Questions tagged [green-function]

In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...

The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...

I'm looking at the linear PDE in 3+1 dimensions, $$ \left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1} $$ Where $\xi$ is generally a ...

Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions. In this case, I ...

The Green function $G(x,y) =G(x-y)$ of the discrete Klein-Gordon operator $-\Delta+p$ on $\mathbb{Z}^{d}$ is given by: \begin{eqnarray} G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{...

Consider the fractional differential equation \begin{align} D_{|x|}^\alpha u(x) +bu(x)=f(x) \end{align} with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees ...