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I'm wondering if my solution is correct. The PDE is ##h_t = h_{zz}## subject to ##h_z(0,t)=0##, ##h(1,t)=-1##, and let's not worry about the initial condition now. To solve I want homogenous boundary conditions, so lets set ##v = h+1##. Then we have the following: ##v_t = v_{zz}## subject to ##v_z(0,t)=0##, ##v(1,t)=0##. To solve take separation of variables where ##v = T(t)Z(z) \implies T'/T= Z''/Z = -\lambda \implies Z''+\lambda Z = 0##. Guess ##Z = A \cos \sqrt{\lambda} z+B\sin \sqrt{\lambda} z##. Then ##Z'(0)=0 \implies B=0##. Thus ##Z = \cos \sqrt{\lambda} z##. ##Z(1)=0 \implies \cos \sqrt{\lambda} = 0 \implies \sqrt{\lambda} = \pi/2+n\pi:n\in0,1,2...##. Thus, ##Z = \cos((\pi/2+n\pi)z)##. Yet this doesn't look right. Any ideas?

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# I PDE Heat Eq

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