Ali M Mubaraki ,Hwajoon Kim ,R I Nuruddeen ,Urooj Akram and Yasir Akbar

1 Department of Mathematics and Statistics,Collage of Science,Taif University,P O Box 11099,Taif 21944,Saudi Arabia

2 Kyungdong University,Yangju,Republic of Korea

3 Department of Mathematics,Faculty of Science,Federal University Dutse,P O Box 7156 Dutse,Jigawa State,Nigeria

4 Department of Mathematics,COMSATS University Islamabad,Lahore,Pakistan

5 Department of Mathematics,COMSATS University Islamabad,44000 Islamabad,Pakistan

Abstract The current study examines the special class of a generalized reaction-advection-diffusion dynamical model that is called the system of coupled Burger’s equations.This system plays a vital role in the essential areas of physics,including fluid dynamics and acoustics.Moreover,two promising analytical integration schemes are employed for the study;in addition to the deployment of an efficient variant of the eminent Adomian decomposition method.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.At the end,certain graphical illustrations and tables are provided to support the reported analytical and numerical results.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.

Keywords: reaction-advection-diffusion model,Burger’s equations,METEM,KM,NLDM

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 reactionadvection-diffusion dynamical equations in porous medium that follows from the modification of [1] as (integer-order)

wherew1=w1(x,t),andw2=w2(x,t),are the respective concentrations,α1,α2are the respective diffusion coefficients,γ1,γ2are the respective advection coefficients,θ1,θ2are the respective reaction coefficients,η1,λ1are real constants,m1,m2are non-negative integers,β1,β2,η2,λ2are coupling parameters;whilef1(x,t) andf2(x,t) are the respective source functions,otherwise known as inhomogeneous terms Additionally,whilem1andm2are 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 theWolfram 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 onM(?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 pattern

More so,upon making use of the following wave transformation

in equation (2),one gets an ordinary differential equation of the form

where in equation (3),kandcare real constants (nonzero);whileDin equation (4) is a differential operator.

Therefore,we now make consideration to two of the aiming analytical methods as follows

(i)METEM

In accordance with METEM,the following finite series is offered [18–20]

for equation (4),whereM(?N)is 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;whilea0,aj,bj,forj=1,2,…,Mare unknown constants that would be determined little later,ensuring thataM≠0,andbM≠0 for allM.

Additionally,the application of METEM requires the function Ψ(ξ) in equation (5) to satisfy a Riccati differential equation of the form

that admits the following solution sets

Remarkably,METEM reveals three different types of solutions,including hyperbolic,periodic and irrational solutions.For the hyperbolic and periodic solutions,equation (7) demonstrated the modality of how these structures are constructed for any given evolution and Schrodinger equation;while we intentionally ignore the irrational solution possibilities atz=0.Complete sets of METEM solutions and applications can be found in[18–20] and the references therewith.

(ii)KM

According to KM,equation (4) is presumed to have a finite series solution as follows [21–23]

whereM(?)Nis 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;whilea0,aj,forj=1,2,…,Mare unknown constants that would be determined,such thataM≠0,for allM.

What’s more,KM requires the function Ψ(ξ) in equation(5)to satisfy the following differential equation

such that the following exponential solution is satisfied by the latter equation is given by the function

wheredis 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.

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 Ψ(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 theWolfram Mathematica 9software will be utilized in the present study for all computational as well as graphical purposes.

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 equation

with the following prescribed initial data

Also,from equations (11)–(12),Lis a linear differential operator,Nis a nonlinear operator,h(x,t)is an inhomogeneous term;whileg(x) is the prescribed nice initial profile.

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]

Therefore,taking the Laplace transform of equation (11) intalongside using equation (12),the equation transforms to the following

Next,we take the inverse Laplace transformL-1of the above equation ins,which gives

More so,via the Adomian’s approach,we decompose the unknown solutionw(x,t) using the following infinite sum[24–26]

and the nonlinear termed operatorN(w(x,t))using the followinginfniite su m of Adomian’s polynomials as follows

whereRn?s are the polynomials devised by Adomian [24–26],and to be computed via the following recurrent formula

So,by rewriting equation(15)in terms of the summations given in equations (16) and (17),one gets

Lastly,identifying the terms arising from the prescribed initial data and the inhomogeneous function with the first componentw0(x,t),and the rest of the terms recurrently follow as suggested by the approach,we thus get the following recurrent scheme

such that upon taking the net sum of them-term approximations yields

where

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

wherew1=w1(x,t),andw2=w2(x,t)are the respective wave profiles,η1,λ1are real constants,η2,λ2are coupling parameters;whilef1(x,t) andf2(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 isf1(x,t)=0=f(x,t).Therefore,we make use of the wave transformation for the coupled model as follows

Therefore,the governing model in equation (23) becomes

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

Thus,we proceed as follows:

(i)METEM

WithM1=1,andM2=1,equation (25) via the application of the METEM admits the following solution

wherea0,a1,b1,c0,c1andd1are unknown constants to be determined.Therefore,substituting the above solution form into equation (25) reveals the following algebraic system of equations,after setting each coefficient of Ψ(ξ) to zero as follows

Now,upon solving the above algebraic system of equations,the unknown constantsa0,a1,b1,c0,c1andd1are determined as follows

Figure 1.3D plots for the solution of coupled Burger’s equation given in equation (33) via METEM.

