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  1. #1
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    IBV

    $\displaystyle \text{D.E.}: \ u_t=ku_{xx}, \ 0<x<L, \ t>0$
    $\displaystyle \displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u_ x(L,t)=0\end{cases}$
    $\displaystyle \text{I.C.}: \ u(x,0)=f(x) \ \ 0<x<L$

    $\displaystyle u(x,t)=\varphi(x)T(t)$

    $\displaystyle \displaystyle\varphi(x)T'(t)=k\varphi''(x)T(t)\Rig htarrow\frac{\varphi''}{\varphi}=\frac{T'}{Tk}=-\lambda$

    $\displaystyle \varphi''+\lambda\varphi=0\Rightarrow \varphi=A\cos(x\sqrt{\lambda})+B\sin(x\sqrt{\lambd a})$

    $\displaystyle \displaystyle T=Ce^{-\lambda kt}$

    $\displaystyle \varphi_1(0)=1 \ \varphi_2(0)=0$
    $\displaystyle \varphi_1'(0)=0 \ \varphi_2'(0)=1$

    $\displaystyle \varphi_1: \ A=1 \ \varphi_2: \ A=0$
    $\displaystyle \displaystyle\varphi_1': \ B=0 \ \varphi_2': \ B=\frac{1}{\sqrt{\lambda}}$

    $\displaystyle \displaystyle\varphi=A_2\cos(x\sqrt{\lambda})+B_2\ frac{\sin(x\sqrt{\lambda})}{\sqrt{\lambda}}$

    $\displaystyle \displaystyle\varphi(0): \ A_2=0$

    $\displaystyle \displaystyle\varphi': \ B_2\cos(x\sqrt{\lambda})=0$

    The eigenvalues must satisfy:

    $\displaystyle \displaystyle\cos(x\sqrt{\lambda})=0$

    $\displaystyle \displaystyle\lambda_n=\pi^2\left(\frac{1+2n}{2L}\ right)^2, \ n\in\mathbb{Z}$

    The eigenfunction is:

    $\displaystyle \displaystyle\varphi_n(x)=\frac{2L}{\pi(1+2n)}\sin \left(\frac{\pi x(1+2n)}{2L}\right), \ n\in\mathbb{Z}^+$

    Is this much correct?

    Thanks.
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  2. #2
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    $\displaystyle \displaystyle u(x,t)=\sum_{n=1}^{\infty}b_n\frac{2L}{\pi(1+2n)}\ exp\left(-\frac{\pi^2}{4}\left(\frac{1+2n}{L}\right)^2kt\rig ht)\sin\left(\frac{\pi x(1+2n)}{2L}\right)$

    I have shown $\displaystyle \displaystyle\int_0^L\varphi_n(x)\varphi_m(x) \ dx=0, \ m\neq n$.

    Before I solve for $\displaystyle b_n$ is u(x,t) correct?

    Thanks.
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  3. #3
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    Quote Originally Posted by dwsmith View Post
    $\displaystyle \text{D.E.}: \ u_t=ku_{xx}, \ 0<x<L, \ t>0$
    $\displaystyle \displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u_ x(L,t)=0\end{cases}$
    $\displaystyle \text{I.C.}: \ u(x,0)=f(x) \ \ 0<x<L$

    $\displaystyle u(x,t)=\varphi(x)T(t)$

    $\displaystyle \displaystyle\varphi(x)T'(t)=k\varphi''(x)T(t)\Rig htarrow\frac{\varphi''}{\varphi}=\frac{T'}{Tk}=-\lambda$

    $\displaystyle \varphi''+\lambda\varphi=0\Rightarrow \varphi=A\cos(x\sqrt{\lambda})+B\sin(x\sqrt{\lambd a})$

    $\displaystyle \displaystyle T=Ce^{-\lambda kt}$

    $\displaystyle \varphi_1(0)=1 \ \varphi_2(0)=0$
    $\displaystyle \varphi_1'(0)=0 \ \varphi_2'(0)=1$
    Where did $\displaystyle \varphi_1$ and $\displaystyle \varphi_2$ suddenly come from?
    Are they $\displaystyle e^{\lambda kt}(Acos(x\sqrt{\lambda}))$ and $\displaystyle e^{\lambda kt}(B sin(x\sqrt{\lambda}))$?

    If so, you cannot, generally, do that. But here it does not hurt.

    $\displaystyle \varphi_1: \ A=1 \ \varphi_2: \ A=0$
    $\displaystyle \displaystyle\varphi_1': \ B=0 \ \varphi_2': \ B=\frac{1}{\sqrt{\lambda}}$

    $\displaystyle \displaystyle\varphi=A_2\cos(x\sqrt{\lambda})+B_2\ frac{\sin(x\sqrt{\lambda})}{\sqrt{\lambda}}$
    All you really need is that $\displaystyle \phi(0, t)= e^{-\lambda kt}(Acos(0)+ Bsin(0))= Ae^{-k\lambda kt}= 0$. Since the exponential is never 0, A= 0.

    Then $\displaystyle \phi(L, t)= e^{-\lambda kt}B cos(L\sqrt{\lambda})= 0$. Again, the exponential is not 0 so we must have either B= 0, which would mean the solution was identically equal to 0, and so not an eigenfunction, or cos(L\sqrt{\lambda})= 0.

