# Numerical solution to a non-linear PDE

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?

• Are there boundary/initial conditions? What kind of problems did you run into when solving? – Dylan May 17 '19 at 13:43
• In my particular case the BC's is C(0,t)=14 and C(xi,t)=2, the IC is C(x,0)=2. And the PDE is the diffusion equation if D linear dependent of C, so D=C*k where k is some constant and C is the concentration of some contaminant. – Peter Panduro Jørgensen May 17 '19 at 20:09
• That doesn't make sense. If $k$ is just a constant, then $D_t = D_{xx} \implies kC_t = kC_{xx}$ which is just the same diffusion equation in $C$ – Dylan May 17 '19 at 20:32
• The Diffusion equation with a concentration depending diffusion coefficient is $$\frac{\partial C}{\partial t}=\frac{\partial }{\partial x} \left(D \frac{\partial C}{\partial x}\right)$$ The k disappears because it is made dimensionless – Peter Panduro Jørgensen May 18 '19 at 5:44
• You should definitely specify that in the question. I was assuming $D$ was the unknown variable in the diffusion equation. – Dylan May 18 '19 at 9:53

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.