Solving Burger's Equation Homework

[squenshl]

Homework Statement

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 u_t + uu_x = vu_xx.
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)).

Homework Equations

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

 

[mighty2000]

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a) To show that the function \phi(x,t) = 1 + sqrt(a/t)exp(-x^2/(4vt)) satisfies the dispersion equation \phit = v\phixx, we can simply plug in the given function into the dispersion equation and show that it holds true.

\phit = \frac{\partial}{\partial t}(1 + \sqrt{\frac{a}{t}}e^{-\frac{x^2}{4vt}}) = -\frac{1}{2\sqrt{at^3}}e^{-\frac{x^2}{4vt}} + \frac{1}{8v^2t^2}x^2e^{-\frac{x^2}{4vt}}

\phixx = \frac{\partial^2}{\partial x^2}(1 + \sqrt{\frac{a}{t}}e^{-\frac{x^2}{4vt}}) = -\frac{1}{2\sqrt{at}}e^{-\frac{x^2}{4vt}} + \frac{1}{16v^2t}x^2e^{-\frac{x^2}{4vt}}

v\phixx = v(-\frac{1}{2\sqrt{at}}e^{-\frac{x^2}{4vt}} + \frac{1}{16v^2t}x^2e^{-\frac{x^2}{4vt}}) = -\frac{v}{2\sqrt{at}}e^{-\frac{x^2}{4vt}} + \frac{1}{16vt}x^2e^{-\frac{x^2}{4vt}}

Therefore, we can see that \phit = v\phixx and the function \phi(x,t) satisfies the dispersion equation.

b) To find the corresponding solution u(x,t) to Burger's equation using the Cole-Hopf transformation, we first need to find the expression for \psi.

\psi = -2vlog(\phi) = -2vlog(1 + \sqrt{\frac{a}{t}}e^{-\frac{x^2}{4vt}}) = -2vlog(1) - 2vlog(\sqrt{\frac{a}{t}}e^{-\frac{x^2}{4vt}}) = -2vlog(\sqrt{\frac{a}{t}}e^{-\