1D heat equation applied to a rod. Dirichlet + Neumann boundary conditions

The temperature profile of a rod, subjected to a thermal loading is derived. The rod has length \(L\), and a cross-sectional area, \(A\).  The thermal loading \(P\) is applied at \(x=0\). At \(x=L\), the temperature is kept constant at \(T_C\).

The one-dimensional heat equation is given by:

$$\frac{d^2T}{dx^2}=0$$
The boundary conditions are as follows:
$$T(x=L)=T_C$$
$$\frac{dT}{dx}(x=0)=-\frac{P}{\lambda A}$$

where, \(P\) is the aplied thermal loading in \([W]\), \(A\) is the cross-sectional area of the beam, and \(\lambda\) the thermal conductivity of the involved material in \([W/mK]\).

The thermal gradients are obtained by integration of equation 1.
$$\frac{dT}{dx} = C_1$$
$$T(x)=C_1 x + C_2$$

Solving for the boundary conditions, we find the temperature profile of the beam to be:
$$T(x) =- \frac{P}{\lambda A} x+T_C+\frac{pL}{\lambda A}$$