Hello. Can anybody solve numerically the following IBVP:

[tex]\begin{cases} \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left ( xt \left ( \frac{\partial u}{\partial t} \right )^2 \right ) + 2t(1-6x^2), \, 0<x<1, \, 0<t<1 \\ u(x;0)=x^2, \, 0 \le x \le 1 \\ u(0;t)=t^2, \, t \ge 0 \\ u(1;t)=t^2+1, \, t \ge 0 \end{cases}[/tex]

using explicit (pure implicit) difference scheme. Compare the numerical solution with the exact one [tex]u_{\operatorname{exact}}=x^2+t^2[/tex]. What is the order of the approximation of the chosen difference scheme?

I tried to construct a pure implicit scheme (all values of the unknown function [tex]y[/tex] in the right side of the respective difference equations I take on the layer [tex]j+1[/tex]) but I have no idea how to solve the difference equations. They are nonlinear and include 3 unknown values - [tex]y_{i-1}^{j+1}, \, y_{i}^{j+1}, \, y_{i+1}^{j+1}[/tex]. Thank you for helping me.