Glossary
j j = 1 corresponds to the first equation, while j = 2 to the second equation
wj = wj(x, t) Corresponds to the concentration for j = 1,2 from equation (1 ),
the coupled system of reaction-advection-diffusion equations
wj = wj(x, t) Corresponds to the wave profile for j = 1, 2 from equation (23 ),
the coupled system of Burger's equations
fj(x, t) Corresponds to the source function for j = 1, 2
mj v Non-negative integers
x spatial variable
t Temporal variable
αj Diffusion coefficient for j = 1, 2
βj Coupling parameter for j = 1, 2
γj Advection coefficient for j = 1, 2
θj Reaction coefficient for j = 1, 2
η1, λ1 Real numbers (constants)
η2, λ2 Coupling parameter for j = 1, 2
1. Introduction
Physical systems involving more than one process are greatly admired in the essential areas of science and engineering. One of these models that describes physical phenomena in a variety of processes is the generalized coupled system of reaction-advection-diffusion dynamical equations in porous medium that follows from the modification of [1] as (integer-order)
$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}{w}_{1} & = & {\alpha }_{1}{\partial }_{{xx}}{w}_{1}+{\beta }_{1}{\partial }_{{xx}}{w}_{2}+{\eta }_{1}{w}_{1}{\partial }_{x}{w}_{1}+{\eta }_{2}{\partial }_{x}({w}_{1}{w}_{2})\\ & & -{\gamma }_{1}{w}_{1}^{{m}_{1}}{\partial }_{x}{w}_{1}+{\theta }_{1}{w}_{1}(1-{w}_{1})+{f}_{1}(x,t),\\ {\partial }_{t}{w}_{2} & = & {\alpha }_{2}{\partial }_{{xx}}{w}_{2}+{\beta }_{2}{\partial }_{{xx}}{w}_{1}+{\lambda }_{1}{w}_{2}{\partial }_{x}{w}_{2}+{\lambda }_{2}{\partial }_{x}({w}_{1}{w}_{2})\\ & & -{\gamma }_{1}{w}_{2}^{{m}_{2}}{\partial }_{x}{w}_{2}+{\theta }_{2}{w}_{2}(1-{w}_{2})+{f}_{2}(x,t),\end{array}\end{eqnarray}$
where w1 = w1(x, t), and w2 = w2(x, t), are the respective concentrations, α1, α2 are the respective diffusion coefficients, γ1, γ2 are the respective advection coefficients, θ1, θ2 are the respective reaction coefficients, η1, λ1 are real constants, m1, m2 are non-negative integers, β1, β2, η2, λ2 are coupling parameters; while f1(x, t) and f2(x, t) are the respective source functions, otherwise known as inhomogeneous terms Additionally, ${\partial }_{t}=\tfrac{\partial }{\partial t},$ ${\partial }_{x}=\tfrac{\partial }{\partial x},$ and ${\partial }_{{xx}}=\tfrac{{\partial }^{2}}{\partial {x}^{2}};$ while m1 and m2 are non-negative integers. This model takes into consideration three important phenomena involving the motion of particles in a given medium, read [2] and the references therein. In particular, the first process, which is a reaction, describes possible processes like the adsorption, decay and reaction of the substances with other components. More so, the advection process refers to the transport of a substance or heat by the flow of a liquid; while diffusion is the movement of a substance from a region of high concentration to a region of low concentration throughout the physical domain of the problem' [3]. Moreover, to highlight some universal or rather significant features of the governing model given in the above equation, several important coupled dynamical models may be extracted from this very model including, for instance, the reaction-advection model, reaction-diffusion model, advection-diffusion model, the system of Burger's equation, and many other linear and nonlinear evolution equations to mention a few.Furthermore, having stated some universal nature of the governing model, and on the other hand, its application in a variety of areas of physics like fluid dynamics, environmental engineering, groundwater, and biological model; the present paper aims further to extract one of the celebrated models from the governing model that is called the system of coupled Burger's equations [4]. This model plays a vital role in modelling fluid flow problems, among others. It is pertinent to recall that the classical Burger's equation plays an eminent part in modeling various phenomena in fluid dynamics, traffic flow, gas dynamics and nonlinear acoustics, to state a few. Additionally, different mathematical methods consisting of analytical and computations have in the past literature been employed to examine the coupled system of Burger's equation. One may, however, read the works of [5, 6] and the references therein with regard to various methods. Equally, one may find different extensions and modifications of this very important extract, including, for example, the coupled inhomogeneous singular Burger's equations [7] among others. Besides, this model is an important coupled nonlinear evolution equation [8–11] that admits various forms of solutions. More so, it is important to recall that various integration schemes have been devised and successfully utilized on nonlinear evolution equations, see [12–17] and the references therewith.
However, the current study examines the system of coupled Burger's equations; being one of the astonishing members of the generalized reaction-advection-diffusion dynamical model with vast applications. In light of this, the study would employ two integration schemes, together with a modification of the famous Adomian decomposition method for the numerical study. More so, the modified extended tanh expansion method (METEM) [18–20], and the Kudryashov method (KM) [21–23] would be considered as the analytical methods; while a version of the Adomian's decomposition method [24] that we call in this study the numerical Laplace decomposition method (NLDM) [25, 26] will be adopted as the numerical approach. In addition, a number of plots and tables will be provided in the end, to portray the results to be acquired. Also, as for the limitations, some interesting cases the inhomogeneous variant of the coupled system of Burger's equation would be analyzed in the present study (see section 3 ); besides, all the computational simulations and graphical illustrations would be implemented and produced on the Wolfram Mathematica 9, sequentially. Lastly, the paper is organized as follows: section 2 gives the methods of solution. Section 3 presents the application of the methods given in section 2 ; while section 4 gives the application of the NLDM approach with examples. Section 5 discusses the obtained results; while section 6 gives some concluding comments.
2. Methodology
This section under consideration gives the outline of the adopted methodology of the present investigation. Basically, two modes of techniques will be employed, consisting of analytical and numerical approaches. For the analytical method, the modified extended tanh expansion method (METEM) [18–20] and the Kudryashov method (KM) [21–23] will be utilized; while the numerical Laplace decomposition method (NLDM) [25, 26] will be adopted for the numerical purpose. The METEM and KM are elegant analytical methods that construct exact solutions to diverse evolution and Schrodinger equations, with great relevance in nonlinear sciences like optics and plasma. On the other hand, KM mainly gives exponential solution structures. For the applications of these solutions, kindly read section 5 . Moreover, the two analytical methods give a series solution that depends on $M(\in {\mathbb{N}})$ to be computed based on the homogeneous balancing way—this way depends on the orders of the highest linear operator and the nonlinear term present in the governing equation.
2.1. Analytical methods
For the analytical methods, let us make consideration to the generalized nonlinear partial differential equation of the following pattern2 ), one gets an ordinary differential equation of the form3 ), k and c are real constants (nonzero); while D in equation (4 ) is a differential operator.
