Numerical solution to a non-linear PDE

Question

I have this Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$

Where C is a function of (x,t) It comes from the diffusion equation where D is concentration depending, and has the linear form $D=k \cdot C$. The PDE made dimensionless for simplicity.

I have tried to find a solution with finite difference methods but without luck, The PDE can be linearized but this will make the numerical solution to inaccurate so no luck there either.

So how can I get a proper numerical solution?

Answer 1

You can look at this article

The way you obtain the result that is shown in the article is to look for a scaling function : $$ C(t,r) = t^{-a s} F(r t^{-a} )$$

You obtain an equation and you impose that only the variable : $x=r t^{-a} $ remains, sinceyou want a separation of variables. You obtain a relation between $s$ and $n$ that are defined in the article : $a=1/(sn+2)$. And you finish the work.

Tell me if you want me to write the whole calculation.