Unsteady-State Heat Conduction in a Rod

Applying the microscopic energy balance to this system gives 

This being a diffusive system or a parabolic partial differential equation, we apply the Crank-Nicolson finite-difference analogs to equation (8.5.1) and obtain 

6 TECHNIQUES FOR PROBLEMS WITH BOTH CONVECTIVE AND DIFFUSION EFFECTS THE STATE-VARIABLE FORMULATION 

Partial differential equations which contain both diffusive and convective terms pose special problems for numerical solution via finite differencing. The basic partial differential equation of interest is given by 

Therefore, the original second-order differential equation is transformed into a coupled set of two first-order state-variable equations, 

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