$\begin{eqnarray}Q(w,{\partial }_{t}w,{\partial }_{x}w,{\partial }_{t}{\partial }_{x}w,{\partial }_{{tt}}w,{\partial }_{{xx}}w,{\partial }_{t}{D}_{{xx}}w,{\partial }_{{xxx}}w,\ldots )=0.\end{eqnarray}$
More so, upon making use of the following wave transformation $\begin{eqnarray}w(x,t)=W(\xi ),\ \xi ={kx}-{ct},\end{eqnarray}$
in equation ( $\begin{eqnarray}\begin{array}{l}R\left(W(\xi ),{DW}(\xi ),{D}^{2}W(\xi ),{D}^{3}W(\xi ),\right.\\ \qquad \left.{D}^{4}W(\xi ),{D}^{5}W(\xi ),...\right)=0,\end{array}\end{eqnarray}$
where in equation (Therefore, we now make consideration to two of the aiming analytical methods as follows
Hence, with the application of the METEM and KM as rightly presented above, the assumed solution in the respective cases given in equations (5 ) and (8 ) is thus substituted into equation (4 ) to yield a polynomial equation in $Psi$(ξ). More so, upon collecting the coefficients of the resulting polynomial, and subsequently setting each coefficient to zero, one gets an algebraic system of equations. Of course, this tedious computation is done via the application of mathematical software. Therefore, we, state here that the Wolfram Mathematica 9 software will be utilized in the present study for all computational as well as graphical purposes.
| i | (i)METEM In accordance with METEM, the following finite series is offered [18–20] $\begin{eqnarray}W(\xi )={a}_{0}+\sum _{j=1}^{M}\left({a}_{j}{{\rm{\Psi }}}^{j}(\xi )+\displaystyle \frac{{b}_{j}}{{{\rm{\Psi }}}^{j}(\xi )}\right),\end{eqnarray}$ for equation (Additionally, the application of METEM requires the function $Psi$(ξ) in equation ( $\begin{eqnarray}D{\rm{\Psi }}-z={{\rm{\Psi }}}^{2},\end{eqnarray}$ that admits the following solution sets $\begin{eqnarray}\begin{array}{rcl}\mathrm{for}\qquad z\lt 0,\qquad {\rm{\Psi }}(\xi ) & = & \left\{\begin{array}{l}-\sqrt{-z}\coth (\sqrt{-z}\xi ),\\ -\sqrt{-z}\tanh (\sqrt{-z}\xi ),\end{array}\right.\\ \mathrm{for}\qquad z\gt 0,\qquad {\rm{\Psi }}(\xi ) & = & \left\{\begin{array}{l}-\sqrt{z}\cot (\sqrt{z}\xi ),\\ \sqrt{z}\tan (\sqrt{z}\xi ).\end{array}\right.\end{array}\end{eqnarray}$ Remarkably, METEM reveals three different types of solutions, including hyperbolic, periodic and irrational solutions. For the hyperbolic and periodic solutions, equation ( |
| ii | (ii)KM According to KM, equation ( $\begin{eqnarray}W(\xi )={a}_{0}+\sum _{j=1}^{M}{a}_{j}{{\rm{\Psi }}}^{j}(\xi ),\end{eqnarray}$ where $M(\in {\mathbb{N}})$ is equally a natural number to be obtained by making a balance between the orders of the highest linear operator, and that of the nonlinear term present; while a0, aj, for j = 1, 2,…,M are unknown constants that would be determined, such that aM ≠ 0, for all M. What's more, KM requires the function $Psi$(ξ) in equation ( $\begin{eqnarray}D{\rm{\Psi }}+{\rm{\Psi }}={{\rm{\Psi }}}^{2},\end{eqnarray}$ such that the following exponential solution is satisfied by the latter equation is given by the function $\begin{eqnarray}{\rm{\Psi }}(\xi )=\displaystyle \frac{1}{1+{{de}}^{\xi }},\end{eqnarray}$ where d is an arbitrary constant. This form of solution expressed in the above equation is basically the only form of solution posed by KM; in comparison with the METEM which gives three types of solutions. |
2.2. NLDM
The current section uses the known Laplace integral transform in conjunction with the famous Adomian decomposition method [24] to present the NLDM procedure. In short, this section derives an iterative closed-form solution to a generalized nonlinear equation in mathematical physics. Thus, to present this approach, we consider the following one-dimensional nonlinear inhomogeneous partial differential equation11 )–(12 ), L is a linear differential operator, N is a nonlinear operator, h(x, t) is an inhomogeneous term; while g(x) is the prescribed nice initial profile.
$\begin{eqnarray}{\partial }_{t}w(x,t)=h(x,t)+{Lw}(x,t)+{Nw}(x,t),\end{eqnarray}$
with the following prescribed initial data $\begin{eqnarray}w(x,0)=g(x).\end{eqnarray}$
Also, from equations (Thus, as the celebrated Laplace integral transform will be utilized, let us define this integral transform and its corresponding inverse transform here as follows [27]11 ) in t alongside using equation (12 ), the equation transforms to the following
$\begin{eqnarray}\begin{array}{l}{\mathscr{L}}\{w(x,t)\}={\displaystyle \int }_{0}^{\infty }w(x,t){{\rm{e}}}^{-{st}}{\rm{d}}{t}={w}^{* }(x,s),\\ \qquad {\mathfrak{R}}(s)\gt 0,\\ {{\mathscr{L}}}^{-1}\{{w}^{* }(x,s)\}=\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{c-i\infty }^{c+i\infty }{{\rm{e}}}^{{st}}{w}^{* }(x,s){\rm{d}}{s}=w(x,t),\\ \qquad c\gt 0.\end{array}\end{eqnarray}$
Therefore, taking the Laplace transform of equation ( $\begin{eqnarray}\begin{array}{rcl}{\mathscr{L}}\{w(x,t)\} & = & {s}^{-1}g(x)+{s}^{-1}{\mathscr{L}}\{h(x,t)\}\\ & & +\,{s}^{-1}{\mathscr{L}}\{{Lw}(x,t)+{Nw}(x,t)\}.\end{array}\end{eqnarray}$
Next, we take the inverse Laplace transform ${{\mathscr{L}}}^{-1}$ of the above equation in s, which gives15 ) in terms of the summations given in equations (16 ) and (17 ), one gets
