Math Help - Solving a PDE ???

1. Solving a PDE ???

Hey all,

I am trying to determine how many solns. the PDE Ut = Uxx has. However, I have no idea how to solve this PDE.

Unlike ODE's which you are given a fairly "standard" y and x variable equation, PDE's just lose me from the beginning. Please show me how to solve such an equation as above, because I have about four others to solve and have no idea how to determine the number of solns to each. I am sure that if I am shown once, I can get the rest.

Thanks.

2. Originally Posted by spearfish
Hey all,

I am trying to determine how many solns. the PDE Ut = Uxx has. However, I have no idea how to solve this PDE.

Unlike ODE's which you are given a fairly "standard" y and x variable equation, PDE's just lose me from the beginning. Please show me how to solve such an equation as above, because I have about four others to solve and have no idea how to determine the number of solns to each. I am sure that if I am shown once, I can get the rest.

Thanks.
Typically with the PDE $u_t = u_{xx}$ you're given boundary conditions and initial conditions. Do you have these?

On a side note - unlike ODEs, PDEs (without BC's and IC's) usually have an infinite number of solutions. Ex.

$y'' + y = 0$ solutions $y = \sin x, y = \cos x$

$u_t = u_{xx}$ has, as one class of solution

$u = e^{-k^2 t} \sin k x ,$ $u = e^{-k^2 t} \cos k x$ for any number $k$.

It also has polynomial solutions

$
u = x^2 + 2t
$

$
u = x^3 +6 x t
$

$
u = x^4 + 12 x^2 t + 12 t^2
$

The list goes on and on.

3. Thanks for the reply Danny. Unfortunately, I was not given any BC's or IC's, I was just told to determine the number of solutions to the equation Ut = Uxx. As you stated above, I am guessing the answer would be infinite, but how do I know if it's infinitely many solutions or just a few. For example, how did you come up (or know) the list of solutions to the equation you listed in your reply above?

4. Originally Posted by spearfish
Thanks for the reply Danny. Unfortunately, I was not given any BC's or IC's, I was just told to determine the number of solutions to the equation Ut = Uxx. As you stated above, I am guessing the answer would be infinite, but how do I know if it's infinitely many solutions or just a few. For example, how did you come up (or know) the list of solutions to the equation you listed in your reply above?
I've studied PDEs for more than twenty years and in that time I've read a lot and work alot with them. That's how I knew of these solutions. I also did some work about 10 years ago where we were able to show given one "seed" solution, we could generate a whole hierarchy of solutions. I can share that with you if you like.

5. Wow, that's a lot of experience with PDE's!

Thanks for the offer Danny. Although sharing this info with me would be nice, I am afraid I would be lost, since I am just beginning my journey to understanding these PDE's. What about solutions in the form of U(x,t) = exp^(ax + bt). How would I go about finding solutions of this form or tell how many solns of this form it has? Last question, I promise, lol.

6. Originally Posted by spearfish
Wow, that's a lot of experience with PDE's!

Thanks for the offer Danny. Although sharing this info with me would be nice, I am afraid I would be lost, since I am just beginning my journey to understanding these PDE's. What about solutions in the form of U(x,t) = exp^(ax + bt). How would I go about finding solutions of this form or tell how many solns of this form it has? Last question, I promise, lol.
Actually, just substitute this form into the PDE and require it be automatically satisfied. So here

$u_t = b e^{ax + bt},\;\;\;u_{xx} = a^2 e^{ax+bt}$

giving $u_t = u_{xx}\;\;\; \Rightarrow\;\;\; b e^{ax + bt} = a^2 e^{ax+bt}$

So what's the condition on $a$ and $b$ such that this is satisfied?

7. There were no conditions on a and b. The problem just said to find as many solutions for Ut = Uxx in the form of

$
U_{x,t} = e^{ax + bt}
$
.

No other information was given.

8. Originally Posted by Danny
Actually, just substitute this form into the PDE and require it be automatically satisfied. So here

$u_t = b e^{ax + bt},\;\;\;u_{xx} = a^2 e^{ax+bt}$

giving $u_t = u_{xx}\;\;\; \Rightarrow\;\;\; b e^{ax + bt} = a^2 e^{ax+bt}$

So what's the condition on $a$ and $b$ such that this is satisfied?
Let me help a bit more. From

$b e^{ax + bt} = a^2 e^{ax+bt}$ we see that this is only satisfied (for nonzero a and b) if $b = a^2$. Thus, we have solution of the form

$
u(x,y) = e^{a x+a^2 t}
$

9. Danny,

Once again, I can't thank you enough for taking the time to help me understand. I have a better understanding now. I am sure I ll have plenty more questions as my study of these PDE's progresses, but I ll jut take it one step at a time for now.