Inhomogeneous Initial, Boundary Value Theorem Proof

1. Mar 22, 2004

I think my lack of background in proofs is showing.

From "Partial Differential Equations of Mathematical Physics and Integral Equations" by Ronald B. Guenther and John W. Lee, pp. 107 problem 4-3.10.

Prove:

For the inhomogeneous, initial, boundary value problem

$$\left\{\begin{array}{lll} u_{tt} - c^2u_{xx} = F(x,t), & 0 < x < L, & t > 0, \\ u(x,0) = 0, & u_t(x,0) = 0, & 0 \leq x \leq L, \\ u(0,t) = 0, & u(L,t) = 0, & t \geq 0 \end{array} \right.$$

Let F(x,t) have continuous third-order partial derivatives for $0 \leq x \leq L$ and $t \geq 0$. Assume that $F(0,t) = F(L,t) = F_{xx}(0,t) = F_{xx}(L,t) = 0 \qquad \forall t \geq 0$. Then the problem above has a unique solution given by

$$u(x,t) = \sum_{n=1}^\infty u_n(t) \sin{(\lambda_n x)}$$
where
$$u_n(t) = \frac{1}{c\lambda_n}\int_0^t F_n(\tau) \sin{[c\lambda_n (t - \tau)]}\,d\tau$$
and
$$\lambda_n = \frac{n\pi}{L}$$

using the following steps. Fix T > 0 and restrict x and t to $0 \leq x \leq L$ and $0 \leq t \leq T$.

(a) Write out the expressions for $u_x,u_{xx},u_t,u_{tt}$ assuming that term-by-term differentiation is permissible.

This I can do.

$$u_x = \sum_{n=1}^\infty u_n(t) \lambda_n \cos{(\lambda_n x)}$$
$$u_{xx} = -\sum_{n=1}^\infty u_n(t) \lambda_n^2 \sin{(\lambda_n x)}$$
$$u_t = \sum_{n=1}^\infty \sin{(\lambda_n x)}\int_0^t F(\tau) \cos{[c\lambda_n (t-\tau)]}\,d\tau$$
$$u_{tt} = \sum_{n=1}^\infty \sin{(\lambda_n x)}[F(t) - c^2\lambda_n^2 u_n(t)]$$

(b) Use the definition of $u_n$ to confirm that

$$|\lambda_n^2 u_n(t)| \leq T \lambda_n \frac{||F_n||}{c}$$
$$|u_n'(t)| \leq T||F_n||$$
$$|u_n''(t)| \leq |F_n(t)| + cT\lambda_n ||F_n||$$
where
$$||F_n|| = \max_{0 \leq t \leq T} |F_n(t)|$$

Can do that, too.

(c) Assume that $F(0,t) = F(L,t) = F_{xx}(0,t) = F_{xx}(L,t) = 0 \qquad \forall t \geq 0$ and deduce that

$$|F_n(t)| \leq \frac{2}{\lambda_n^3 L}\int_0^L |F_{xxx}(x,t)\,dx$$

This I haven't pulled off yet. I don't quite see how the partial derivatives of u and the derivatives of $u_n$ relate to $F_{xxx}$.

The Fourier sine series expansion of F(x,t) relates F(x,t) with $F_n(t)$, where $F_n(t)$ is the sine coefficient of the Fourier series. This puts the two on the same page, but I couldn't take it anywhere. It seems to me like bringing u(x,t) and un(t) into the proof is going out of the scope of F and Fn.

Honestly, I really can't find much use to the three inequalities in part (b).

Anyway, I'd appreciate any help. This is homework due tomorrow, but frankly I don't care if it gets done by then because I have nothing on the line in this class (I have a running bet for \$50 that I'll just have to retake it in 2 or 3 years). I'm more concerned with figuring out how the proof works (for 2-3 years down the road).

Sorry for dragging anybody through crap they never wanted to remember.

2. Mar 23, 2004

Nevermind. I should have guessed. Integration by parts.

3. Mar 27, 2004

Integral

Staff Emeritus
WOW! someone is using this book! Excellent. I took a PDE course from Dr. Lee in '87. We used the rough draft of his book.

Dr. Guenther was my Major prof. I consider both of them friends.

