## 3 Numerical Results  
We are comparing our results with the exact solution, which can be obtained by the Hopf-Cole transformation mentioned on Equations (2) and (3). For doing this, we first calculate  
\[\Phi(x,t)=A_{0}+\sum_{p=1}^{\infty}A_{p}e^{-p^{2}\varepsilon^{2}t}\cos\left( p\pi x\right)\]  
and after it, by Equation (2), we obtain  
\[u(x,t)=2\pi\varepsilon\frac{\sum_{p=1}^{\infty}A_{p}e^{-p^{2}\varepsilon\pi ^{2}t}p\ \ \sin\left(p\pi x\right)}{A_{0}+\sum_{p=1}^{\infty}A_{p}e^{-p^{2}\varepsilon \pi^{2}t}\cos\left(p\pi x\right)} \tag{14}\]  
where  
\[A_{0}=\int_{a}^{b}e^{-\frac{1}{2t}\int_{a}^{x}u(\xi,0)d\xi}dx,\]  
\[A_{p}=2\int_{a}^{b}e^{-\frac{1}{2t}\int_{a}^{x}u(\xi,0)d\xi}\cos\left(p\pi x \right)dx,\ \ p=1,2,\ldots\]  
Our goal is to obtain at least five exact decimals from the exact solution in order to prove the precision of our results, and for that we have seen that it is enough with two summands of the Fourier series, thereby only \(A_{0}\), \(A_{1}\) and \(A_{2}\) will have to be calculated. In the following examples we consider that the boundary conditions are zero, so we have that \(g_{1}(t)=0\) and \(g_{2}(t)=0\), what means that \(u_{0,j}=0\) and \(u_{n,j}=0\), \(\forall j\). We can compare our results with the exact solution in this case, however our method has been found numerically to be unconditionally stable whatever the boundary conditions are. We use the R2013a version of Matlab with double precision for obtaining both approximated and exact solutions. The stopping criterion used is  
\[\|u^{(k+1)}(x,t_{j})-u^{(k)}(x,t_{j})\|+\|F(u^{(k+1)}(x,t_{j}))\|<10^{-15}\]  
**Example 1. ** The initial and boundary conditions of [PERSON] equation for this example are  
\[u(x,0)=\frac{2\upvarepsilon\upbeta\pi sin(\pi x)}{\upalpha+\upbeta cos(\pi x )},\qquad 0\leq x\leq 2\]  
and \(u(0,t)=u(2,t)=0,t\geq 0\) where \(\upalpha=5\) and \(\upbeta=4\). In the following tables we show, for different values of \(n\), \(m\) and \(\upvarepsilon\), the average number of iterations by using [PERSON]', [PERSON]'s and M5 methods. We also show the greatest error, GE, that we obtain for every value of \(x_{i}\) and \(t_{j}\), where \(i=0,1,...,n-1\) and \(j=0,1,...,m-1\). The initial approximation of each nonlinear system to be solved is the solution of the previous one (and for the first system the initial approximation is the exact value of the initial condition). After that, we obtain the matrix of errors as the absolute difference between the approximated solution and the exact solution on each node, finally we call the greatest error as the greatest element from that matrix. While other authors prefer to only compare and give the value of the error in some points of the mesh we have preferred a more rigorous criterium, it consists on comparing all the error values from the matrix of errors and giving the value of GE, in this way we get a better idea about how bad is the worst possible situation and we know for sure that the error of the rest of the points will always be below this GE, the other way we can not know if there is some point that has a greater one. An interesting result emerges from that error matrix if we consider a sufficiently high upper bound of the studied time. If we obtain the GE between all values of \(x\) on each time we observe that the GE increase really fast at first but after the peak is reached it starts to decrease. This result can be seen on the Figure 1.  
In Table 1 we set the maximum time of the study, the number of time nodes and the Reynolds constant as \(t_{max}=1\), \(m=100\) and \(\upvarepsilon=\frac{1}{Re}=0.1\) and we try different amount of space nodes. In each case we obtain the already mentioned GE and the average number of iterations (Avg. iter.) needed to solve each nonlinear system by using [PERSON]', Traub's and M5 methods. From these results we observe that, in general, the more space subintervals we take (smaller space step), the less is GE.