# Solve the following Partial Differential Equation with homogeneous BC's

Problem:

$\frac{\partial u}{\partial t}+9u-\frac{\partial u^2}{\partial x^2}=0$ in 0<$x$<1 and $t$>0

With initial and boundary conditions

$u(0,t)=0$, $u(1,t)=0$, $u(x,0)=cos(\pi$x)

My attempt consisted in using the method of separation of variables to and solving each Strum-Liouville problems and to find their eigenvalues and their eigenfunctions.

$u(x,t)=X(x)T(t)$ so the pde changes to

$X(x)T'(t) + 9X(x)T(t)-X''(x)T(t)=0$

separate the variables

$\frac{T'(t)}{T(t)}=\frac{X''(x)-9X(x)}{X(x)}=-\lambda$

which leaves two equations to solve

$X''(x)-(9-\lambda)X(x)=0$

$T'(t)+(\lambda)T(t)=0$

It is the first of these which is giving me trouble. I am struggling to find the eigenvalues and thus the eigen funcions in order to get to the general solution.

Any pointers would be appreciated.

• Your BCs are $X(0) = X(1) = 0$ and the solution to the ODE is $$X = A \cos(\gamma x) + B \sin(\gamma x)$$ where $\lambda - 9 = \gamma^{2}$. The BCs imply $X(x) = B \sin(\gamma x)$ and your eigenvalues end up being $\lambda = n^{2} \pi^{2} + 9$. – Mattos Dec 2 '16 at 4:03