where η, ε, α and β are constants depending on the system parameters.
Differential Quadrature Method (DQM)
Differential quadrature method first introduced by  is one of the most efficient numerical method to solve partial differential equations. The key procedure in differential quadrature approximation is to determine the weighting coefficient.
 and  obtain explicit formulations to compute the weighting coefficient of the first and second order derivative.  presented a simple algebraic formulation to compute the weighting coefficient of the first order derivative without any restriction on the choice of the grid points and a recurrence relationship to compute the weighting coefficient of the DQ method and its applications were rapidly developed after the late 1980s, thanks to the innovative work in the computation of the weighting coefficients by other researchers and the author. As a result, the DQ method has emerged as a powerful numerical discretization tool in the past decade. As compared to the conventional low order finite difference and finite element methods, the DQ method can obtain very accurate numerical results using a considerably smaller number of grid points and hence requiring relatively little computational effort. So far, the DQ method has been efficiently employed in a variety of problems in engineering and physical sciences.
Based on the analysis of function approximation and the analysis of linear vector space, the DQ method can be classified as polynomial-based DQ (PDQ) and Fourier series expansion-based DQ (FDQ) methods. PDQ is usually applied to non-periodic problems while FDQ can be applied to both periodic and non-periodic problems. But its performance for periodic problems is much better. The details of PDQ and FDQ methods and their application in engineering can be referred to the book of  which is the first book in the area to systematically describe the DQ method and its application in engineering.
Fourier Expansion Basis
The polynomial approximation is suitable for most of the engineering problems, but for some problems, especially for those with periodic behaviors, Fourier series expansion could be a better choice for the true solution instead of polynomial approximate the interval [0, 2π]  and  , the Fourier series expansion can be given by
where the coefficient
are expressed as
For practical application, the truncated Fourier series expansion is usually used. Thus Equation (1.4) can be described as
Moreover, the convergence of the above Equation (1.8) of
is guaranteed by Weirstrass’s second theorem. i.e.
be continuous on the interval
. Then for any
there exists an integer n and a trigonometric Sn such that the inequality
is satisfied for all values of x. (1.9)
1.2. Statement of the Problem
Consider the Coupled Viscous burgers’ Equation in Equations (1.1)-(1.3) is a nonlinear partial differential equation, by using Fourier basis based on weighted average differential quadrature method one can expect to find its’ solution numerically by using some approximation method.
In this regard  presented numerical simulations for the Coupled Viscous Burgers’ Equation and compared the results with experimental data recently;  proposed a Fourier Pseudospectral method for solving Coupled Viscous Burgers’ Equation;  and  applied differential quadrature method to solve the viscous Burgers’ Equation;  use finite difference and cubic spline finite element methods to solve Burgers’ equation;  and  used polynomial Differential quadrature method for numerical solution of coupled viscous Burger’ equations;  transform the Burgers’ equation to linear heat equation using Hopf-Cole transformation and then use explicit finite difference and exact explicit finite difference method to solve the transformed linear heat equation with Neumann boundary condition;  solve one dimensional Burgers’ Equation by using differential quadrature method based on Fourier Expansion basis;  used cubic B-spline collocation scheme based on Crank-Nicolson formulation for time integration and cubic B-spline functions for space integration by linearizing the nonlinear terms to solve coupled viscous Burgers’ Equation.
In this study, we look for the solution of coupled viscous Burger’ equation by applying the differential quadrature method based on Fourier Expansion basis. As a result, this study attempted to answer the following basic research questions.
1) How do we describe the differential quadrature method based on Fourier Expansion basis for coupled viscous Burgers’ equation?
2) How the present method is applied to solve coupled viscous Burgers’ equation?
3) To what extent the proposed method is approximate the exact solution?
1.3. Objectives of the Study
1.3.1. General Objective
The general objective of this study is to find the numerical solution of coupled viscous burger equation using differential quadrature method based on Fourier expansion basis.
1.3.2. Specific Objectives
The specific objectives of the present study are:
1) To describe the differential quadrature method based on fourier expansion basis in solving coupled viscous Burgers’ equation numerically.
2) To solve the coupled viscous Burgers’ equation using differential quadrature method based on a Fourier expansion.
3) To determine the accuracy that the proposed method approximate the exist solutions.
1.4. Significance of the Study
The outcomes of this study have the following importance to:
1) Find an alternative numerical solution of coupled viscous Burgers’ Equation.
2) Apply differential quadrature method based on Fourier expansion basis to find the solution for some practical problems like coupled viscous Burgers’ equation.
1.5. Delimitation of the Study
This study was delimited to the numerical solution of the coupled viscous Burgers’ equation.
