Diffusion equation issue

Question

I need help, I’m stuck on this problem. Below is my progress.

Answer 1

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• $\ds{\on{u}_{x}\pars{x,t} = -\sum_{n = 1}^{\infty}\on{a}_{n}\pars{t}\,n\sin\pars{nx}}$ already satisfies $\ds{\on{u}_{x}\pars{0,t} = \on{u}_{x}\pars{\pi,t} = 0}$. Note that \begin{equation} \int_{0}^{\pi}\cos\pars{mx}\cos\pars{nx}\,\dd x = {\pi \over 2}\,\delta_{mn}\label{1}\tag{1} \end{equation}
• $\ds{\on{u}\pars{x,t} = \sum_{n = 1}^{\infty}\on{a}_{n}\pars{t}\cos\pars{nx} + \on{f}\pars{t}}$ which must satisfies the above partial differential equation. Namely, \begin{align} &\sum_{n = 1}^{\infty}\dot{\on{a}}_{n}\pars{t}\cos\pars{nx} + \dot{\on{f}}\pars{t} + \sum_{n = 1}^{\infty}\on{a}_{n}\,n^{2}\pars{t}\cos\pars{nx} = \expo{-x}\label{2}\tag{2} \end{align}
• Integrate both members of (\ref{2}} over $\ds{x \in \pars{0,\pi}}$: $$ \dot{\on{f}}\pars{t}\pi = -\expo{-\pi} + 1 \implies \on{f}\pars{t} = {1 - \expo{-\pi} \over \pi}\,t + c $$ $\ds{c}$ is a constant.
• Use (\ref{1}) and (\ref{2}): \begin{align} &\dot{\on{a}}\pars{t} + n^{2}\on{a}\pars{t} = {2 \over \pi}\int_{0}^{\pi}\expo{-x}\cos\pars{nx}\dd x = {2 \over \pi}\,{1 - \expo{-\pi}\pars{-1}^{n} \over 1 + n^{2}} \equiv \varphi_{n} \\[5mm] &\ \totald{\bracks{\expo{n^{2}\,t}\on{a}_{n}\pars{t}}}{t} = \varphi_{n}\expo{n^{2}\,t} \implies \expo{n^{2}\,t}\on{a}_{n}\pars{t} - \on{a}_{n}\pars{0} = {\varphi_{n} \over n^{2}}\pars{\expo{n^{2}\,t} - 1} \\[5mm] &\ \on{a}_{n}\pars{t} = \on{a}_{n}\pars{0}\expo{-n^{2}\,t} + {\varphi_{n} \over n^{2}}\pars{1 - \expo{-n^{2}\,t}} \\[5mm] &\ \on{u}\pars{x,t} = {1 - \expo{-\pi} \over \pi}\,t + c + \sum_{n = 1}^{\infty}\on{a}_{n}\pars{0} \expo{-n^{2}\,t}\cos\pars{nx} \\[2mm] + &\ \sum_{n = 1}^{\infty} {\varphi_{n} \over n^{2}}\pars{1 - \expo{-n^{2}\,t}} \cos\pars{nx} \end{align}
• Since $\ds{\on{u}\pars{x,0} = 0}$: $$ 0 = c + \sum_{n = 0}^{\infty}\on{a}_{n}\pars{0}\cos\pars{nx} \implies c = 0\ \mbox{and}\ \on{a}_{n}\pars{0} = 0\ \forall n $$
• Finally, $$ \on{u}\pars{x,t} = {1 - \expo{-\pi} \over \pi}\,t + {2 \over \pi}\sum_{n = 1}^{\infty} {1 - \expo{-\pi}\pars{-1}^{n} \over n^{2}\pars{n^{2} + 1}}\pars{1 - \expo{-n^{2}\,t}} \cos\pars{nx} $$