Method of separation of variables for wave equation

[sigh1342]

Homework Statement

$$u_{tt} = a^2u_{xx} , 0<x< l , t>0 , $$a is constant
$$ u(x,0)=sinx , u_{t} (x,0) = cosx , 0<x< l , t>0 $$
$$ u(0,t)=2t , u(l,t)=t^2 , t>0 $$

Homework Equations

The Attempt at a Solution

I can solve the eigenvalue problem of X(x), and then solve for T(t), but I don't 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.

 

[mighty2000]

Hello there,

To solve the initial value problem for u(x,0) and u_t(x,0), you can use the method of separation of variables. This involves assuming a solution of the form u(x,t) = X(x)T(t), and then substituting this into the given equation. This will result in two separate ordinary differential equations, one for X(x) and one for T(t). You can then solve these equations individually to find the general solution for u(x,t).

For the boundary conditions, you can use the method of Fourier series to find the coefficients of the solution. This involves expanding the given boundary conditions into a Fourier series and then equating the coefficients to the general solution found above. This will give you a set of equations that you can solve to find the specific values of the coefficients.

Hope this helps! Let me know if you have any further questions.