11.5.2.30 Fourier method

Problem: Solve the Laplace equation, using the Fourier method: \[ \left\{\begin{array}{l} \frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}=0, \quad\left\{\begin{array}{l} 0 \leq x \leq 1 \\ 0 \leq y \leq 3 \end{array}\right. \\ T(0, y)=T(1, y)=0 \\ T(x, 0)=T(x, 3)=x-1 \end{array}\right. \]