- #1
BrainHurts
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Homework Statement
Consider the nonlinnear diffusion problem
[itex] u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 [/itex]
with the constraint and boundary conditions
[itex] \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0[/itex]
Investigate the existence of scaling invariant solutions for the equation and then use the constraint condition to show that
[itex] u=t^{- \frac{1}{3}}y(z)[/itex] and [itex] z=xt^{- \frac{1}{3}} [/itex]
Homework Equations
The Attempt at a Solution
So this is what I did
Let
[itex] x = AX [/itex] and [itex] t = BT [/itex] and [itex] u=CU [/itex]
[itex] \frac{\partial}{\partial x} = \frac{1}{A} \frac{\partial}{\partial X} [/itex]
[itex] \frac{\partial^2}{\partial x^2} = \frac{1}{A^2} \frac{\partial^2}{\partial X^2} [/itex]
[itex] \frac{\partial}{\partial t} = \frac{1}{B} \frac{\partial}{\partial T} [/itex]
so
[itex] u_t = \frac{1}{B} \frac{\partial u}{\partial T} [/itex]
[itex] u_x = \frac{1}{A} \frac{\partial u}{\partial X} [/itex]
[itex] u_{xx} = \frac{1}{A^2} \frac{\partial^2 u}{\partial X^2} [/itex]
and
[itex] u_t - (u_x)^2 + uu_{xx} = 0 [/itex] becomes
[itex] u_T - \frac{B}{A^2} (u_X)^2 - \frac{BC}{A^2}uu_{XX} = 0 [/itex]
so if [itex] \frac{B}{A^2} = 1[/itex] and [itex] A = \lambda, then B = \lambda^2 [/itex]
and I want [itex] \frac{BC}{A^2} = 1[/itex] means [itex] C = 1 [/itex]
so a scaling invariant solution is [itex] u(\lambda x, \lambda^2 t) [/itex]
but I'm getting that [itex] z = \frac{x}{t^{\frac{1}{2}}} [/itex]
so I'm not really sure where the
[itex] z=xt^{- \frac{1}{3}} [/itex]