Consider the heat conduction in a thin insulated bar of length \(3 \mathrm{~m}\) where the initial temperature at \(t=0\) is \(25^{\circ} \mathrm{C}\) and the ends of the bar are kept at \(10^{\circ} \mathrm{C}\) at \(x=0\) and \(40^{\circ} \mathrm{C}\) at \(x=3\). The partial differential equation for the temperature distribution \(u(x, t)\) at the distance \(x\) and time \(t\) in the bar is given by \(\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\). Here the thermal diffusivity is simply equal to \(1 \mathrm{~m}^{2} \mathrm{~s}^{-1}\). The boundary conditions for this problem are given by \(u(0, t)=10, t>0, u(3, t)=40, t>0\), and the initial condition is \(u(x, 0)=\) 25 , for \(0
(a) Explain why \(u(x, t)\) can be written as two separate functions. Show that the boundary value problem can be rewritten as two separate problems: \(\psi^{\prime \prime}(x)=0\), \(\psi(0)=10, \psi(3)=40\), and \(\frac{\partial v}{\partial t}=\frac{\partial^{2} v}{\partial x^{2}}\) with \(v(0, t)=0, v(3, t)=0, v(x, 0)=\) \(15-10 x\).

(b) What is the solution for \(\psi(x)\) ? The solution for \(\psi(x)\) is in fact needed to obtain the initial condition for \(v(x, 0)\).

(c) The partial differential equation problem for the function \(v(x, t)\) is in fact the same boundary value problem as given in Problems 5.9 and 5.10. Using this solution for \(v(x, t)\), give the complete solution for \(u(x, t)\).

(d) What is the physical significance of the solution for \(\psi(x)\) ?