$\begin{eqnarray}\begin{array}{rcl}w(x,t) & = & g(x)+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{h(x,t)\}\}\\ & & +\,{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{Lw}(x,t)+{Nw}(x,t))\}\}.\end{array}\end{eqnarray}$
More so, via the Adomian's approach, we decompose the unknown solution w(x, t) using the following infinite sum [24–26] $\begin{eqnarray}w(x,t)=\sum _{n=0}^{\infty }{w}_{n}(x,t),\end{eqnarray}$
and the nonlinear termed operator $N\left(w(x,t)\right)$ using the following infinite sum of Adomian's polynomials as follows $\begin{eqnarray}{Nw}(x,t)=\sum _{n=0}^{\infty }{R}_{n},\end{eqnarray}$
where Rn's are the polynomials devised by Adomian [24–26], and to be computed via the following recurrent formula $\begin{eqnarray}{R}_{n}=\displaystyle \frac{1}{n!}\displaystyle \frac{{{\rm{d}}}^{n}}{{\rm{d}}{\zeta }^{n}}{\left[N\left(\sum _{j=0}^{\infty }{\zeta }^{j}{w}_{j}\right)\right]}_{\zeta =0},\quad n=0,1,2,\ldots \end{eqnarray}$
So, by rewriting equation ( $\begin{eqnarray}\begin{array}{rcl}w(x,t) & = & g(x)+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{h(x,t)\}\}\\ & & +\,{{\mathscr{L}}}^{-1}\left[{s}^{-1}{\mathscr{L}}\left[L\sum _{n=0}^{\infty }{w}_{n}(x,t)+\sum _{n=0}^{\infty }{R}_{n}\right]\right].\end{array}\end{eqnarray}$
Lastly, identifying the terms arising from the prescribed initial data and the inhomogeneous function with the first component w0(x, t), and the rest of the terms recurrently follow as suggested by the approach, we thus get the following recurrent scheme $\begin{eqnarray}\left\{\begin{array}{l}{w}_{0}(x,t)=g(x)+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{h(x,t)\}\},\\ {w}_{k+1}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{{Lw}}_{k}(x,t)+{R}_{k}\}\},\quad k\ \geqslant 0,\end{array}\right.\end{eqnarray}$
such that upon taking the net sum of the m-term approximations yields $\begin{eqnarray}{{\rm{\Phi }}}_{m}(x,t)=\sum _{j=0}^{m-1}{w}_{j}(x,t),\end{eqnarray}$
where $\begin{eqnarray}w(x,t)=\mathop{\mathrm{lim}}\limits_{m\to \infty }{{\rm{\Phi }}}_{m}(x,t)=\sum _{j=0}^{\infty }{w}_{j}(x,t).\end{eqnarray}$
3. Application
This section gives the application of the analytical and numerical approaches outlined in the above section. Additionally, the section makes consideration to yet another important nonlinear model that emanates from the generalized coupled reaction-advection-diffusion dynamical model earlier given in equation (1 ). Precisely, we make consideration of the coupled inhomogeneous Burger's equation upon setting the respective diffusion coefficients, advection coefficients, and the reaction coefficients to zero, that is, α1 = 0 = α2, γ1 = 0 = γ2, and θ1 = 0 = θ2. More so, the coupled inhomogeneous Burger's equations of interest reads
$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}{w}_{1} & = & {\partial }_{{xx}}{w}_{1}+{\eta }_{1}{w}_{1}{\partial }_{x}{w}_{1}+{\eta }_{2}{\partial }_{x}({w}_{1}{w}_{2})+{f}_{1}(x,t),\\ {\partial }_{t}{w}_{2} & = & {\partial }_{{xx}}{w}_{2}+{\lambda }_{1}{w}_{2}{\partial }_{x}{w}_{2}+{\lambda }_{2}{\partial }_{x}({w}_{1}{w}_{2})+{f}_{2}(x,t),\end{array}\end{eqnarray}$
where w1 = w1(x, t), and w2 = w2(x, t) are the respective wave profiles, η1, λ1 are real constants, η2, λ2 are coupling parameters; while f1(x, t) and f2(x, t) are the respective source functions. Additionally, Burger's equation has been used to describe a variety of phenomena, including a mathematical model of turbulence and an approximate theory of shock wave flow in viscous fluid. It is a simple model of sedimentation or evolution of scaled volume concentrations of two types of particles in fluid suspensions or colloids under gravity's influence. This equation is interesting from a numerical viewpoint because analytical solutions are not generally available. Furthermore, we demonstrate the applicability of the presented analytical and numerical approaches on the governing coupled inhomogeneous Burger's equation given above in what follows.3.1. Analytical methods
To start off, let us consider the coupled homogeneous Burger's equation from the above model, that is f1(x, t) = 0 = f(x, t). Therefore, we make use of the wave transformation for the coupled model as follows23 ) becomes
$\begin{eqnarray}{w}_{1}(x,t)={W}_{1}(\xi )\qquad {w}_{2}(x,t)={W}_{2}(\xi ),\qquad \xi ={kx}-{ct}.\end{eqnarray}$
Therefore, the governing model in equation ( $\begin{eqnarray}\begin{array}{rcl}{{cD}}_{\xi }{W}_{1} +\,{k}^{2}{D}_{\xi \xi }{W}_{1}+{\eta }_{1}{{kW}}_{1}{D}_{\xi }{W}_{1}+{\eta }_{2}{{kD}}_{\xi }({W}_{1}{W}_{2})=0,\\ {{cD}}_{\xi }{W}_{2} +\,{k}^{2}{D}_{\xi \xi }{W}_{2}+{\lambda }_{1}{{kW}}_{2}{D}_{\xi }{W}_{2}+{\lambda }_{2}{{kD}}_{\xi }({W}_{1}{W}_{2})=0,\end{array}\end{eqnarray}$
such that upon making a balance between the orders of the highest linear operator, and that of the nonlinear term present in the respective equation gives that $\begin{eqnarray}{M}_{1}=1,\qquad \mathrm{and}\qquad {M}_{2}=1.\end{eqnarray}$
