Page 2 of 2 FirstFirst 12
Results 16 to 25 of 25

Math Help - Eigenvalue problem

  1. #16
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Once we obtain \exp(-\lambda kt), where does the n come from in the exponential for the summation?
    Follow Math Help Forum on Facebook and Google+

  2. #17
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    This is what I have.

    \displaystyle u(x,t)=\sum_{n=0}^{\infty}a_n\exp(-\lambda_nkt)\cos\left(\frac{(2n+1)\pi x}{2L}\right), \ \ \ \ \ \lambda_n=\left[\frac{(2n+1)\pi }{2L}\right]^2
    Follow Math Help Forum on Facebook and Google+

  3. #18
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by dwsmith View Post
    This is what I have.

    \displaystyle u(x,t)=\sum_{n=0}^{\infty}a_n\exp(-\lambda_nkt)\cos\left(\frac{(2n+1)\pi x}{2L}\right), \ \ \ \ \ \lambda_n=\left[\frac{(2n+1)\pi }{2L}\right]^2
    Right Now you just need to solve for the a_n's.
    Follow Math Help Forum on Facebook and Google+

  4. #19
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    I understand the use of orthogonality because everything goes to zero accept for when m = n, but why is it used here?
    Follow Math Help Forum on Facebook and Google+

  5. #20
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by dwsmith View Post
    I understand the use of orthogonality because everything goes to zero accept for when m = n, but why is it used here?
    \displaystyle \varphi_m(x)u(x,0)=\varphi_{m}(x)(L-x)=\sum_{n=0}^{\infty}a_n\varphi_m(x)\varphi_n(x)

    So as you said when m=n

    \displaystyle \int_{0}^{L}\varphi_{n}(x)(L-x)dx=a_n\int_{0}^{L}\varphi_n(x)\varphi_n(x)dx

    Solving for what we want gives

    \displaystyle a_n=\frac{\int_{0}^{L}\varphi_{n}(x)(L-x)dx}{\int_{0}^{L}[\varphi_n(x)]^2dx}
    Follow Math Help Forum on Facebook and Google+

  6. #21
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by TheEmptySet View Post
    \displaystyle \varphi_m(x)u(x,0)=\varphi_{m}(x)(L-x)=\sum_{n=0}^{\infty}a_n\varphi_m(x)\varphi_n(x)

    So as you said when m=n

    \displaystyle \int_{0}^{L}\varphi_{n}(x)(L-x)dx=a_n\int_{0}^{L}\varphi_n(x)\varphi_n(x)dx

    Solving for what we want gives

    \displaystyle a_n=\frac{\int_{0}^{L}\varphi_{n}(x)(L-x)dx}{\int_{0}^{L}[\varphi_n(x)]^2dx}
    a_n looks pretty brutal but here is what I obtained:

    \displaystyle a_n=(-1)^n\frac{4L}{(2m+1)\pi}+\left[(-1)^n-1\right]\left(\frac{8L(x-1)}{\left[(2m+1)\pi\right]^2}\right)

    Correct?
    Follow Math Help Forum on Facebook and Google+

  7. #22
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    \displaystyle \int_{0}^{L} \cos\left( \frac{(2n+1)\pi x}{2L}\right)(L-x)dx=\frac{4L^2}{\pi^2(2n+1)^2}
    Follow Math Help Forum on Facebook and Google+

  8. #23
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by TheEmptySet View Post
    \displaystyle \int_{0}^{L} \cos\left( \frac{(2n+1)\pi x}{2L}\right)(L-x)dx=\frac{4L^2}{\pi^2(2n+1)^2}
    \displaystyle \int_0^L L\cos\left(\frac{(2m+1)\pi x}{2L}\right)dx-\int_0^Lx\cos\left(\frac{(2m+1)\pi x}{2L}\right)dx

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

    Is this much correct?
    Follow Math Help Forum on Facebook and Google+

  9. #24
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    \displaystyle u(x,t)=\frac{8L}{(2n+1)^2\pi^2}\sum_{n=0}^{\infty}  \exp(-\lambda_n kt)\cos\left(\frac{(2n+1)\pi x}{2L}\right), \ \ \ \lambda_n=\left[\frac{(2n+1)\pi}{2L}\right]^2

    Correct?
    Follow Math Help Forum on Facebook and Google+

  10. #25
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by dwsmith View Post
    \displaystyle u(x,t)=\frac{8L}{(2n+1)^2\pi^2}\sum_{n=0}^{\infty}  \exp(-\lambda_n kt)\cos\left(\frac{(2n+1)\pi x}{2L}\right), \ \ \ \lambda_n=\left[\frac{(2n+1)\pi}{2L}\right]^2

    Correct?
    I like to use tabular integration when I am doing Fourier series. This works if you of the functions will differentiate to 0 eventally.

    \begin{array}{|c|c|}\hline f(x) & g(x) \\ \hline  (-1)\frac{d}{dx}f(x) & \int g(x) \\ \hline  \frac{d^2}{dx^2}f(x) & \int\int g(x) \\ \hline \vdots & \vdots \\  \hline  (-1)^n\frac{d^n}{dx^n}f(x) & \int_{\text{n times}} g(x) \\ \hline 0 & \int_{\text{n+1 times}} g(x)\\ \hline \end{array}

    So in your case we have

    \begin{array}{|c|c|}\hline L-x & \cos(\lambda_n x) \\  \hline(-1)[-1] & \frac{1}{\lambda_n}\sin(\lambda_n x) \\ \hline 0 & - \frac{1}{\lambda_n^2}\cos(\lambda_n x) \\ \hline \end{array}

    Now multiply diagonally down to get the antiderivative

    \displaystyle \frac{(L-x)}{\lambda_n }\sin(\lambda_n x)-\frac{1}{\lambda_n^2} \cos(\lambda_n x) \bigg|_{0}^{L}=-\frac{1}{\lambda_n^2}\left[ \cos(\lambda_n L)-1\right]=\frac{1}{\lambda_n^2}

    This is the same as yours
    Follow Math Help Forum on Facebook and Google+

Page 2 of 2 FirstFirst 12

Similar Math Help Forum Discussions

  1. Eigenvalue Problem
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: July 5th 2011, 01:35 AM
  2. eigenvalue problem
    Posted in the Differential Equations Forum
    Replies: 13
    Last Post: January 3rd 2011, 08:39 AM
  3. Eigenvalue problem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 4th 2010, 07:52 PM
  4. eigenvalue problem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 14th 2009, 10:10 AM
  5. eigenvalue problem
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 10th 2007, 06:29 AM

/mathhelpforum @mathhelpforum