# Burger's equation

1. Oct 18, 2011

### squenshl

1. The problem statement, all variables and given/known data
a) Show that the function $\phi$(x,t) = 1 + sqrt(a/t)exp(-x^2/(4vt)) with a > 0 satisfies the dispersion equation $\phi$t = v$\phi$xx
b) Use the Cole-Hopf transformation to find the corresponding solution u(x,t) to Burger's equation ut + uux = vuxx.
c) Show that this solution is anti-symmetric (odd) and that the area under the solution for x > 0 A(t) is given by A(t) = 2v log(1+sqrt(a/t)).

2. Relevant equations

3. The attempt at a solution
a) is easy.
For b) the Cole-Hopf transformation is $\psi$ = -2vlog($\phi$)
and to find u we solve u(x,t) = (-2v$\phi$x(x,t))/$\phi$(x,t)
so do we just use the $\phi$(x,t) in part a).
c) For oddness do we show that -u(-x,t) = u(x,t) and to find A(t) do we use A(t) = $\int_0^{∞}$ u(x,t) dx