A rectangular bar of dimensions \(L_{x} \times L_{y} \times L_{z}\), constant material properties \(\kappa, ho, C\), and initial temperature \(T_{0}\) is immersed in cold water maintained at constant temperature \(T_{w}

a) Write the complete PDE problem (heat equation and initial and boundary conditions) for conduction heat transfer within the bar. Assume that the boundaries of the bar have the same temperature as water at all times.

b) Develop the simple explicit and Crank-Nicolson schemes. Include the finite difference approximations of the boundary conditions.

c) What are the truncation error and stability conditions for each scheme?