Non-Homogeneous Heat Equation Problem

[smellymoron]

Hi, I'm not sure how to solve problems of this form:

Uxx - Ut = h(x,t)
where Uxx is second derivative of U(x,t) wrt x and Ut is first derivative of U(x,t) wrt t.
Boundary conditions are as follows:
U(0,t)=U(a,t)=U(x,0)=0
and h(x,t) is a fairly simple function, or even constant, say h=1.

I'm fine with the homogeneous case except my coefficients don't seem to work when the solution is plotted. No idea how to solve when h(x,t) is anything other than 0 though. I have B&DPi textbook here but it isn't very clear. I'm going to keep trying but in the meantime if anyone gets a chance to help me, please do. Particularly for the coefficients as I need to assess the convergence of the solution.
I checked out the tutorial at the top of this forum but it didnt seem to quite have what I'm after. Ah, how I miss the days of solving a simple 1st order ODE.
Thanks in advance.

 

[Feynman]

u can choose the " seperation's variable method"

 

[saltydog]

Hi, I'm not sure how to solve problems of this form:

Uxx - Ut = h(x,t)
where Uxx is second derivative of U(x,t) wrt x and Ut is first derivative of U(x,t) wrt t.
Boundary conditions are as follows:
U(0,t)=U(a,t)=U(x,0)=0
and h(x,t) is a fairly simple function, or even constant, say h=1.

Would be an interesting problem to work on although I'd have to review. I did find a link on the net for a start:

http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_nonhomo_D.pdf

And a plot of the solution too would be nice.

 

[saltydog]

Would be an interesting problem to work on although I'd have to review. I did find a link on the net for a start:

http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_nonhomo_D.pdf

And a plot of the solution too would be nice.

I checked a bit. How about checking out Duhamel's principle for inhomogeneous heat equations. This seems to be the simplest approach.

Edit:
How about trying one (a simple one) using this method and posting it with the solution. Know LaTex? And a plot too. Or just do what you want, whatever.

 

[smellymoron]

Thanks for the link Saltydog.
Thats exactly what i spent an hour looking for and couldn't find.. a simple runthrough of the procedure for solving the problem. Much appreciated.

I've got latex on the computer here but never used it. I'll see if I can import a Mathematica image to show you a plot of the final solution.

 

[saltydog]

We have a LaTex-enabled compiler here. You just write the LaTex commands in the post and the compiler formats it. For example:

$$\text{DE:}\quad u_t-u_{xx}=tSin(x) \qquad 0\leq x\leq \pi$$

$$\text{BC:}\quad u(0,t)=0 \qquad u(\pi,t)=0$$

$$\text{IC:}\quad u(x,0)=0$$


See how nice that looks. You can just double-click on any of the equations in here and a window will pop-up with the latex commands or just go to the Physics forum and see the "introducing LaTex" thread. Now, the equation above is just a rod with an internal heat source that eventually develops into a sine wave. This one is easily solved using Duhamel's principle. It looks like:

Edit: Suppose the heat source is a sine wave. It just gets bigger (hotter) and bigger with time.

 

[smellymoron]

$$\text{DE:}\quad u_t+u_{xx}=1 \qquad 0\leq x\leq \pi$$

ah got it thanks.
i have so many variations of the same types of questions, i think I'm getting the hang of them now, though i haven't gotten one answer fully correct yet.

 

[saltydog]

\!\(4\/\(n\^3\ π\^3\)\ Exp[\(-n\^2\)\ π\^2\ t]\)

testing testing

How about going to the Physics/General Physics forum and look at the thread "Introducting LaTex" and check out the varous examples. You can experiment there. Remember to enclose the commands in the prefix and postfix bracket tex operators. Just double click on some equation and you'll see what I mean.

 

[Feynman]

We can use also Duhamel formula in this case :
$$u(x,t)=e^{t\Delta}u(x,0)+\int_{0}{t}e^{{t-s}\Delta}h(t,s}ds$$
where $$e^{t\Delta}$$ is the semi group associate to Uxx

 

[Boulayman]

Heat equation problem

Hi everyone,

We want to study the distribution of temperature inside a room filled with computers, provided all kind of conditions.

And we are trying to figure out how to modelize a room with various conditions ( BCs, initial conditions, heat generation)

For example, we are trying to come up with a heat generation term for the computers. Our understanding is that these computers would diffuse heat in a circular fashion ( we don;t to get all the way to the molecular level :) ). So, our very first guess would be to have some function g(x,y,t)=gamma/(x-x0) +gamma/(y-y0)

where gamma would be some kind of constant related to the kind of computer we are using and x0,y0 would be some kind location in a grid modelizing the room.

So what we want to know is are we on the right track, where can we find some info about heat generated by a computer, is our equation entirely wrong ?

Also, when we to solve the heat equation, we also have a term with a k coefficient that is liinked to the thermal conductivity, density and heat capacity. Since all this stuff varies a lot thoughout the room, this does look to be a great source of complexity. Is there anny way/reason to simplify/neglect that ?

Cheers, and thanks for taking the time to read that