11.5.5.4 Mixed problems

Problem: Solve the mixed problem, using the Fourier method: \[ \left\{\begin{array}{l} u_{t}=u_{x x}, 00 \\ u(x, 0)=\cos ^{4} x-\sin ^{4} x \\ u(0, t)=u_{x}(\pi, t)=0 \end{array}\right. \]