Partial differential equation with variable BC and IC

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

Re: Partial differential equation with variable BC and IC

From your conditions it seems like separation of variables is probably not going to work. When you do separation of variables, your solution will be of the form u(x,t) = X(x)T(t). Now whatever T(t) is (in this case it will be an exponential), u(x1,t) and u(x2,t) are only going to differ by a multiplicative constant factor (i.e X(x2)/X(x1)). But you have 2 different functions of time for the 2 boundaries. So you can see that it can't work. I also have a feeling doing this in an analytical fashion might not be easy. (Speechless)

Re: Partial differential equation with variable BC and IC

twizter,

Thanks for your reply. After quite extensive reading on the topic, I have seen that there is a way to turn the previous homogeneous PDE with inhomogeneous BCs into an inhomogeneous PDE with homogeneous BCs that can be solved by means of eigenfunctions and eigenvalues.

However, to get to this point, every single attempt of solving it I have been able to find requires an assumption of a "steady state temperature distribution shape", and the solutions I have been able to find use a straight line as an approach for this.

Due to the nature of my experiment, I can't wait until it reaches steady state (it would take way over 24 hours, and in such a long time, the boundary conditions would change, so the results would be inconsistent). For this reason, I must focus to a range in the dominion of definition where (1) the disturbance in the output due to variations in the input are negligible, and (2) where all my thermocouples have registered variation in the temperature versus initial steady state.

I don't know if you follow what I mean, but following the line in which I stated the problem above, I can define the following dominion where the PDE is strictly valid:

BC 1: $\displaystyle u\left(x_1,t\right)=f\left(t\right)$ for $\displaystyle t_1<t<t_2$

BC 2: $\displaystyle u\left(x_2,t\right)=g\left(t\right)$ for $\displaystyle t_1<t<t_2$

IC: $\displaystyle u\left(x,t_1\right)=h_1\left(x\right)$ for $\displaystyle x_1<x<x_2$

FC: $\displaystyle u\left(x,t_2\right)=h_2\left(x\right)$ for $\displaystyle x_1<x<x_2$

Now, although theoretically not necessary, I have added a "Final condition" (FC), because it was measured in the lab and it can be modeled using the data obtained. Although both $\displaystyle h_1\left(x\right)$ and $\displaystyle h_2\left(x\right)$ are, in general, third-degree polynomials, the trend observed shows that in the region of study, the order of the distribution along x decreases in the dominion of study from a third-degree polynomial to a second-degree polynomial. (Note: indeed, it fits a third-degree polynomial with $\displaystyle R^2=1$, although a second-degree polynomial could be used for $\displaystyle h_1\left(x\right)$ with $\displaystyle R^2=0.998$. In the case of $\displaystyle h_2\left(x\right)$, it strictly satisfies $\displaystyle R^2=1$ for a second-degree polynomial)

Finally, I just need all this to be able to calculate the coefficients k and c that fit the equation above using an iterative procedure.

I don't know if this explanation made it easier or more difficult, but I really appreciate any help that I can get.

Thanks,

Bernardo

Re: Partial differential equation with variable BC and IC

I am not really sure why you need this analytical expression for the solution based on experimental data. If you have good amount of experimental data (i.e if you know u(x,t) for x1<x<x2 for t1<t<t2), why don't you plot a surface plot and use the surface fitting tool on matlab. You can select polynomial fit or whatever you want. Also make sure your initial condition meets the boundary conditions here. The reason I am saying this is, at t = t1, u(x) = h1(x). Now you need to make sure h(x1) = f(t1). This might not be true when you got f(t) from an experimental fit i.e f(t1) might not be equal to the actual value of u at the boundary x1 at t1 because the fit curve might not be going through all the data points perfectly. Just a note of caution.

Re: Partial differential equation with variable BC and IC

I see what you mean. Indeed, fitting the surface with the experimental data was one of my first options, but after reading some papers on the issue, I think I still need to solve the equation.

The reason is that I am trying to determine the parameters k and c that fit the temperature field separately. I can determine the diffusivity $\displaystyle \alpha$ as follows:

$\displaystyle \alpha=\frac{{k}}{{\rho c}}=\frac{{\frac{{\partial u}}{{\partial t}}}}{{\frac{{d^2 u}}{{dx^2}}}}$

This value can be determined immediately using the data gathered in the lab, for any range in x and any range in t.

However, to analyze the elements of the diffusivity (conductivity $\displaystyle k$ and the capacity $\displaystyle c$) an iterative procedure needs to be used, that actually requires the analytical solution of the temperature field that allows the prediction of temperature values for random values of $\displaystyle k$ and $\displaystyle c$, and that also would allow the determination of the influence of the variation of those parameters in order to determine a new approach for the parameters, that eventually would converge to the actual values of the parameters I am looking for.

Indeed, I would have enough being able to determine just one of the parameters. The density $\displaystyle \rho$ is assumed to be constant (the laboratory procedure is designed to make sure that the density is close to constant all throughout the specimen) and thus, since the diffusivity $\displaystyle \alpha$ can be determined, the remaining parameter can be determined immediately.

I hope this clarifies.

Thanks for your attention to this issue.