Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Solve $$\frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2}$$ with $0 < x < 3, t > 0$, given that $u(0,t) = u(3,t) = 0$, and $$u(x,0) = 5\sin 4\pi x - 3\sin 8\pi x + 2\sin 10 \pi x.$$ Note that $u(x,t)$ is bounded.

share|improve this question
1  
Welcome to Math.SE! A few guidelines on getting the best help from the community possible: What have you done so far? What do you know? If you can support your questions with answers to these two questions, you will be more likely to receive the best help. –  Arkamis Nov 28 '12 at 16:46
1  
Use separation of variables. –  dcs24 Nov 28 '12 at 16:46
    
Do you know the general form of the solution to the heat equation? If so, then it will be very quick to read off the Fourier coefficients and get a full exact solution. –  Matt Nov 28 '12 at 17:50
    
This question has been solved perfectly. Hope that the asker has been diving enough and accept the answer at an early date. –  doraemonpaul Apr 21 '13 at 2:13

1 Answer 1

Let $u(x,t)=\sum\limits_{n=1}^\infty C(n,t)\sin\dfrac{n\pi x}{3}$ so that it automatically satisfies $u(0,t)=u(3,t)=0$ ,

Then $\sum\limits_{n=1}^\infty\dfrac{\partial C(n,t)}{\partial t}\sin\dfrac{n\pi x}{3}=-\dfrac{2n^2\pi^2}{9}\sum\limits_{n=1}^\infty C(n,t)\sin\dfrac{n\pi x}{3}$

$\therefore\dfrac{\partial C(n,t)}{\partial t}=-\dfrac{2n^2\pi^2}{9}C(n,t)$

$\dfrac{dC(n,t)}{C(n,t)}=-\dfrac{2n^2\pi^2}{9}dt$

$\int\dfrac{dC(n,t)}{C(n,t)}=\int-\dfrac{2n^2\pi^2}{9}dt$

$\ln C(n,t)=-\dfrac{2n^2\pi^2t}{9}+f(n)$

$C(n,t)=F(n)e^{-\frac{2n^2\pi^2t}{9}}$

$\therefore u(x,t)=\sum\limits_{n=1}^\infty F(n)e^{-\frac{2n^2\pi^2t}{9}}\sin\dfrac{n\pi x}{3}$

$u(x,0)=5\sin4\pi x-3\sin8\pi x+2\sin10\pi x$ :

$\sum\limits_{n=1}^\infty F(n)\sin\dfrac{n\pi x}{3}=5\sin4\pi x-3\sin8\pi x+2\sin10\pi x$

$F(n)=\begin{cases}5&\text{when}~n=12\\-3&\text{when}~n=24\\2&\text{when}~n=30\\0&\text{otherwise}\end{cases}$

$\therefore u(x,t)=5e^{-32\pi^2t}\sin4\pi x-3e^{-128\pi^2t}\sin8\pi x+2e^{-200\pi^2t}\sin10\pi x$

Note that this solution suitable for $x,t\in\mathbb{C}$ , not only suitable for $0<x<3$ and $t>0$ .

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.