11.5.2.12 Fourier method

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=4 u_{x x}, \\ u(x, 0)=15 \sin 3 \pi x, u(0, t)=0, u_{x}(4.5, t)=0 . \end{array} \]