In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function:
$$
a \, \frac{\partial^2 w}{\partial x^2}
+ b \, \frac{\partial^2 w}{\partial y^2}
+ \delta^2(x,y) = 0 \, ,
$$
subject to the boundary conditions $w(x = \pm 1, y) = w(x, y = \pm 1) = 0$.
Here $a, b \in \mathbb{R}_+$ and $\delta^2(x,y) = \delta(x)\delta(y)$ is the two-dimensional delta Dirac function.
While solutions for ODEs with delta Dirac functions can readily be obtained using the standard approach, I am not aware of any resolution recipe for PDEs with delta Dirac functions.
Any help or hint is highly desirable and appreciated.
Thank you
Best Answer
Use an ansatz of the form
$$ w(x,y) = \sum_{n,m=1}^\infty c_{n,m} \sin \left(n\pi \frac{1+x}{2}\right)\sin\left(m\pi \frac{1+y}{2}\right) $$
Decomposing the delta function into its Fourier series gives
$$ \delta(x,y) = \sum_{n,m=1}^\infty \sin \left(\frac{n\pi}{2}\right)\sin\left( \frac{m\pi}{2}\right)\sin \left(n\pi \frac{1+x}{2}\right)\sin\left(m\pi \frac{1+y}{2}\right) $$
Plugging the above expressions into the equation gives
$$ -\left[a\left(\frac{n\pi}{2}\right)^2 + b\left(\frac{m\pi}{2}\right)^2\right]c_{n,m} = -\sin \left(\frac{n\pi}{2}\right)\sin\left( \frac{m\pi}{2}\right) $$