Equation (3.15) is often called the fully-implicit form of the one-dimensional transient conduction difference equation because all quantities in the equation, except for the temperatures in the energy storage term, are evaluated at the new time step, m + 1. In an alternate form called Crank-Nicholson, these quantities are evaluated at both time step m and m + 1 and then averaged. This has the effect of significantly improving accuracy of the numerical solution relative to the fully-implicit from without increasing complexity of the solution method. Derive the one-dimensional transient conduction difference equation in the Crank-Nicholson form.

GIVENOne-dimensional transient conduction difference equation in the implicit form

Ax? At < 2a (3.15)