Formulate complete PDE problems (specify the equation, space domain, time interval, and boundary and initial conditions) for the following model situations.

a) Conduction heat transfer occurs in a thin rod of length \(L\) with insulated sidewalls (see Figure 3.2). Temperature is initially constant \(T(0)=T_{0}\). We are asked to find the temperature distribution in the time period \(0

b) Equilibrium temperature distribution is to be found in a rectangular metal plate \(0 \leq x \leq L_{x}, 0 \leq y \leq L_{y}\). The boundaries \(x=0\) and \(x=L_{x}\) are kept at constant temperatures \(T_{1}\) and \(T_{2}\). The boundaries \(y=0\) and \(y=L_{y}\) are thermally insulated.

c) One-dimensional string stretched between the points \((x, y)=\) \((0,0)\) and \((x, y)=\left(L_{x}, 0\right)\) oscillates elastically during the time period \(0

d) The situation is almost the same as in part

(a) of this problem, but with two changes. The temperature of the left end of the rod is not constant but a function of time: \(T_{0}+\sin (\pi t)\). Also, there are internal heat sources of constant and uniform volumetric heat generation rate \(Q\) distributed along the rod.

e) The situation is almost the same as in part

(b) of this problem, but the temperatures of the boundaries at \(x=0\) and \(x=L\) are now functions of time: \(T_{1}=\sin (\pi t), T_{2}=\sin (2 \pi t)\). The solution is to be found at \(0 \leq t \leq t_{1}\). The temperature is equal to 0 in the entire plate at \(t=0\).