3
$\begingroup$

I'm not able to give a general solution of this pde:

$ {\partial \over \partial t} {\Phi(x,t)}={k^2\Phi(x,t)^2}{\partial^2 \over \partial x^2}{\Phi(x,t)}$ Can someone help me?

  • 0
    @Willie Wong: You are right. In fact this is a diffusion equation of atoms doping a particular kind of material in which part of the atoms form a bound with the material itself.2012-02-10

1 Answers 1

3

Here's my try. It's a standard trick that is used in physics, but I can't guarantee that the solution is the most general. Actually, I'm certain that it is not.

First of all, define $\tilde t = t k^2$ so that the equation does not have $k$ anymore. Now guess an ansatz of the form $\Phi=X(x)T(t)$, where $X$ is a function of $x$ only and $T$ is a function of $T$ only. Plugging that into the equation yields

X(x) T'(t)-T(t)^3 X(x)^2 X''(x)=0

which can be transformed, assuming all $T,X$ do not vanish to

\frac{T'(t)}{T(t)^3}=X(x)''X(x)

Since the left hand side depends on $t$ only, and the right one on $x$, they both must be a constant, which we'll denote by $g$. So you need to solve the two ODE's:

\begin{align} T'&=g T^3\\ X''&=\frac{1}{X} \end{align}

The solution of the first equation is $T=\pm\frac{1}{\sqrt{-2(g t+C)}}$ where $C$ is an integration constant. If you choose it negative enough then $T$ will be real. The solution of the $X$ equation is a bit more difficult, but Mathematica finds one: $X=A^{-2} e^{ \operatorname{InverseErf}\left[\pm i\sqrt{A \frac{2g}{\pi } (x+B)^2}\right]^2}$ where $A,B$ are integration constants. Is this any good?

  • 2
    @Riccardo.Alestra Thx. By the way, if your equation really describes "diffusion with a diffusion coefficient proportional to the square of the concentration itself", then it should read $\partial_t \Phi=\nabla\cdot\left(\Phi^2\vec\nabla\Phi\right)$ Written like that it will conserve the particle number.2012-02-12