How can I solve this nonlinear Partial differential equation? Can anyone help me, how to find the exact solution of this PDE:
$$
\begin{align}
u_{tt} &amp;= u^2 u_{xx} \\ 
u(x,0)&amp;=g(x) \\ 
u_t(0, t)&amp;=0
\end{align}
$$
Give me roughly idea or method please.
Thank you.
Let $u(x,t) = X(x)T(t)$
$$\implies XT'' = X^2T^3X''$$
$$\implies XX'' = \frac{T''}{T^3} = \lambda$$
Now solving each equation in $X$ and $T$ separately, we have
$$(1) \space \space X'' = \frac{\lambda}{X} $$
$$(2) \space \space T'' = \lambda T^3$$
Can you solve each of these independently, apply the boundary conditions to get your solution?
Let $u(x,t) = X(x)T(t)$
$$\implies XT'' = X^2T^3X''$$
$$\implies XX'' = \frac{T''}{T^3} = \lambda$$
Now solving each equation in $X$ and $T$ separately, we have
$$(1) \space \space X'' = \frac{\lambda}{X} $$
$$(2) \space \space T'' = \lambda T^3$$
Can you solve each of these independently, apply the boundary conditions to get your solution?