Thus, we proceed as follows:| i | (i)METEM With M1 = 1, and M2 = 1, equation ( $\begin{eqnarray}\begin{array}{rcl}{W}_{1} & = & {a}_{0}+{a}_{1}{\rm{\Psi }}(\xi )+\displaystyle \frac{{b}_{1}}{{\rm{\Psi }}(\xi )},\\ {W}_{2} & = & {c}_{0}+{c}_{1}{\rm{\Psi }}(\xi )+\displaystyle \frac{{d}_{1}}{{\rm{\Psi }}(\xi )},\end{array}\end{eqnarray}$ where a0, a1, b1, c0, c1 and d1 are unknown constants to be determined. Therefore, substituting the above solution form into equation ( $\begin{eqnarray*}\begin{array}{l}-2{b}_{1}{d}_{1}{\eta }_{2}k+2{b}_{1}{k}^{2}z-{b}_{1}^{2}{\eta }_{1}k=0,\\ \quad -\,2{b}_{1}{d}_{1}{\lambda }_{2}k+2{d}_{1}{k}^{2}z-{d}_{1}^{2}{\lambda }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}{a}_{0}{c}_{1}k{\lambda }_{2}z+{a}_{1}{c}_{0}k{\lambda }_{2}z-{a}_{0}{d}_{1}k{\lambda }_{2}-{b}_{1}{c}_{0}k{\lambda }_{2}\\ -\,{c}_{0}{d}_{1}k{\lambda }_{1}-{{cd}}_{1}+{c}_{0}{c}_{1}k{\lambda }_{1}z+{{cc}}_{1}z=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}-{a}_{0}{b}_{1}{\eta }_{1}k+{a}_{1}{c}_{0}{\eta }_{2}{kz}+{a}_{0}{c}_{1}{\eta }_{2}{kz}+{a}_{1}{cz}\\ \quad -\,{a}_{0}{d}_{1}{\eta }_{2}k+{a}_{0}{a}_{1}{\eta }_{1}{kz}-{b}_{1}{c}_{0}{\eta }_{2}k-{b}_{1}c=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}-2{b}_{1}{d}_{1}{\eta }_{2}{kz}+2{b}_{1}{k}^{2}{z}^{2}-{b}_{1}^{2}{\eta }_{1}{kz}=0,\\ -\,{a}_{0}{b}_{1}{\eta }_{1}{kz}-{a}_{0}{d}_{1}{\eta }_{2}{kz}-{b}_{1}{c}_{0}{\eta }_{2}{kz}-{b}_{1}{cz}=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}-2{b}_{1}{d}_{1}k{\lambda }_{2}z+2{d}_{1}{k}^{2}{z}^{2}-{d}_{1}^{2}k{\lambda }_{1}z=0,\\ -\,{a}_{0}{d}_{1}k{\lambda }_{2}z-{b}_{1}{c}_{0}k{\lambda }_{2}z-{c}_{0}{d}_{1}k{\lambda }_{1}z-{{cd}}_{1}z=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}2{a}_{1}{c}_{1}{\lambda }_{2}{kz}+2{c}_{1}{k}^{2}z+{c}_{1}^{2}{\lambda }_{1}{kz}=0,\\ {a}_{0}{c}_{1}k{\lambda }_{2}+{a}_{1}{c}_{0}k{\lambda }_{2}+{c}_{0}{c}_{1}k{\lambda }_{1}+{{cc}}_{1}=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}2{a}_{1}{c}_{1}{\eta }_{2}{kz}+2{a}_{1}{k}^{2}z+{a}_{1}^{2}{\eta }_{1}{kz}=0,\\ {a}_{1}{c}_{0}{\eta }_{2}k+{a}_{0}{c}_{1}{\eta }_{2}k+{a}_{1}c+{a}_{0}{a}_{1}{\eta }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}2{a}_{1}{c}_{1}{\lambda }_{2}k+2{c}_{1}{k}^{2}+{c}_{1}^{2}{\lambda }_{1}k=0,\\ 2{a}_{1}{c}_{1}{\eta }_{2}k+2{a}_{1}{k}^{2}+{a}_{1}^{2}{\eta }_{1}k=0.\end{array}\end{eqnarray*}$ Now, upon solving the above algebraic system of equations, the unknown constants a0, a1, b1, c0, c1 and d1 are determined as follows $\begin{eqnarray}\begin{array}{cc}{a}_{0}=-\displaystyle \frac{c{\lambda }_{1}-2c{\eta }_{2}}{k\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)}, & {c}_{0}=\displaystyle \frac{2c{\lambda }_{2}-c{\eta }_{1}}{k\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{cc}{a}_{1}=-\displaystyle \frac{2\left(2{\eta }_{2}k-k{\lambda }_{1}\right)}{4{\eta }_{2}{\lambda }_{2}-{\eta }_{1}{\lambda }_{1}}, & {c}_{1}=-\displaystyle \frac{2\left({\eta }_{1}k-2k{\lambda }_{2}\right)}{{\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{cc}{b}_{1}=\displaystyle \frac{{d}_{1}{\lambda }_{1}-2{d}_{1}{\eta }_{2}}{{\eta }_{1}-2{\lambda }_{2}}, & {d}_{1}={d}_{1},\end{array}\end{eqnarray}$ with $\begin{eqnarray}z=\displaystyle \frac{{d}_{1}\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)}{2k\left({\eta }_{1}-2{\lambda }_{2}\right)}.\end{eqnarray}$ Therefore, this method of solution posed four (4) different sets of solutions via the application of equation ( $\begin{eqnarray}\begin{array}{rcl}{\,}^{1}{w}_{1}(x,t) & = & {a}_{0}-{a}_{1}\sqrt{-z}\coth (\sqrt{-z}({kx}-{ct}))\\ & & -\displaystyle \frac{{b}_{1}}{\sqrt{-z}\tanh (\sqrt{-z}({kx}-{ct}))},\\ {}^{1}{w}_{2}(x,t) & = & {c}_{0}-{c}_{1}\sqrt{-z}\coth (\sqrt{-z}({kx}-{ct}))\\ & & -\displaystyle \frac{{d}_{1}}{\sqrt{-z}\tanh (\sqrt{-z}({kx}-{ct}))},\end{array}\end{eqnarray}$ and $\begin{eqnarray}\begin{array}{rcl}{\,}^{2}{w}_{1}(x,t) & = & {a}_{0}-{a}_{1}\sqrt{-z}\tanh (\sqrt{-z}({kx}-{ct}))\\ & & -\displaystyle \frac{{b}_{1}}{\sqrt{-z}\coth (\sqrt{-z}({kx}-{ct}))},\\ {\,}^{2}{w}_{2}(x,t) & = & {c}_{0}-{c}_{1}\sqrt{-z}\tanh (\sqrt{-z}({kx}-{ct}))\\ & & -\displaystyle \frac{{d}_{1}}{\sqrt{-z}\coth (\sqrt{-z}({kx}-{ct}))},\end{array}\end{eqnarray}$ while for $z=\tfrac{{d}_{1}\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)}{2k\left({\eta }_{1}-2{\lambda }_{2}\right)}\gt 0,$ we get the following wave solutions $\begin{eqnarray}\begin{array}{rcl}{\,}^{3}{w}_{1}(x,t) & = & {a}_{0}-{a}_{1}\sqrt{z}\cot (\sqrt{z}({kx}-{ct}))\\ & & -\displaystyle \frac{{b}_{1}}{\sqrt{z}\tan (\sqrt{z}({kx}-{ct}))},\\ {\,}^{3}{w}_{2}(x,t) & = & {c}_{0}-{c}_{1}\sqrt{z}\cot (\sqrt{z}({kx}-{ct}))\\ & & -\displaystyle \frac{{d}_{1}}{\sqrt{z}\tan (\sqrt{z}({kx}-{ct}))},\end{array}\end{eqnarray}$ and $\begin{eqnarray}\begin{array}{rcl}{\,}^{4}{w}_{1}(x,t) & = & {a}_{0}+{a}_{1}\sqrt{z}\tan (\sqrt{z}({kx}-{ct}))\\ & & +\displaystyle \frac{{b}_{1}}{\sqrt{z}\cot (\sqrt{z}({kx}-{ct}))},\\ {\,}^{4}{w}_{2}(x,t) & = & {c}_{0}+{c}_{1}\sqrt{z}\tan (\sqrt{z}({kx}-{ct}))\\ & & +\displaystyle \frac{{d}_{1}}{\sqrt{z}\cot (\sqrt{z}({kx}-{ct}))}.\end{array}\end{eqnarray}$ Therefore, for k = 0.61, c = 0.25, d1 = 1, and d1 = − 1, we give the three-dimensional (3D) plots for the solution given in equations ( |
| ii | (ii)KM With M1 = 1, and M2 = 1, equation ( $\begin{eqnarray}{W}_{1}={a}_{0}+{a}_{1}{\rm{\Psi }}(\xi ),\qquad {W}_{2}={b}_{0}+{b}_{1}{\rm{\Psi }}(\xi ),\end{eqnarray}$ where a0, a1, b0 and b1 are unknown constants to be determined. Therefore, substituting the above solution form into equation ( $\begin{eqnarray*}\begin{array}{l}-{a}_{1}{b}_{0}{\eta }_{2}k-{a}_{0}{b}_{1}{\eta }_{2}k-{a}_{1}c+{a}_{1}{k}^{2}\\ \quad -\,{a}_{0}{a}_{1}{\eta }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}{a}_{1}{b}_{0}{\eta }_{2}k+{a}_{0}{b}_{1}{\eta }_{2}k-2{a}_{1}{b}_{1}{\eta }_{2}k+{a}_{1}c\\ \quad -\,3{a}_{1}{k}^{2}-{a}_{1}^{2}{\eta }_{1}k+{a}_{0}{a}_{1}{\eta }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}2{a}_{1}{b}_{1}{\eta }_{2}k+2{a}_{1}{k}^{2}+{a}_{1}^{2}{\eta }_{1}k=0,\ \ -{a}_{1}{b}_{0}{\lambda }_{2}k\\ \quad -\,{a}_{0}{b}_{1}{\lambda }_{2}k-{b}_{1}c+{b}_{1}{k}^{2}-{b}_{0}{b}_{1}{\lambda }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{l}{a}_{1}{b}_{0}{\lambda }_{2}k+{a}_{0}{b}_{1}{\lambda }_{2}k-2{a}_{1}{b}_{1}{\lambda }_{2}k+{b}_{1}c\\ \quad -\,3{b}_{1}{k}^{2}-{b}_{1}^{2}{\lambda }_{1}k+{b}_{0}{b}_{1}{\lambda }_{1}k=0,\end{array}\end{eqnarray*}$ $\begin{eqnarray*}2{a}_{1}{b}_{1}{\lambda }_{2}k+2{b}_{1}{k}^{2}+{b}_{1}^{2}{\lambda }_{1}k=0.\end{eqnarray*}$ Then, on solving the above algebraic system of equations, the unknown constants a0, a1, b0, and b1 are determined as follows $\begin{eqnarray}\begin{array}{rcl}{a}_{0} & = & \displaystyle \frac{\left(c-{k}^{2}\right)\left(2{\eta }_{2}-{\lambda }_{1}\right)}{k\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)},\\ {b}_{0} & = & -\,\displaystyle \frac{\left(c-{k}^{2}\right)\left({\eta }_{1}-2{\lambda }_{2}\right)}{k\left({\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}\right)},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & -\,\displaystyle \frac{2\left(2{\eta }_{2}k-k{\lambda }_{1}\right)}{4{\eta }_{2}{\lambda }_{2}-{\eta }_{1}{\lambda }_{1}},\\ {b}_{1} & = & -\,\displaystyle \frac{2\left({\eta }_{1}k-2k{\lambda }_{2}\right)}{{\eta }_{1}{\lambda }_{1}-4{\eta }_{2}{\lambda }_{2}},\end{array}\end{eqnarray}$ which yields only one wave solution set which is as follows $\begin{eqnarray}\begin{array}{rcl}{\,}^{1}{w}_{1}(x,t) & = & {a}_{0}+\displaystyle \frac{{a}_{1}}{1+{{d}{\rm{e}}}^{{kx}-{ct}}},\\ {\,}^{1}{w}_{2}(x,t) & = & {b}_{0}+\displaystyle \frac{{b}_{1}}{1+{{d}{\rm{e}}}^{{kx}-{ct}}},\end{array}\end{eqnarray}$ where d is an arbitrary constant. Thus, for d = 0.1, k = 1.5, and c = 0.25, we give the 3D plots for the above solution of the coupled Burger's equation via the application of the KM in figure 3. Again, the solution given in equation ( |
Figure 1. 3D plots for the solution of coupled Burger's equation given in equation ( |
Figure 2. 3D plots for the solution of coupled Burger's equation given in equation ( |
Figure 3. 3D plots for the solution of coupled Burger's equation given in equation ( |
3.2. NLDM
To successfully apply the NLDM, let us prescribe the following generalized initial conditions23 ) admits the following recursive solution18 ) as follows
$\begin{eqnarray}{w}_{1}(x,0)={r}_{1}(x),\qquad \ \ {w}_{2}(x,0)={r}_{2}(x).\end{eqnarray}$
Therefore, without further delay, the coupled inhomogeneous Burger's equation given in equation ( $\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}{w}_{{1}_{0}}(x,t)={r}_{1}(x)+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{f}_{1}(x,t)\}\},\\ {w}_{{1}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{1}_{{k}_{{xx}}}}+{\eta }_{1}{A}_{k}+{\eta }_{2}{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\\ \left\{\begin{array}{l}{w}_{{2}_{0}}(x,t)={r}_{2}(x)+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{f}_{2}(x,t)\}\},\\ {w}_{{2}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{2}_{{k}_{{xx}}}}+{\lambda }_{1}{B}_{k}+{\lambda }_{2}{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\end{array}\end{eqnarray}$
with Ak's, Bk's and Ck's, respectively denote the polynomials by Adomian in favour of the following nonlinear terms $\begin{eqnarray}\begin{array}{rcl}{F}_{1}({w}_{1}) & = & {w}_{1}{\partial }_{x}{w}_{1},\qquad {F}_{2}({w}_{2})={w}_{2}{\partial }_{x}{w}_{2},\\ {F}_{3}({w}_{1},{w}_{2}) & = & {\partial }_{x}({w}_{1}{w}_{2}),\end{array}\end{eqnarray}$
