Hello everyone!

I have a problem with a PDE that I have to solve.

I need to solve the heat equation (2nd order PDE) with respect to time and distance in a region defined between two distances namely x1 and x2, between two times t1 and t2. The problem I have is that both BC and the IC are dependent on time and distance respectively. The BCs and the IC have been determined from experimental data, and all of them are fittings to 3rd order polynomials.

Once the equation is simplified, it is quite easy to solve the heat equation using variable separation, but since the BCs and the IC are fittings to 3rd order polynomials, I am quite confused on how to apply them to the problem in order to solve it.

To be more mathematically rigorous, the problem can be defined as:

$\displaystyle \rho c \frac{{\partial u}}{{\partial t}} = k \frac{{d^2 u}}{{dx^2}}, x_1<x<x_2, t_1<t<t_2$

With the following boundary and initial conditions:

$\displaystyle u\left(x_1, t\right)=f\left(t\right), t_1<t<t_2$

$\displaystyle u\left(x_2, t\right)=g\left(t\right), t_1<t<t_2$

$\displaystyle u\left(x, t_1\right)=h\left(x\right), x_1<x<x_2$

Where $\displaystyle f\left(t\right)$, $\displaystyle g\left(t\right)$, and $\displaystyle h\left(x\right)$ have been determined from experimental data and are third-order polynomials.

Anyone can help me with this issue??

Thank you very much in advance!

Bernie