Results 1 to 4 of 4

Math Help - Can't get PDE solution in textbook

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    56

    Can't get PDE solution in textbook

    Hey guys having trouble getting the same answer in the text book.

    Question: Consider solving wave equation (c=1) for string length Pi with fixed endpoints u(0,t)=0 and u(pi,t)=0 and initial velocity = 0 and displacement = 0.02sinx.

    So after doing the usual steps i arrived at u_n(x,t) = \sum_{n = 1}^\infty (Ancos(nt) + Bnsin(nt))*sin(nx)

    Using the initial conditions and fourier coefficient formula's i have Bn=0

    and An=\frac{2}{\pi  } \int 0.02sin(x)*sin(nx) dx

    Can i treat these both as sin(x)? and use the trig indentity 1/2 -1/2cos(2x) to solve?

    because the answer is u(x,t)=0.02costsinx and if i do that it will work.

    but i just dont understand where the 'n' went in my solution :

    u_n(x,t) = \sum_{n = 1}^\infty (A_ncos(nt) + B_nsin(nt))*sin(nx)

    seeing that it should be an infinite series and the answer given is not!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,326
    Thanks
    11
    Realize that you want to calculate

    \int_0^{\pi} \sin x \sin nx\,dx.

    There will two answers depending of the value of n.

    n = 1\;\; \int_0^{\pi} \sin^2 x \,dx = \dfrac{\pi}{2} (this is where your identity comes in)

    n \ne 1\;\; \int_0^{\pi} \sin x \sin nx\,dx = \dfrac{-\sin n \pi}{n^2-1} = 0
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,997
    Thanks
    1133
    Quote Originally Posted by olski1 View Post
    Hey guys having trouble getting the same answer in the text book.

    Question: Consider solving wave equation (c=1) for string length Pi with fixed endpoints u(0,t)=0 and u(pi,t)=0 and initial velocity = 0 and displacement = 0.02sinx.

    So after doing the usual steps i arrived at u_n(x,t) = \sum_{n = 1}^\infty (Ancos(nt) + Bnsin(nt))*sin(nx)

    Using the initial conditions and fourier coefficient formula's i have Bn=0

    and An=\frac{2}{\pi  } \int 0.02sin(x)*sin(nx) dx
    A standard technique for integrating such a thing is to use the trig identity
    sin(a)sin(b)= (1/2)(cos(a-b)- cos(a+b) so that sin(x)sin(nx)= (1/2)(cos((1-n)x)- cos((1+n)x)

    But, in fact, the functions "sin(nx)" and "sin(mx)" are "orthogonal" if m\ne n then \int_{-\pi}^\pi sin(x)sin(nx)dx= 0 as well as sin(nx) being orthogonal to cos(mx) for any m.

    [/quote]Can i treat these both as sin(x)? and use the trig indentity 1/2 -1/2cos(2x) to solve?

    because the answer is u(x,t)=0.02costsinx and if i do that it will work.

    but i just dont understand where the 'n' went in my solution :

    u_n(x,t) = \sum_{n = 1}^\infty (A_ncos(nt) + B_nsin(nt))*sin(nx)

    seeing that it should be an infinite series and the answer given is not![/QUOTE]
    \int_{-\pi}^\pi sin(x)sin(nx) dx= 0
    as long as n\ne 1 while for n= 1
    \int_{-\pi}^\pi sin^2(x)= \pi
    You have A_1= 0.002 while A_n= 0 for n> 1 and B_n= 0 for all n.

    Since
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    With what the experts have said in mind, if you can't recall the product-to-sum identities, let

    I = \int_{-\pi}^{\pi}\sin{x}\sin{nx}\;{dx}, ~~ J = \int_{-\pi}^{\pi}\cos{x}\cos{nx}\;{dx}

    Then calculate J+I and J-I. Of course, you will need to know the cosine double angle formulas.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: February 8th 2010, 05:58 PM
  2. Textbook question.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 11th 2009, 07:23 AM
  3. Textbook Suggestion
    Posted in the Geometry Forum
    Replies: 8
    Last Post: July 14th 2009, 02:54 PM
  4. Is this a textbook error or...
    Posted in the Algebra Forum
    Replies: 3
    Last Post: June 1st 2007, 10:20 AM
  5. PDE textbook things...
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: September 25th 2006, 08:38 PM

Search Tags


/mathhelpforum @mathhelpforum