where we express a few terms from these nonlinear terms using the application of equation ( $\begin{eqnarray*}\begin{array}{rcl}{A}_{0} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ {B}_{0} & = & {w}_{{2}_{0}}{\partial }_{x}{w}_{{2}_{{0}_{}}},\\ {A}_{1} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{1}_{{1}_{}}}+{w}_{{1}_{1}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ {B}_{1} & = & {w}_{{2}_{0}}{\partial }_{x}{w}_{{2}_{{1}_{}}}+{w}_{{2}_{1}}{\partial }_{x}{w}_{{2}_{{0}_{}}},\\ {A}_{2} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{1}_{{2}_{}}}+{w}_{{1}_{1}}{\partial }_{x}{w}_{{1}_{{1}_{}}}+{w}_{{1}_{2}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ {B}_{2} & = & {w}_{{2}_{0}}{\partial }_{x}{w}_{{2}_{{2}_{}}}+{w}_{{2}_{1}}{\partial }_{x}{w}_{{2}_{{1}_{}}}+{w}_{{2}_{2}}{\partial }_{x}{w}_{{2}_{{0}_{}}},\\ & & \vdots \\ \,\, & & \vdots \end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{C}_{0} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{2}_{0}}+{w}_{{2}_{0}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ {C}_{1} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{2}_{{1}_{}}}+{w}_{{1}_{1}}{\partial }_{x}{w}_{{2}_{{0}_{}}}+{w}_{{2}_{0}}{\partial }_{x}{w}_{{1}_{{1}_{}}}+{w}_{{2}_{1}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ {C}_{2} & = & {w}_{{1}_{0}}{\partial }_{x}{w}_{{2}_{{2}_{}}}+{w}_{{1}_{1}}{\partial }_{x}{w}_{{2}_{{1}_{}}}+{w}_{{1}_{2}}{\partial }_{x}{w}_{{2}_{{0}_{}}}\\ & & +\,{w}_{{2}_{0}}{\partial }_{x}{w}_{{1}_{{2}_{}}}+{w}_{{2}_{1}}{\partial }_{x}{w}_{{1}_{{1}_{}}}+{w}_{{2}_{2}}{\partial }_{x}{w}_{{1}_{{0}_{}}},\\ & & \vdots \end{array}\end{eqnarray*}$
Lastly, upon taking the net sum of m-term components, one obtains $\begin{eqnarray}{{\rm{\Phi }}}_{{1}_{m}}(x,t)=\sum _{j=0}^{m-1}{w}_{{1}_{j}}(x,t),\qquad {{\rm{\Phi }}}_{{2}_{m}}(x,t)=\sum _{j=0}^{m-1}{w}_{{2}_{j}}(x,t),\end{eqnarray}$
where $\begin{eqnarray}\displaystyle \begin{array}{rcl}{w}_{1}(x,t) & = & \mathop{\mathrm{lim}}\limits_{m\to \infty }{{\rm{\Phi }}}_{{1}_{m}}(x,t)=\sum _{j=0}^{\infty }{w}_{{1}_{j}}(x,t),\\ {w}_{2}(x,t) & = & \mathop{\mathrm{lim}}\limits_{m\to \infty }{{\rm{\Phi }}}_{{2}_{m}}(x,t)=\sum _{j=0}^{\infty }{w}_{{2}_{j}}(x,t).\end{array}\end{eqnarray}$
4. Numerical validation
In relation to the derived numerical scheme that is associated with the NLDM, this section further makes consideration to certain numerical examples that will give more light to the scheme. More so, we try to present some comparisons of results, where necessary. What's more, the obtained recursive solution in equations (41 ) becomes42 ), accordingly. More so, we evaluate a few solution components from the above scheme as follows Accordingly, the recursive scheme in this particular example takes the following form42 ). Further, we simulate this example and the absolute error difference between the exact solution (w1(x, t), w2(x, t)) and that of the approximate solution via the sum of the first 20 components $({{\rm{\Phi }}}_{{1}_{20}},{{\rm{\Phi }}}_{{2}_{20}})$ are reported in table 2. Furthermore, we give the 3D and 2D plots for the obtained approximate solution in figures 6 and 7, respectively. What's more, a very good agreement between the exact solution and the reported numerical solution has been graphically shown in figure 7, just after using the sum of the first 20 components. In fact, this further proves the efficacy of the used method. Accordingly, the following recursive solution is thus obtained42 ). More so, we evaluate a few solution components from the above scheme as follows
Considering the coupled homogeneous Burger's equation given in equation (
$\begin{eqnarray}\begin{array}{rcl}{\eta }_{1} & = & 2={\lambda }_{1},\qquad {\eta }_{2}=-1={\lambda }_{2},\\ {f}_{1}(x,t) & = & 0={f}_{2}(x,t),\end{array}\end{eqnarray}$
together with the following prescribed initial data $\begin{eqnarray}{w}_{1}(x,0)=\sin (x),\qquad \ \ {w}_{2}(x,0)=\sin (x).\end{eqnarray}$
This problem further satisfies the following exact solution $\begin{eqnarray}{w}_{1}(x,t)={{\rm{e}}}^{-t}\sin (x),\qquad \ \ {w}_{2}(x,t)={{\rm{e}}}^{-t}\sin (x).\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}{w}_{{1}_{0}}(x,t)=\sin (x),\\ {w}_{{1}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{1}_{{k}_{{xx}}}}+2{A}_{k}-{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\\ \left\{\begin{array}{l}{w}_{{2}_{0}}(x,t)=\sin (x),\\ {w}_{{2}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{2}_{{k}_{{xx}}}}+2{B}_{k}-{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\end{array}\end{eqnarray}$
where the Adomian polynomials Ak, Bk and Ck are determined from equation ( $\begin{eqnarray*}\begin{array}{ll}{w}_{{1}_{0}}(x,t)=\sin (x), & {w}_{{2}_{0}}(x,t)=\sin (x),\\ {w}_{{1}_{1}}(x,t)=-t\sin (x), & {w}_{{2}_{1}}(x,t)=-t\sin (x),\\ {w}_{{1}_{2}}(x,t)=\displaystyle \frac{{t}^{2}}{2!}\sin (x), & {w}_{{2}_{2}}(x,t)=\displaystyle \frac{{t}^{2}}{2!}\sin (x),\\ {w}_{{1}_{3}}(x,t)=-\displaystyle \frac{{t}^{3}}{3!}\sin (x), & {w}_{{2}_{3}}(x,t)=-\displaystyle \frac{{t}^{3}}{3!}\sin (x),\\ {w}_{{1}_{4}}(x,t)=\displaystyle \frac{{t}^{4}}{4!}\sin (x), & {w}_{{2}_{4}}(x,t)=\displaystyle \frac{{t}^{4}}{4!}\sin (x),\\ {w}_{{1}_{5}}(x,t)=-\displaystyle \frac{{t}^{5}}{5!}\sin (x), & {w}_{{2}_{5}}(x,t)=-\displaystyle \frac{{t}^{5}}{5!}\sin (x),\\ {w}_{{1}_{6}}(x,t)=\displaystyle \frac{{t}^{6}}{6!}\sin (x), & {w}_{{2}_{6}}(x,t)=\displaystyle \frac{{t}^{6}}{6!}\sin (x),\\ \vdots & \vdots \end{array}\end{eqnarray*}$