The study also delimited to the practically most important (linear) differential quadrature based on Fourier expansion that are of the same type as differential quadrature and Fourier expansion formulas, i.e., linear combinations of weighted differential function evaluations and their error analysis for quadrature.
2.1. Study Area and Period
The study was conducted in Jimma University under the department of mathematics from September 2016 G.C. to September 2017 G.C. Conceptually the study focus on the solution of coupled viscous Burgers’ equation using differential quadrature method based on Fourier expansion basis.
2.2. Study Design
This study was employed mixed-design (documentary review design and experimental design) on the coupled viscous Burgers’ equation type.
2.3. Source of Information
The relevant sources of information for this study is books, published articles and related studies from internet and the experimental results were obtained by writing MATLAB code for the present methods.
2.4. Mathematical Procedures
The study is an experimental as it evolves entirely laboratory work with the help of computer and MATLAB software. Farther, important materials for the study were collected by the researcher using the documentary analysis. The required numerical data was collected by coding and running using MATLAB software to get the numerical results and table of some examples that have exact solution, to show the validity and efficiency of the method. Hence, in order to achieve the stated objectives, the study was followed the procedures
1) Problem preparation or formulation.
2) Discretizing the space variable.
3) Replacing the partial differential equation (or the Coupled Viscous Burgers’ Equation) by using differential quadrature method based on Fourier expansion basis to obtain a system of first order ordinary differential equations.
4) The obtained systems of ordinary differential equations was solved by classical fourth order Runge-Kutta method.
5) Validating the schemes using numerical examples.
6) Writing MATLAB code for the method to solve the systems obtained.
3. Result and Discussion
3.1.1. Differential Quadrature Method
In seeking an efficient discretization technique to obtain accurate numerical solution using a considerably small number of grid points,  and  introduced the method of differential quadrature (DQ).where a partial derivative of a function with respect to a coordinate direction is expressed as a linear weighted sum of all the functional values at all mesh points along that direction. The key to DQ is to determine the weighting coefficient for the discretization of a derivative of any order.  suggested two methods to determine the weighting coefficients of first order derivative. The first method is based on an ill-conditioned algebraic equation system and the second method uses a simple algebraic formulation, but the coordinates of the grid points are fixed by the roots of the shifted Legendre polynomial. In earlier applications of the DQ method, Bellman’s first method was usually used because it allows the use of arbitrary grid points distribution. However, since the algebraic equation system of this method is ill-conditioned, the number of the grid points usually used is less than 13  . After that,  obtain explicit formulations to compute the weighting coefficient for the first and second order derivative.  presented a simple algebraic formulation to compute the weighting coefficient of the first order derivative without any restriction on the choice of the grid points and a recurrence relationship to compute the weighting coefficient of the DQ method and  and  have further developed some simple algebraic formulation to compute the weighting coefficient. Recently, the most frequently used DQ procedures to solve one and two dimensional differential equations are Lagrange interpolation polynomials based differential quadrature method (PDQM). The DQ method approximates the derivative of a smooth function at a grid point by a linear weighted summation of all the functional value in the whole computational domain.
The coupled viscous Burgers’ equation is a nonlinear partial differential equation of the form
With initial condition
The boundary condition
where η, α, ε and β are arbitrary constants depending on the system parameters.
3.1.2. Differential Quadrature Method Based on the Fourier Expansion basis
Let the first and second order derivative of
at a point
be approximated by the following equations
represents the weighting coefficients of first and second derivative respectively, N is the number of grid points. And the key procedure in DQ is to determine the weighting coefficient.
For, the solution of any partial differential equation has to be convergent and bounded, so the solution
of the viscous Burger equation be approximated by a Fourier expansion of the form
in this equation constitutes an
dimensional linear vector space since
are linearly independent vectors.
Thus, we can consider (4.4a) as a set of base vectors.
Actually FDQ uses two sets of base functions, one is the base functions (4.4a) and the other is the terms of the Lagrange interpolating trigonometric polynomial given by
By using the above two sets of base vectors to drive explicit formulations to compute the weighting coefficients of the first and second order derivatives, for the non-diagonal weighting coefficient we need to use the second set of base vectors. For simplicity, we set
is the Kronecker delta operator.
Then Equation (4.4b) can be reduced to
Using the same approach as in a polynomial differential quadrature method  we let all the base vectors given by Equation (4.10) satisfy Equations ((4.1) and (4.2)) and obtain
It is observed from Equations ((4.11) and (4.12)), the computation of
is equivalent to the evaluations of
can be calculated by Equation (4.9). We successively differentiate Equation (4.8) to get