]= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\lbrace v_{m-1}(x\mbox{,}t)\rbrace -\left(1-\frac{k_m}{n}\right)\frac{1}{s}x^3- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{s}L\left\{\frac{{\partial }^2v}{\partial x^2}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 2\sum_{i=0}^{m-1}v_i\frac{{\partial }^{\beta }v_{m-i}}{\partial x^{\beta }}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{\partial }{\partial x}\left(\sum_{i=0}^{m-1}u_iv_{m-1-i}\right)\right\}\mbox{.}

Obviously, the solution of the mth-order deformation Eqs. (35) and (36) for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\geqslant 1}

becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(38)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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{,}

(39)

On solving the above equations, it gives the following results

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_0=x^2\mbox{,}\quad v_0=x^3\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_1=-\hslash \left(2-5x^4+4\frac{x^{4-\alpha }}{\Gamma (3-\alpha )}\right)t\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v_1=-\hslash \left(6x-5x^4+12\frac{x^{6-\beta }}{\Gamma (4-\beta )}\right)t\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_2=(\hslash +n)u_1+{\hslash }^2(-84x^2+30x^5+35x^6+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4(4-\alpha )(3-\alpha )+8)\frac{x^{2-\alpha }}{\Gamma (3-\alpha )}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(\frac{4(7-\alpha )}{\Gamma (3-\alpha )}+\frac{240}{\Gamma (5-\alpha )}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{20}{\Gamma (3-\alpha )}\right)x^{6-\alpha }+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(\frac{8\Gamma (5-\alpha )}{\Gamma (5-2\alpha )}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \frac{16}{\Gamma (3-\alpha )}\right)\frac{x^{6-2\alpha }}{\Gamma (3-\alpha )}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 12(8-\beta )\frac{x^{7-\beta }}{\Gamma (4-\beta )})\frac{t^2}{\Gamma (3-1)}\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v_2=(\hslash +n)v_1+{\hslash }^2\left(-84x^2+30x^5+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 35x^6+\left(\frac{12(6-\beta )(5-\beta )}{\Gamma (4-\beta )}+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. \frac{12}{\Gamma (2-\beta )}+\frac{72}{\Gamma (4-\beta )}\right)x^{4-\beta }-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left(\frac{12(8-\beta )}{\Gamma (4-\beta )}+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. \frac{240}{\Gamma (5-\beta )}+\frac{60}{\Gamma (4-\beta )}\right)x^{7-\beta }+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left(\frac{24\Gamma (7-\beta )}{\Gamma (7-2\beta )}+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. \frac{144}{\Gamma (4-\beta )}\right)\frac{x^{9-2\beta }}{\Gamma (4-\beta )}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. 4(7-\alpha )\frac{x^{6-\alpha }}{\Gamma (3-\alpha )}\right)\frac{t^2}{\Gamma (3-1)}\mbox{,}

(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. (31) are presented as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u(x\mbox{,}t)=u_0(x\mbox{,}t)+\sum_{m=1}^{\infty }u_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): v(x\mbox{,}t)=v_0(x\mbox{,}t)+\sum_{m=1}^{\infty }v_m(x\mbox{,}t){\left(\frac{1}{n}\right)}^m

(41)

Eq. (41) represents the family of q-HATM solutions of system of Eq. (31), which converges rapidly. Although exact solution of system of Eq. (31) is not available so diagrammatical representations to elucidate proposed method. If we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1}

in Eq. (41), we get the solutions obtained by using HAM as a special case of q-HATM solution. On the other hand, if we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash =-1}

, then we arrive at the results obtained by HPM [36], DTM [37] and VIM [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hslash }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

-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.

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