Consider the temperature distribution \(u(x, t)\) in an insulated bar of length \(L\) as governed by the heat conduction equation \(\frac{\partial u}{\partial t}=\kappa \frac{\partial^{2} u}{\partial x^{2}}\), subject to the boundary conditions \(\left.\frac{\partial u}{\partial x}ight|_{x=0}=0 t>0, u(L, t)=0, t>0\), and initial condition \(u(x, 0)=u_{o}\), \(0
(a) Consider the transformation of the above parabolic heat conduction equation. Using the nondimensional variables \(v=\frac{u}{u_{o}}, \eta=\frac{x}{L}\), and \(\tau=\frac{\kappa t}{L^{2}}\), give the transformed partial differential equation for \(u(\eta, \tau)\) with the corresponding boundary and initial conditions.

(b) Consider the direct method for the numerical solution of the parabolic partial differential equation. Show that with \(\kappa=1\) and \(L=1\), with a reflexive boundary condition at \(i=0\), the following formula results: \(u_{0, j+1}=(1-2 r) u_{0, j}+2 r u_{1, j}\), where \(r=k / h^{2}\).