I am attempting to solve the diffusion equation
$$\left( \partial / \partial t – D (\partial/\partial x)^2 \right) p = J$$
where $p$ is the probability density, $J$ is a source, and $D$ is the diffusion coefficient.
In particular, I'd like to solve this over the finite interval $[0, 1]$ with boundary conditions such that $(\partial p/\partial x)(x\in\{0,1\})=0$, i.e. zero space derivative at the boundaries.
Furthermore, we consider a point source
$$J(x, t) = \delta(x – x_0)\delta(t – t_0) \, .$$
As the equation is linear with constant coefficients, it seems we should solve it via Fourier transformation (also known as "separation of variables").
To that end, write
$$p(x, t) = \int \frac{d\omega}{2\pi} e^{i \omega t} \sum_{k=-\infty}^\infty e^{i 2\pi k x} \tilde{p}_k(\omega)$$
where
$$\tilde{p}_k(\omega) = \int_{-\infty}^\infty p_k(t) e^{-i \omega t} \, dt
$$
is the time-domain Fourier transform of $p_k(t)$, which are the coefficients of the space-domain Fourier series
$$p(x, t) = \sum_{k=-\infty}^\infty e^{i 2\pi k x} p_k(t) \, .$$
Also note that
$$J(x, t) = \int \frac{d\omega}{2\pi} \sum_{k=-\infty}^\infty e^{i \omega (t – t_0)} e^{i2\pi k (x – x_0)} \, .$$
Here's where I get fuzzy.
I believe that the following are true:
- I do not need to do anything special to ensure that the final solution is real. The source $J$ is real and the diffusion equation is such that it will not create any imaginary content from that real source.
- I do need to do something to enforce the boundary conditions. In particular, I think it's convenient to rewrite the Fourier series in terms of $\sin$ and $\cos$.
So, let's rewrite the sum over $k$ like this:
$$
\tilde{p}_0(\omega) + \sum_{k=1}^\infty \left( \tilde{p}_k(\omega) + \tilde{p}_{-k}(\omega) \right) \cos(2 \pi k x) + \left(\tilde{p}_k(\omega) – \tilde{p}_{-k}(\omega) \right) \sin(2\pi k x)
\, .
$$
I think that the boundary conditions are satisfied if the $\sin$ terms vanish, which I think requires $\tilde{p}_k(\omega) = \tilde{p}_{-k}(\omega)$.
Question: Have I made a mistake yet?
Now let's stuff our Fourier representations of $p$ and $J$ into the diffusion equation.
Let $A_k(\omega) \equiv \tilde{p}_k(\omega) + \tilde{p}_{-k}(\omega)$ to clean up the notation.
\begin{align}
\left( \partial / \partial t – D (\partial / \partial x)^2 \right) \int \frac{d\omega}{2\pi} e^{i \omega t} \left[
\tilde{p}_0(\omega) + \sum_{k=1}^\infty A_k(\omega) \cos(2\pi k x)
\right]
&= \underbrace{\int \frac{d \omega}{2\pi} \sum_{k=-\infty}^\infty e^{i \omega (t – t_0)} e^{i 2\pi k (x – x_0)}}_{J(x, t)} \\
\text{let $A_0 = p_0$} \quad \left( \partial / \partial t – D (\partial / \partial x)^2 \right) \int \frac{d\omega}{2\pi} \sum_{k=0}^\infty A_k(\omega) e^{i \omega t} \cos(2\pi k x)
&= \\
\int \frac{d\omega}{2\pi} \sum_{k=0}^\infty (i\omega + D(2\pi k)^2 )A_k(\omega) e^{i \omega t} \cos(2\pi k x)
&= \\
\end{align}
Now normally at this point we invoke an orthonormality statement (or in the language of separation of variables, we multiply each side by $\cos(2\pi k x)$ and integrate) to match coefficients term-by-term on the left and right, giving us an algebraic equation in the Fourier domain.
The problem, however, is that the sums over $k$ on the left and right run over different values.
In particular, the sum on the right (the expansion of $J$) includes the $\sin$ terms that I claimed have to be zero to satisfy the boundary conditions.
Question: What is going on here?
It seems that, maybe, a delta function source simply violates the boundary conditions, i.e. because if we write the delta function as a sequence of e.g. increasingly narrow Gaussian functions, each element in that sequence does not satisfy the boundary conditions (unless $x_0 = 1/2$).
Am I on the right track, or have I made a mistake already?
This is a crosspost. The original post is here on the math site.
I'm reposting here because while I got an answer on the Math site, the author of that answer hasn't responded to what I think is an important comment on that answer, and no further answers have appeared.
Best Answer
This is a case where using hardcore Fourier series in terms of sines and cosines would be less confusing than starting with a more general Fourier integral.
Another approach would be to use the separation of variables to find the eigenmodes of this equation (there is relation between the mode number $k$ and the corresponding frequency). The delta-function source than can be dealt with using the resolution of identity in terms of the eigenmodes.
Finally, let me note that this problem is a very close relative of the well-known problem about a particle in an infinite quantum well, with only difference that the time-dependent Schrödinger equation is replace by the diffusion equation (i.e., there is no imaginary unit $\imath$). And you probably will find a lot of discussions about that one in this community.