Partial Differential Equation

Consider a thin rod of length L. Fourier's law, relating heat flux q'' and temperature T is then q(x;T)=-kdT/dx, where 0<=x<=L is the position along rod,  t is time, and k  is the thermal conductivity. Using this expression, and applying conservation of energy to a small element of the rod between x and x+ \delta x  yields the heat equation

where: alpha=k/cc is the thermal diffusivity, in which \rho is the mass density and cc is the specific heat capacity;  Ta is the ambient temperature; and c is the heat transfer coefficient between the rod and the ambient air. The last term in Eq. (1) represents the distributed heat flux due to the temperature difference between the rod and air. In general, the parameters \alpha and c can depend on position, and Ta can depend on position and time.

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