11.5.2.26 Fourier method

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \begin{array}{l} \frac{\partial u}{\partial t}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0,0 \leq x \leq 1, u(0, t)=u(1, t)=0, \\ u(x, 0)=f(x)=\left\{\begin{array}{ll} x, & 0 \leq x \leq 0.5 \\ 1-x, & 0.5 \leq x \leq 1 \end{array}\right. \end{array} \]