Therefore, taking the net sum of the components, one gets the following expressions $\begin{eqnarray}\begin{array}{l}{w}_{1}(x,t)=\left(1-t+\displaystyle \frac{{t}^{2}}{2!}-\displaystyle \frac{{t}^{3}}{3!}+\displaystyle \frac{{t}^{4}}{4!}-\displaystyle \frac{{t}^{5}}{5!}+\displaystyle \frac{{t}^{6}}{6!}+...\right) \quad \sin (x),\\ {w}_{2}(x,t)=\left(1-t+\displaystyle \frac{{t}^{2}}{2!}-\displaystyle \frac{{t}^{3}}{3!}+\displaystyle \frac{{t}^{4}}{4!}-\displaystyle \frac{{t}^{5}}{5!}+\displaystyle \frac{{t}^{6}}{6!}+...\right) \quad \sin (x),\end{array}\end{eqnarray}$
that obviously converge to the following exact analytical solution $\begin{eqnarray}{w}_{1}(x,t)={w}_{2}(x,t)={e}^{-t}\sin (x).\end{eqnarray}$
Hence, we give the 3D and 2D plots for the obtained approximate solution of coupled homogeneous Burger's equation in figures 4 and 5, respectively. Additionally, the absolute error difference between the exact solution $({w}_{1}(x,t),{w}_{2}(x,t))=({{\rm{e}}}^{-t}\sin (x),{{\rm{e}}}^{-t}\sin (x))$ and that of the approximate solution via the sum of the first 15 components $({{\rm{\Phi }}}_{{1}_{15}},{{\rm{\Phi }}}_{{2}_{15}})$ are reported in table 1. What's more, a very good agreement between the exact solution and the reported numerical solution has been graphically shown in figure 5, just for using the sum of the first 15 components. In fact, this further proves the efficacy of the used method.Considering the coupled homogeneous Burger's equation given in equation (
$\begin{eqnarray}\begin{array}{l}{\eta }_{1}=2={\lambda }_{1},\,{\eta }_{2}=-0.1,\,{\lambda }_{2}\\ =\,-0.3,\,{f}_{1}(x,t)=0={f}_{2}(x,t),\end{array}\end{eqnarray}$
with the following initial data $\begin{eqnarray}\begin{array}{rcl}{w}_{1}(x,0) & = & 0.05(1.0-\tanh (0.0275(20(x-0.5)))),\\ {w}_{2}(x,0) & = & 0.05(0.5-\tanh (0.0275(20(x-0.5)))).\end{array}\end{eqnarray}$
The problem further satisfies the following exact solution $\begin{eqnarray}\begin{array}{rcl}{w}_{1}(x,t) & = & 0.05(1.0-\tanh (0.0275(20(x-0.5)\\ & & -0.055t))),\\ {w}_{2}(x,t) & = & 0.05(0.5-\tanh (0.0275(20(x-0.5)\\ & & -0.055t))).\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}{w}_{{1}_{0}}(x,t)=0.05(1.0-\tanh (0.0275(20(x-0.5)))),\\ {w}_{{1}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{1}_{{k}_{{xx}}}}+2{A}_{k}-0.1{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\\ \left\{\begin{array}{l}{w}_{{2}_{0}}(x,t)=0.05(0.5-\tanh (0.0275(20(x-0.5)))),\\ {w}_{{2}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{w}_{{2}_{{k}_{{xx}}}}+2{B}_{k}-0.3{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\end{array}\end{eqnarray}$
where the Adomian polynomials Ak, Bk and Ck are equally obtained from equation (Considering the coupled singular inhomogeneous Burger's equation with [4, 7]
$\begin{eqnarray}\begin{array}{l}{\eta }_{1}=2={\lambda }_{1},\qquad {\eta }_{2}=-1={\lambda }_{2},\\ \qquad {f}_{1}(x,t)={x}^{2}{e}^{t}-4{e}^{t}={f}_{2}(x,t),\end{array}\end{eqnarray}$
as follows $\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}{w}_{1} & = & {x}^{-1}{\partial }_{x}({{xw}}_{1})+2{w}_{1}{\partial }_{x}{w}_{1}-{\partial }_{x}({w}_{1}{w}_{2})+{x}^{2}{{\rm{e}}}^{t}-4{{\rm{e}}}^{t},\\ {\partial }_{t}{w}_{2} & = & {x}^{-1}{\partial }_{x}({{xw}}_{2})+2{w}_{2}{\partial }_{x}{w}_{2}-{\partial }_{x}({w}_{1}{w}_{2})+{x}^{2}{{\rm{e}}}^{t}-4{{\rm{e}}}^{t},\end{array}\end{eqnarray}$
together with the following prescribed initial data $\begin{eqnarray}{w}_{1}(x,0)={x}^{2},\qquad \ \ {w}_{2}(x,0)={x}^{2}.\end{eqnarray}$
This problem further satisfies the following exact solution $\begin{eqnarray}{w}_{1}(x,t)={x}^{2}{{\rm{e}}}^{t},\qquad \ \ {w}_{2}(x,t)={x}^{2}{{\rm{e}}}^{t}.\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}{w}_{{1}_{0}}(x,t)={x}^{2}+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{x}^{2}{{\rm{e}}}^{t}-4{{\rm{e}}}^{t}\}\},\\ {w}_{{1}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{x}^{-1}{\partial }_{x}({{xw}}_{{1}_{{k}_{x}}})+2{A}_{k}-{C}_{k}\}\},\ k\geqslant 0,\end{array}\right.\\ \left\{\begin{array}{l}{w}_{{2}_{0}}(x,t)={x}^{2}+{{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{x}^{2}{{\rm{e}}}^{t}-4{{\rm{e}}}^{t}\}\},\\ {w}_{{2}_{k+1}}(x,t)={{\mathscr{L}}}^{-1}\{{s}^{-1}{\mathscr{L}}\{{x}^{-1}{\partial }_{x}({{xw}}_{{2}_{{k}_{x}}})+2{B}_{k}-{C}_{k}\}\},\ \ k\geqslant 0,\end{array}\right.\end{array}\end{eqnarray}$
where the Adomian polynomials Ak, Bk and Ck are equally obtained from equation ( $\begin{eqnarray*}\begin{array}{rcl}{w}_{{1}_{0}}(x,t) & = & {x}^{2}+\left({{\rm{e}}}^{t}-1\right)\left({x}^{2}-4\right),\\ {w}_{{2}_{0}}(x,t) & = & {x}^{2}+\left({{\rm{e}}}^{t}-1\right)\left({x}^{2}-4\right),\\ {w}_{{1}_{1}}(x,t) & = & 4\left({{\rm{e}}}^{t}-1\right),\\ {w}_{{2}_{1}}(x,t) & = & 4\left({{\rm{e}}}^{t}-1\right),\\ {w}_{{1}_{k}}(x,t) & = & 0,\quad k\geqslant 2\\ {w}_{{2}_{k}}(x,t) & = & 0,\quad k\geqslant 2,\end{array}\end{eqnarray*}$
Therefore, taking the net sum of the components, one gets the following expressions $\begin{eqnarray}{w}_{1}(x,t)={w}_{2}(x,t)={x}^{2}{{\rm{e}}}^{t},\end{eqnarray}$
which is, in fact, the obvious available exact analytical solution.
