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PDE:

$$ u_t (x,t) - u_{xx} (x,t) = 0, \quad u(0,t)=0, \quad u_x(2,t)=0, \quad u(x,0)=\delta(x-x_0) $$

where $0<x_0<2$ is a given constant.

I worked out my solution by using separation of variables $$u(x,t)=X(x)T(t) \implies \frac{X''(x)}{X(x)}=\frac{T'(t)}{T(t)}=\lambda$$

For $\lambda>0$ and $\lambda=0$ we get trivial solutions, but for $\lambda=-k^2<0$ we get the solutions

$$ X(x)=c_1 sink_nx, \quad k_n=\frac{\pi}{2}(\frac{1}{2}+n), \quad n=0,1,... $$ $$ T(t)=c_2e^{-k_n^2t} $$

Superposition gives the general solution to the PDE as

$$ \sum_{n=0}^\infty c_n sin(k_nx)e^{-k_n^2t}, \quad k_n=\frac{\pi}{2}(\frac{1}{2}+n)$$

Now for my question, I don't understand how to handle the initial condition to calculate the fourier coefficients $c_n$. I have the solution for it though, and what they have done is this (I understand this step);

$$ u(x,0)=\sum_{n=0}^\infty c_n sin(k_n x) = \delta(x-x_0) \quad (1)$$

This is the step I don't understand;

$$ (1) \implies c_n = \frac{\int_0^2 \delta(x-x_0)sin(k_nx)dx}{\int_0^2 sin^2(k_nx)dx}=sin(k_n x_0) $$

Any explanation as to why and how they got that expression for $c_n$, and also why it equals $sin(k_n x_0)$ (it's like they're discounting the denominator and only looking at the numerator?) would be greatly appreciated.

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Starting from your last equation

$$ \sum_n c_n \sin (k_n x) = \delta(x - x_0) $$

Multiply by $\sin (k_m x)$ on both sides and integrate

$$ \sum_n c_n \color{blue}{\int \sin (k_m x) \sin (k_n x)~{\rm d}x} = \int \delta(x - x_0)\sin (k_m x)~{\rm d}x \tag{1} $$

The blue term is zero unless $n = m$, which means

$$ \sum_n c_n \color{blue}{\delta_{nm}\int \sin (k_n x) \sin (k_n x)~{\rm d}x} = c_m\color{blue}{\int \sin^2 (k_m x) ~{\rm d}x} = \int \delta(x - x_0)\sin (k_m x)~{\rm d}x $$

and from here

$$ c_m = \frac{\int \delta(x - x_0)\sin (k_m x)~{\rm d}x}{\int \sin^2 (k_m x) ~{\rm d}x} $$

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  • $\begingroup$ Thanks for the help! I will give you the green tick, but if you can share any intuition as to why this procedure is done, that would be great aswell. $\endgroup$ – armara May 19 at 9:38
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    $\begingroup$ @armara Intuitively speaking, imagine you have a vector ${\bf x}$ which you can represent in an orthonormal basis ${\bf x} = \sum_k a_k {\bf u}_k$, if you want to know the components $a_k$, you just need to project the vector on the elements of the basis, ${\bf x} \cdot {\bf u}_k = a_k$. In your case, the elements of the basis are the functions $\sin(k_n x)$, the inner product $\cdot$ is an integral, and the vector ${\bf x}$ is the distribution $\delta(x - x_0)$. Hope this makes sense. The big difference is that the basis is orthogonal, but not orthonormal, so you end up with the denominator $\endgroup$ – caverac May 19 at 10:45

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