## 1 Introduction  
The numerical solution of nonlinear equations and systems is widely used in different branches of Science and Technology, as in the analysis of diffusion phenomena (see, for example, [1, 2]), the study of dynamical models of chemical reactors [3], the numerical computing of the dominant eigenvalue of the neutron transport criticality problem (see [4]), the study of radioactive transfer [5], or in the orbit determination of artificial satellites (see [6]), among others. Burgers' equation arises in the theory of shock waves, in turbulence problems and in continuous stochastic processes. It has a large variety of applications: modeling of water in unsaturated oil, gas dynamics, heat conduction, elasticity, statics of flow problems, mixing and turbulent diffusion, cosmology, seismology, _etc._ (see [7] and the references therein). We consider the one-dimensional Burgers' equation  
\[\frac{\partial u}{\partial t}-\frac{1}{Re}\frac{\partial^{2}u}{\partial x^{2}} +u\frac{\partial u}{\partial x}=0,\qquad a<x<b,\qquad t>0 \tag{1}\]  
with initial condition \(u(x,0)=f(x)\), \(a<x<b\), and boundary conditions \(u(a,t)=g_{1}(t)\), \(u(b,t)=g_{2}(t)\), \(t\geq 0\); where \(Re\) is the Reynolds number of the viscous fluid flow problem and \(f(x)\), \(g_{1}(t)\) and \(g_{2}(t)\) are given functions. Usually \(\varepsilon\) instead of \(\frac{1}{Re}\) is used, for simplicity. Numerical difficulties have been experienced in getting the solution of Burgers' equation for big values of \(\varepsilon\). This problem shows a roughly similar structure to that of Navier-Stokes equations due to the form of the nonlinear convection term and the occurrence of the viscosity term. So, this can be considered as a simplified form of the Navier-Stokes equation. In recent years, many researchers have used different numerical methods specially based on finite difference, finite element boundary techniques and direct variational methods to solve this problem (see, for example, [7; 8; 9] and the references therein). It has been shown by [PERSON] and [PERSON] that [PERSON] equation can be solved exactly for an arbitrary initial function \(f(x)\). The transformation  
\[u=-\frac{2\varepsilon}{\Phi}\frac{\partial\Phi}{\partial x} \tag{2}\]  
relates \(u=u(x,t)\) and \(\Phi=\Phi(x,t)\), and if \(\Phi\) is a solution of the linear diffusion equation  
\[\frac{\partial\Phi}{\partial t}=\varepsilon\frac{\partial^{2}\Phi}{\partial x ^{2}} \tag{3}\]  
then \(u\) is a solution of the quasilinear Burgers' Equation (1). This transformation let us obtain the exact values of \(u(x,t)\) because Equation (3) has a Fourier series solution, however its computational cost is very high and we only use it for comparing the precision of our results (as we will see in the Numerical Section, the integrals of the Fourier coefficients must be calculated). As we can see in [7], some researchers have taken advantage of this transformation and have applied the Crank-Nicolson scheme to the linear Equation (3) in order to obtain the values of \(\Phi(x,t)\) and after it the values of \(u(x,t)\). Other authors have used the implicit scheme, where \(\frac{\partial u}{\partial t}\) is approximated by backward differences, however, due to its truncation error has order one, the precision of the results decreases. In this paper, we apply a particular difference scheme called Crank-Nicolson type method, obtained by applying divided differences directly on the nonlinear Equation (1), needing to solve a nonlinear system of equations for obtaining the solution in each value of \(t\). These nonlinear systems are solved by using fixed point iterative methods.