4. Mar 28, 2004

Really? I rather like the book. But that's only because it's the cheapest textbook I've ever had to buy--I can't very well judge other ones since I've never seen them. Dover books are saving my bank account.

5. Mar 28, 2004

Integral

Staff Emeritus
I am glad to hear that you like the book. These guys were my favorite math profs.

John Lee is pretty tough as a prof, he does not give away grades. I took Systems of ODEs from him as a sophomore, got a D, he expected us to be able to get the correct numbers out of matrix problems, gave no partial credit for method, if the numbers were wrong, it was wrong. At that stage in my life, while I could do algebra fine, I could not add 2 and 2 to consistently get 4. :)

Ron Guenther gave me perhaps the most interesting problem I have ever been assigned. We were handed a Styrofoam ball, told that there was a concentric inhomogeneity, we were to find the thermal properties of the core with surface measurements only. (This was a Mathematical modeling class).

6. Mar 28, 2004

"At that stage in my life" sounds like good news for me! I couldn't add to save my life, much to the chagrin of me and whomever I'm working with. I've always hoped it'd just kinda go away someday, or I'd at least be able to do something about it. It's quite frustrating.

I've never really liked the whole emphasis on right answers thing. My high school physics teacher had the same kind of thing, except he gave partial credit up until you made that mistake. Everything after snowballed. It worked okay for me because I just did everything symbolically--which, for some strange reason I can do just fine--until the very last step, so even though I only managed to add correctly about 70% of the time, it never hurt me much. Now if it were based solely on the final answer, I think I'd be in a bit of trouble.

My current PDEs professor is a good guy. He's a little heavy on manually repeating the proofs in the book during the lectures, but he throws in some problems, too, so we see some of the curveballs that might be thrown at us. I have no idea how he grades, since the class is a midterm (Tuesday!) and a final, but from what he says he seems more concerned with preparing us for some kind of Qualifier Exam (no idea what it is, since it doesn't much concern me) as opposed to handing out grades.

The styrofoam ball problem sounds like the heat equation in three dimensions with the boundary conditions being whatever you measure on the surface.

7. Mar 28, 2004

Integral

Staff Emeritus
Pretty much, the interesting boundary was the interior one as we also needed to find the radius of the core. Our group made the best run at it. We had access to a Physics lab with thermal couples and A/D capabilities. We were able to read the surface temp several times a minute.

I am not sure if my addition problems have gone away or simply are not challenged any more. For me a lot of it was simply attention to detail. I am somewhat worried about the up coming generation, I see a lot of solution by calculator with no effort to apply pencil and paper to a problem. Guess as long as you have a calculator at hand this is not an issue. Unfortunately, the solution by calculator method often precludes the algebra to arrive at a complete solution before you plug in the numbers.

Last edited: Mar 28, 2004
8. Mar 28, 2004

What do you mean by the radius of the core? Just the center of the ball? Or was there something inside the ball and you didn't know what it was?

Guess you'd have to start weighing the ball and rolling it around if there were something inside it.

Eh, I'm going to bet that it's just gone away. Mostly because I don't like the alternative.

9. Mar 28, 2004

Integral

Staff Emeritus
Yes, the concentric inhomgeneity mentioned in the problem description was a steel ball bearing. I cheated, one of the first things I did was roll the styrofoam across the table with a magnet. AH! HA! iron. This lead to later troubles as I made to good of a guess at the thermal properties so my methods were unable to improve them much.

10. Mar 28, 2004

Hm. That does sound interesting.

What tools were you supposed to have available to you? Anything you could find as long as you didn't penetrate the surface?

11. Mar 28, 2004

Integral

Staff Emeritus
Experimental methods were left completely unspecified. One team attempted to HEAT their ball, they also learned about the core when the styrofoam melted! Another team attempted to use a thermometer for the surface measurements. But the surface temp of the styrofoam changed faster then the thermometer could react. All they got was the rate of change of the thermometer.

Our group got far and away the best data, but the team could not agree on how to set up the model, I ended up pretty much doing it alone on my Apple II+ (this was 1983!) I attempted to match all 3 parameters simultaneously, when I should have improved them individually. So we really did not get as good of results as we should have. Of course to do a run of the model I had to walk away from the Apple and let it run over night. I think a modern computer would have allowed enough play time that we could have gotten a better solution.

12. Mar 28, 2004