11.5.2.27 Fourier method

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \left\{\begin{array}{l} \frac{\partial T}{\partial t}=a^{2} \frac{\partial^{2} T}{\partial x^{2}}, \quad(0 \leq x \leq 4) \\ \left.T\right|_{x=0}=T(0, t)=0 \\ \left.T\right|_{x=4}=T(4, t)=0 \\ T(x, 0)=f(x)=\left\{\begin{array}{ll} 0, & 0 \leq x<1 \\ 1, & 1 \leq x \leq 4 \end{array}\right. \end{array}\right. \]