Chapter 1.7, Problem 62P

Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length $\Delta x$, derive the heat equation, $\frac{\partial T}{\partial t}=K\frac{{\partial }^{2}T}{\partial x^2},$ where $K=k_t/cp,$ c is the specific heat of the material, and p is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is $T\left(x,t\right)=T_0+\frac{A}{\sqrt{t}}{e}^{-x^2/4Kt},$ where $T_0$ is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.