Figure 2.3D plots for the solution of coupled Burger’s equation given in equation (35) via METEM.

Therefore,fork=0.61,c=0.25,d1=1,andd1=-1,we give the three-dimensional (3D) plots for the solution given in equations (33) and (35),respectively,of the coupled homogeneous Burger’s equation via the application of the METEM in figures 1 and 2,sequentially.Specifically,the solutions given in equations(33)and (32) are dark-singular solutions;while those reported in equations(34)and(35)are singular periodic solutions.

(ii)KM

WithM1=1,andM2=1,equation (25) via the application of the KM admits the following solution

wherea0,a1,b0andb1are unknown constants to be determined.Therefore,substituting the above solution form into equation (25) reveals the following algebraic system of equations,after setting each coefficient of Ψ(ξ) to zero as follows

Then,on solving the above algebraic system of equations,the unknown constantsa0,a1,b0,andb1are determined as follows

which yields only one wave solution set which is as follows

Figure 3.3D plots for the solution of coupled Burger’s equation given in equation (39) via KM.

wheredis an arbitrary constant.Thus,ford=0.1,k=1.5,andc=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(39)through the KM is an exponential wave solution.As can be observed in figure 3,the plot is a kink-shaped profile.

3.2.NLDM

To successfully apply the NLDM,let us prescribe the following generalized initial conditions

Therefore,without further delay,the coupled inhomogeneous Burger’s equation given in equation(23)admits the following recursive solution

withAk?s,Bk?s andCk?s,respectively denote the polynomials by Adomian in favour of the following nonlinear terms

where we express a few terms from these nonlinear terms using the application of equation (18) as follows

and

Lastly,upon taking the net sum ofm-term components,one obtains

where

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.

Example 4.1.Considering the coupled homogeneous Burger’s equation given in equation (23) with [1]

together with the following prescribed initial data

This problem further satisfies the following exact solution

What’s more,the obtained recursive solution in equations(41)becomes

where the Adomian polynomialsAk,BkandCkare determined from equation (42),accordingly.More so,we evaluate a few solution components from the above scheme as follows

Therefore,taking the net sum of the components,one gets the following expressions

that obviously converge to the following exact analytical solution

Figure 4.3D plot for the approximate solution of coupled Burger’s equation given in Example 4.1.

Figure 5.2D comparison between the exact and approximate solution of coupled Burger’s equation given in Example 4.1.

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(w1(x,t),w2(x,t))=(e-tsin (x),e-tsin(x))and that of the approximate solution via the sum of the first 15 componentsare 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.

Example 4.2.Considering the coupled homogeneous Burger’s equation given in equation (23) with [1]

Table 1.Error difference between the exact solution of (w1(x,t),w2(x,t)) and the obtained numerical solutionat t=1.

Table 1.Error difference between the exact solution of (w1(x,t),w2(x,t)) and the obtained numerical solutionat t=1.

with the following initial data

The problem further satisfies the following exact solution

Accordingly,the recursive scheme in this particular example takes the following form

where the Adomian polynomialsAk,BkandCkare equally obtained from equation (42).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 componentsare 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.

Example 4.3.Considering the coupled singular inhomogeneous Burger’s equation with [4,7]

together with the following prescribed initial data

Table 2.Error difference between the exact solution of (w1(x,t),w2(x,t)) and the obtained numerical solutionat t=1..

Table 2.Error difference between the exact solution of (w1(x,t),w2(x,t)) and the obtained numerical solutionat t=1..

This problem further satisfies the following exact solution

Accordingly,the following recursive solution is thus obtained

where the Adomian polynomialsAk,BkandCkare equally obtained from equation (42).More so,we evaluate a few solution components from the above scheme as follows

Therefore,taking the net sum of the components,one gets the following expressions

which is,in fact,the obvious available exact analytical solution.

5.Discussion of results

Figure 6.3D plots for the approximate solution of coupled Burger’s equation given in Example 4.2.

Figure 7.2D comparisons between the exact and approximate solution of coupled Burger’s equation given in Example 4.2.

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 darksingular 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 theWolfram Mathematica 9software 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.

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