1.1.4 Pure diffusion. Left end insulated, right end at non-zero temperature

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A bar length \(L=2\)\[ \frac{\partial u}{\partial t}=\alpha ^{2}\frac{\partial ^{2}u}{\partial x^{2}}\qquad 0<x<L,t>0 \] Boundary conditions, left end only insulated\begin{align*} \left . \frac{\partial u}{\partial x}\right \vert _{x=0} & =0\\ u\left ( L,t\right ) & =T_{0}=100^{0} \end{align*}

Initial conditions\[ u\left ( x,0\right ) =0^{0}\] Solution, \(\lambda _{n}=\left ( \frac{n\pi }{2L}\right ) ^{2},n=1,3,5,\cdots \)\[ u\left ( x,t\right ) =T_{0}-\frac{4T_{0}}{\pi }\sum _{n=1,3,5,\cdots }^{\infty }\frac{\left ( -1\right ) ^{n}}{2n+1}\exp \left ( -\alpha ^{2}\lambda _{n}t\right ) \cos \left ( \sqrt{\lambda _{n}}x\right ) \]

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