    $\displaystyle \displaystyle\varphi(0): \ A_2=0$

    $\displaystyle \displaystyle\varphi': \ B_2\cos(x\sqrt{\lambda})=0$

    The eigenvalues must satisfy:

    $\displaystyle \displaystyle\cos(x\sqrt{\lambda})=0$
    You mean $\displaystyle cos(L\sqrt{\lambda})= 0$

    $\displaystyle \displaystyle\lambda_n=\pi^2\left(\frac{1+2n}{2L}\ right)^2, \ n\in\mathbb{Z}$
    The eigenfunction is:

    $\displaystyle \displaystyle\varphi_n(x)=\frac{2L}{\pi(1+2n)}\sin \left(\frac{\pi x(1+2n)}{2L}\right), \ n\in\mathbb{Z}^+$

    Is this much correct?

    Thanks.
    Yes, that is correct and your second post is correct.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Where did $\displaystyle \varphi_1$ and $\displaystyle \varphi_2$ suddenly come from?
    Are they $\displaystyle e^{\lambda kt}(Acos(x\sqrt{\lambda}))$ and $\displaystyle e^{\lambda kt}(B sin(x\sqrt{\ambda}))$?

    If so, you cannot, generally, do that. But here it does not hurt.
    My book solves all PDEs in the same manner. I don't know why it is done that way but it works in all cases.

    Here is what the book says about $\displaystyle \varphi$ 1 and 2.

    The theorem in the theory of ODE which explicitly states the proposition above is: if the coefficients $\displaystyle c_0(x), \ c_1(x), \ c_2(x)$ are continuous on the interval [a,b] and $\displaystyle c_0(x)$ is different from zero throughout the interval, then for any fixed $\displaystyle x_0$ in the interval [a,b] and for any constants $\displaystyle \alpha, \ \beta$. The DE has one and only one solution $\displaystyle \varphi(x)=\varphi(x,\lambda)$ satisfying at $\displaystyle x_0$ the initial conditions $\displaystyle \varphi(x_0)=\alpha, \ \varphi'(x_0)=\beta$.
    Moreover, $\displaystyle \varphi(x,\lambda)$ and $\displaystyle \varphi'(x,\lambda)$ are continuous and continuously differentiable functions of both x for $\displaystyle a\leq x\leq b$ and all lambda.
    It is convenient to single out the two special solutions $\displaystyle \varphi_1=\varphi_1(x,\lambda), \ \varphi_2=\varphi_2(x,\lambda)$ of which satisfy the DE
    $\displaystyle \varphi_1(0)=1, \ \varphi_2(0)=0$
    $\displaystyle \varphi'_1 (0)=0, \ \varphi'_2(0)=1$.

    We call 1 and 2 the basic solutions at x nought.
    Last edited by dwsmith; Mar 19th 2011 at 03:48 PM.
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  5. #5
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    $\displaystyle \displaystyle \int_0^Lf(x)\sin\left(\frac{\pi x(1+2m)}{L}\right) \ dx=\int_0^Lb_m\sin^2\left(\frac{\pi x(1+2m)}{L}\right) \ dx$

    $\displaystyle \displaystyle\int_0^Lf(x)\sin\left(\frac{\pi x(1+2m)}{L}\right) \ dx=b_m\frac{L}{2}\Rightarrow b_m=\frac{2}{L}\int_0^Lf(x)\sin\left(\frac{\pi x(1+2m)}{L}\right) \ dx$
    Last edited by dwsmith; Mar 20th 2011 at 11:29 AM.
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  6. #6
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    For the case $\displaystyle f(x)=1$

    $\displaystyle \displaystyle b_m=\frac{2}{L}\int_0^L\sin\left(\frac{\pi x(1+2m)}{L}\right) \ dx=\frac{2}{L}\left[\frac{-2L}{\pi(1+2m)}\cos\left(\frac{\pi x(1+2m)}{2L}\right)\right]_0^L$

    $\displaystyle \displaystyle b_m=\frac{4}{\pi(1+2m)}, \ \ m=n$

    $\displaystyle \displaystyle u(x,t)=\frac{8}{\pi^2}\sum_{n=0}^{\infty}\frac{L}{ (1+2n)^2}\exp\left[\frac{-\pi^2}{4}\left(\frac{1+2n}{L}\right)^2kt\right]\sin\left(\frac{\pi x(1+2n)}{2L}\right)$

    If everything above is correct, can I do this?

    $\displaystyle \displaystyle\sum_0^{\infty}\frac{1}{(1+2m)^2}=\fr ac{\pi^2}{8}$

    $\displaystyle \displaystyle u(x,t)=\frac{8}{\pi^2}\sum_{n=0}^{\infty}\frac{1}{ (1+2n)^2}\cdot L\sum_{n=0}^{\infty}\exp\left[\frac{-\pi^2}{4}\left(\frac{1+2n}{L}\right)^2kt\right]\sin\left(\frac{\pi x(1+2n)}{2L}\right)$

    $\displaystyle \displaystyle\Rightarrow u(x,t)=L\sum_{n=0}^{\infty}\exp\left[\frac{-\pi^2}{4}\left(\frac{1+2n}{L}\right)^2kt\right]\sin\left(\frac{\pi x(1+2n)}{2L}\right)$
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  7. #7
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    Ok, I pretty sure the above answer is wrong.

    I now have:

    $\displaystyle \displaystyle u(x,t)=\frac{4}{\pi}\sum_{n=0}^{\infty}\frac{1}{1+ 2n}\exp\left[\frac{-\pi^2}{4}\left(\frac{1+2n}{L}\right)^2kt\right]\sin\left(\frac{\pi x(1+2n)}{2L}\right)$

    How do I show this is true for all x with the IC:

    $\displaystyle \displaystyle u(x,0)=\frac{4}{\pi}\sum_{n=0}^{\infty}\frac{1}{1+ 2n}\sin\left(\frac{\pi x(1+2n)}{2L}\right)=1$
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