Solve $$U_{t}=U_{xx}+u$$ with mixed boundary conditions $$U_x(0,t)=0, U(l,t)=0$$ and initial condition $$U(x,0)=\varphi(x)$$
I know that I have to use separation of variables and I have an idea of how to do it when its either just Dirichlet or just Neumann but both together and with a source I have no idea any help would be appreciated.
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First, we look for a lineal combination $\ds{A\sin\pars{kx} + B\cos\pars{kx}}$ which satisfies the homogeneous boundary conditions at $\ds{x = 0\ \mbox{and}\ x = L}$.
$\ds{\left.\vphantom{\large A}0 = k\bracks{A\cos\pars{kx} - B\sin{kx}} \right\vert_{\ x\ =\ 0}\,\,\, =\,\,\, kA}$ is satisfied with $\ds{k = 0}$ or $\ds{A = 0}$. $\ds{k = 0}$ just adds a constant term which vanishes out because $\ds{\mrm{U}\pars{L,t} = 0}$.
$\ds{B\cos\pars{kL} = 0}$ is satisfied whenever $\ds{k \in W \equiv \braces{\pars{n + \half}\,{\pi \over L}\,,\ n = 0,1,2,\ldots}}$
Now, we are ready to write the general solution as $\ds{\mrm{U}\pars{x,t} = \sum_{k}A_{k}\pars{t}\cos\pars{kx}}$ where $\ds{k \in W}$. It satisfies: \begin{equation} \sum_{k}\totald{A_{k}\pars{t}}{t}\,\cos\pars{kx} = -\sum_{k}A_{k}\pars{t}k^{2}\cos\pars{kx} + \mrm{u}\pars{x,t} \end{equation} Multiply both sides by $\ds{\cos\pars{qx}}$, where $\ds{q \in W}$, and integrate over $\ds{\pars{0,L}}$: \begin{equation} \totald{A_{q}\pars{t}}{t} = -q^{2}A_{q}\pars{t} + \hat{\mrm{u}}_{q}\pars{t}\,,\qquad \hat{\mrm{u}}_{q}\pars{t} \equiv {2 \over L}\int_{0}^{L}\mrm{u}\pars{x,t}\cos\pars{qx}\,\dd x\tag{1} \end{equation} Also, \begin{equation} \varphi\pars{x} = \mrm{U}\pars{x,0} = \sum_{k}A_{k}\pars{0}\cos\pars{kx}\quad\imp\quad A_{k}\pars{0} = {2 \over L}\int_{0}^{L}\varphi\pars{x}\cos\pars{kx}\,\dd x \tag{2} \end{equation}
Eqtn. $\ds{\pars{1}}$ is easily solved: \begin{align} \totald{\bracks{\exp\pars{q^{2}\, t}A_{q}\pars{t}}}{t} & = \exp\pars{q^{2}\, t}\hat{\mrm{u}}_{q}\pars{t}\,,\qquad \pars{~A_{k}\pars{0}\ \mbox{is given by}\ \pars{2}~} \\[5mm] \imp A_{q}\pars{t} & = A_{q}\pars{0}\exp\pars{-q^{2}\, t} + \int_{0}^{t}\exp\pars{-q^{2} \pars{t - \tau}}\hat{\mrm{u}}_{q}\pars{\tau}\,\dd\tau \\ & \hat{\mrm{u}}_{q}\pars{\tau}\ \mbox{is given in}\ \pars{1}. \end{align}
The $\ds{\underline{final\ solution}}$ which satisfies the conditions at the top is given by: \begin{align} \color{#66f}{\mrm{U}\pars{x,t}} & = \color{#66f}{\sum_{q}\bracks{A_{q}\pars{0}\exp\pars{-q^{2}\, t} + \int_{0}^{t}\exp\pars{-q^{2} \pars{t - \tau}}\hat{\mrm{u}}_{q}\pars{\tau}\,\dd\tau}\cos\pars{qx}}\,,\quad q \in W \\[5mm] A_{q}\pars{0} & = {2 \over L}\int_{0}^{L}\varphi\pars{x}\cos\pars{qx}\,\dd x \,,\qquad \hat{\mrm{u}}_{q}\pars{t} \equiv {2 \over L}\int_{0}^{L}\mrm{u}\pars{x,t}\cos\pars{qx}\,\dd x\,,\quad q \in W \end{align}