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M=8; NN=8; m = M - 1; n = NN - 1; alpha = 5.; 
beta = 4.; c = 0.05; T = 2.; h = (1. - 0.)/M; 
k =T/NN

The problem is Subscript[u, t] + u Subscript[u, x] = c Subscript[u, xx] with initial condition

Subscript[u, x, 0] = (2. c beta Pi Sin[Pi Subscript[x, i]])
                   / (alpha + beta Cos[Pi Subscript[x, i]]) 

and boundary condition

Subscript[u, 0, t] = 0; Subscript[u, 1, t] = 0;

I tried the backward finite difference (FD) for Subscript[u, t] and the central FD for the others. I wrote the following code, but I think there are some gaps in it. Because, the approximate solutions do not match the exact one

u[x_, t_] := (2 c beta Pi Exp[-c Pi^2 t] Sin[Pi x])
            / (alpha + beta Exp[-c Pi^2 t] Cos[Pi x]);

Note that Subscript[w,i,j] stands for the approximation in the grid point $(x_i,t_j)$.

The equation is

$$\frac{\partial u(x,t)}{\partial t}+u(x,t) \frac{\partial u(x,t)}{\partial x}=c \frac{\partial ^2u(x,t)}{\partial x^2}$$

with initial condition

$$u(x,0)=\frac{2 \pi \beta c \sin (\pi x)}{\alpha +\beta \cos (\pi x)}$$

and boundary conditions

$$u(0,t)=u(1,t)=0$$

I tried the backward finite difference (FD) for $\frac{\partial u(x,t)}{\partial t}$ and the central FD for the others. I wrote the following code, but I think there are some gaps in it. Because, the approximate solutions do not match the exact one

$$u(x,t)=\frac{2 \pi \beta c e^{-c \pi ^2 t} \sin (\pi x)}{\alpha +\beta e^{-c \pi ^2 t} \cos (\pi x)}$$

Note that Subscript[w, i, j] stands for the approximation in the grid point $(x_i,t_j)$.