11.5.2.3 Fourier method

Problem: Find the temperature distribution \( u(r ; t) \) inside the ball \( r \leq l \), if the initial temperature distribution inside the ball \( \varphi(r) \) is given. \begin{tabular}{|c|c|} \hline\( \left.u\right|_{l} \) & \( \left.u\right|_{t=0}=\varphi \) \\ \hline\( \left.u\right|_{r=l}=0 \) & \( \left.u\right|_{t=0}=\frac{A}{l}\left(\frac{r^{2}}{12 l^{2}}-\frac{r}{4 l^{2}}\right) \) \\ \hline \end{tabular}