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Differential EquationsEasyBoundary Values & Fourier Series

Boundary Values & Fourier Series · Easy problem

The mode u(x,t)=e−rtsin⁡πx2u(x, t) = e^{-rt}\sin\dfrac{\pi x}{2} solves the heat equation ut=uxxu_t = u_{xx}. How long does it take for this mode's amplitude to fall to half its initial value?