Given the following nonlinear PDE: $\partial_{tt} u(x,t)=ku(x,t)\partial_{xx}u(x,t)$, with $u(0,t)=u(L,t)=0$, is it possible to solve it analytically?

Asked Apr 03 '12 at 11:26

Active Feb 08 '24 at 20:53

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Could the solutions have singularities that can be interpreted as shock waves?

edited Feb 08 '24 at 20:53

amWhy

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asked Apr 03 '12 at 11:26

Riccardo.Alestra

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Did you try that way? $$u(x,t)=P(x)T(t)$$ $$\partial_{tt} u(x,t)=ku(x,t)\partial_{xx}u(x,t)$$
$$P(x)T''(t)=kP(x)T(t)P''(x)T(t)$$ $$P(x)T''(t)=kP(x)P''(x)T^2(t)$$ $$\frac{T''(t)}{T^2(t)}=kP''(x)=c$$ c is a constant.
– Mathlover Apr 03 '12 at 12:24

Given the following nonlinear PDE: $\partial_{tt} u(x,t)=ku(x,t)\partial_{xx}u(x,t)$, with $u(0,t)=u(L,t)=0$, is it possible to solve it analytically?

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