Figure 4. 3D plot for the approximate solution of coupled Burger's equation given in Example |
Figure 5. 2D comparison between the exact and approximate solution of coupled Burger's equation given in Example |
Figure 6. 3D plots for the approximate solution of coupled Burger's equation given in Example |
Figure 7. 2D comparisons between the exact and approximate solution of coupled Burger's equation given in Example |
Table 1. Error difference between the exact solution of (w1(x, t), w2(x, t)) and the obtained numerical solution $({{\rm{\Phi }}}_{{1}_{15}},{{\rm{\Phi }}}_{{2}_{15}})$ at t = 1. |
| x | $\left|{w}_{1}(x,t)-{{\rm{\Phi }}}_{{1}_{15}}\right|$ | $\left|{w}_{2}(x)-{{\rm{\Phi }}}_{{2}_{15}}\right|$ |
|---|---|---|
| 0.0 | 0 | 0 |
| 0.1 | 4.51028 × 10−15 | 4.51028 × 10−15 |
| 0.2 | 8.96505 × 10−15 | 8.96505 × 10−15 |
| 0.3 | 1.33366 × 10−14 | 1.33366 × 10−14 |
| 0.4 | 1.75693 × 10−14 | 1.75693 × 10−14 |
| 0.5 | 2.16493 × 10−14 | 2.16493 × 10−14 |
| 0.6 | 2.54796 × 10−14 | 2.54796 × 10−14 |
| 0.7 | 2.90601 × 10−14 | 2.90601 × 10−14 |
| 0.8 | 3.23630 × 10−14 | 3.23630 × 10−14 |
| 0.9 | 3.53606 × 10−14 | 3.53606 × 10−14 |
| 1.0 | 3.79696 × 10−14 | 3.79696 × 10−14 |
Table 2. Error difference between the exact solution of (w1(x, t), w2(x, t)) and the obtained numerical solution $({{\rm{\Phi }}}_{{1}_{20}},{{\rm{\Phi }}}_{{2}_{20}})$ at t = 1. |
| x | $\left|{w}_{1}(x,t)-{{\rm{\Phi }}}_{{1}_{3}}\right|$ | $\left|{w}_{2}(x)-{{\rm{\Phi }}}_{{2}_{3}}\right|$ |
| 0.0 | 7.00966 × 10−5 | 7.00966 × 10−5 |
| 0.1 | 7.20085 × 10−5 | 7.20085 × 10−5 |
| 0.2 | 7.35479 × 10−5 | 7.35479 × 10−5 |
| 0.3 | 7.467 97 × 10−5 | 7.46797 × 10−5 |
| 0.4 | 7.53775 × 10−5 | 7.53775 × 10−5 |
| 0.5 | 7.56246 × 10−5 | 7.56246 × 10−5 |
| 0.6 | 7.54151 × 10−5 | 7.54151 × 10−5 |
| 0.7 | 7.47540 × 10−5 | 7.47540 × 10−5 |
| 0.8 | 7.36572 × 10−5 | 7.36572 × 10−5 |
| 0.9 | 7.21502 × 10−5 | 7.21502 × 10−5 |
| 1.0 | 7.02674 × 10−5 | 7.02674 × 10−5 |
5. Discussion of results
The current study examines the system of coupled Burger's equations [4–7]; being one of the astonishing members of the generalized reaction-advection-diffusion dynamical model with vast applications. In doing so, two promising analytical integrations are employed; in addition to the deployment of an efficient variant of the eminent Adomian decomposition method. More specifically, the analytical methods involved are the METEM [18–20], and the KM [21–23]; while the numerical method is a version of the Adomian's method called NLDM [25, 26]. Three sets of analytical wave solutions are revealed including exponential, periodic, and dark-singular wave solutions; while an amazed rapidly convergent approximate solution is acquired on the other hand via NLDM. Equally, self-explanatory graphical illustrations and tables are provided to support the reported analytical and numerical results. The reported exponential wave solution graphically turned out to be a kink-shaped profile. One would recall the application of kink solitons in polarization switches among dissimilar optical logic domains; while, a dark soliton solution is a localized wave that gives rise to a transitory decrease in wave amplitude (think also of the bright soliton solutions which are known to amplify the ocean waves). Also, periodic wave solutions are used in self-oscillatory and excitable systems, respectively, among others. More so, the numerical test examples considered reveal amazing closed form solutions, which get hold of their respective exact solutions when the net sums of the solution components are taken. In addition, the present study affirms the findings of the available literature. More comparatively, the present study generalizes many studies, including the integer-order derivative versions of the models considered in [4, 6, 7]. Moreover, the employed numerical scheme which doubly serves as a semi-analytical method competes with so many methods in the literature, including [28–37]; while some known robust integration schemes that happen to contend with the two analytical methods of interest in tackling different class of evolution equations include [38–43].
6. Conclusion
In conclusion, the current study examined the class of a reaction-advection-diffusion dynamical model that plays a vital role in essential areas of physics like fluid dynamics and acoustics. More specifically, we comprehensively studied the coupled system of Burger's equations; being a special case of the above-mentioned governing model after setting the diffusion, advection, and reaction coefficients, respectively, to zero. Two known analytical integration schemes by the name METEM and KM were employed to scrutinize the model; in addition to the deployment of an efficient variant of the decomposition method, NLDM. Three sets of exact analytical wave solutions were revealed by the analytical methods, including exponential, periodic, and dark-singular wave solutions. Additionally, the derived numerical schemes revealed amazing approximate solutions that rapidly converged to the available exact solutions. Furthermore, certain graphical illustrations and tables were provided to support the presented analytical and numerical results; a perfect agreement between the exact solutions and the acquired numerical results, on the other hand, has been achieved with low number of iterations - see figures 5 and 7, for instance. We also state here that the Wolfram Mathematica 9 software was used for the simulation of numerical results, and their graphical depictions. No doubt, the present study is set to bridge the existing gap between the analytical and numerical approaches with regard to the solution validity of various models of mathematical physics; also, the employed methods can equally be used to tackle various higher-order dynamical systems of evolution equations. Moreover, in the future, we intended to make use of other reliable numerical approaches like the finite difference and finite element methods to further validate the exactitude of the reported exact analytical solutions in this study.