1.1.10 Diffusion-Reaction. Both ends at zero temperature, no source. reaction term is \(xu(x,t)\)

\begin{equation} u_{t}=ku_{xx}-ux \tag{1} \end{equation} BC are\begin{align*} u\left ( 0,t\right ) & =0\\ u\left ( \pi ,t\right ) & =0 \end{align*}

Initial conditions\[ u\left ( x,0\right ) =\sin \left ( \pi \right ) \qquad 0<x<\pi \] This problem can not be currently solved in Maple nor by Mathematica analytically.

solution

Trying separation of variables. Let \(u=XT\), then the PDE becomes\begin{align*} T^{\prime }X & =kX^{\prime \prime }T-XTx\\ \frac{1}{k}\frac{T^{\prime }}{T} & =\frac{X^{\prime \prime }}{X}-\frac{x}{k}=-\lambda \end{align*}

Where \(\lambda \) is separation constant. Hence\[ T^{\prime }+k\lambda T=0 \] Which has solution \(T\left ( t\right ) =e^{-k\lambda t}\) and\begin{align*} \frac{X^{\prime \prime }}{X}-\frac{x}{k} & =-\lambda \\ X^{\prime \prime }+X\left ( \lambda -\frac{x}{k}\right ) & =0\\ X\left ( 0\right ) & =0\\ X\left ( L\right ) & =0 \end{align*}

In the above \(k>0\).  This ODE involves Airy functions. Will try to solve later when have more time. Too hard now.