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Method of separation of variables for wave equation

  1. Apr 25, 2013 #1
    1. The problem statement, all variables and given/known data
    $$u_{tt} = a^2u_{xx} , 0<x< l , t>0 , $$a is constant
    $$ u(x,0)=sinx , u_{t} (x,0) = cosx , 0<x< l , t>0 $$
    $$ u(0,t)=2t , u(l,t)=t^2 , t>0 $$

    2. Relevant equations



    3. The attempt at a solution
    I can solve the eigenvalue problem of X(x), and then solve for T(t), but I dont know how to solve the initial value problem for $$u(x,0) =sinx , and u_{t} (x,0) = cosx $$
    with I can only compute the fourier expansion of $$ sinx $$ and $$ cosx $$ with $$ λ_{n} = \frac { (n \pi)^2} {l^2} $$ , but the ans looks like ugly, and compare term is fail.
    by the way I'm sorry for my poor english.
     
  2. jcsd
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