2.4.1 Beam PDE \(u_{tt} + u_{xxxx} = 0\)

problem number 90

Added January 20, 2018.

Beam PDE with zero initial velocity. Solve \[ u_{tt} + u_{xxxx} = 0 \] With boundary conditions \begin {align*} u(0,t) &= -12 t^2\\ f(1,t) &=1-12 t^2\\ \frac {\partial ^2 u}{\partial x^2}u(0,t) &=0 \\ \frac {\partial ^2 u}{\partial x^2}u(1,t) &=12 \end {align*}

And initial conditions \begin {align*} u(x,0) &= x^4\\ \frac {\partial u}{\partial t}u(x,0) &=0 \end {align*}

Mathematica


\[\left \{\left \{u(x,t)\to x^4-12 t^2\right \}\right \}\]

Maple


\[u \left (x , t\right ) = x^{4}-12 t^{2}\]