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### Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....
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### Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...
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### Green's function for fractional Laplacian

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 ...
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### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as \begin{align} \mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega \end{align} where the ...
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### Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...
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### Existence of Green's functions for PDEs

Here is what I think I know: Given a symmetric linear differential operator $\mathcal{L}$ that is positive definite on a function space(/space of distributions) $\mathcal{H}$, we can find its inverse, ...
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### Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk $\Omega$ with subject to Robin boundary conditions. The formulation is as ...
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### Green's function of the Ornstein-Uhlenbeck operator

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 $d\geq 2$ ...
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### Reference request for a treatment of Schwinger–Dyson equations

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?
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### What's Known About the Green's Function to the 1D Diffusion Equation with Position-dependent Diffusion Coefficient?

Consider a one-dimensional diffusion equation $$C(x) \partial_t \Phi(t,x) = \partial_x^2 \Phi(t,x),$$ on the interval $[0,1]$. The function $C(x)$ has a pole of order 1 at $x=0$ and a pole of finite ...
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### What can we say about the left inverse of the Green's function?

Let $\mathbb{D}$ be an self-adjoint elliptic operator of a compact manifold and $G(x,y)$ the Green's function of $\mathbb{D}$. By definition $G(x,y)$ is the right inverse of $\mathbb{D}$ in the sense ...
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### Is Green's function of an elliptic operator always symmetric?

Let $D$ be an elliptic operator of a compact Riemannian manifold and $G(x_0,x_1)$ the Green's function of $D$. Is $G$ always symmetric in variables $x_0$ and $x_1$, i.e. $G(x_0,x_1)=G(x_1,x_0)$? If ...
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### Green function and translational symmetry

I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ...
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### Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
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### Solvable PDEs and their Green's functions

I have a class of PDEs of the form $$-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0$$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
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### Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
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### direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta$ function. I can show that the integral is constant, but it ...
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Let $n$ - dimension $\geq 3$. Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \... 2answers 384 views ### Green's function - Hyperbolic Riemann surface A Riemann surface is said to be: -Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function. -Poincaré hyperbolic if it is covered by the unid disk. Are this ... 3answers 470 views ### Hyperbolic Riemann Surface Let$X$be a compact Riemann surface and$x\in X$. Is$X - \overline{D(x,r_x)}$hyperbolic? 0answers 417 views ### Green's function of coupled ODEs For functions$a(x)$and$b(x)$and "sources"$S_1(f,g)$,$S_2(f,g)$and$S_3(f,g)$lets say one has the differential equations for functions$f(x)$and$g(x)$,$f' + af + bg = S_1(f,g) + S_2(f,g)$... 1answer 173 views ### dilation operator green function how can i solve$ -ixDG(x,s)-iG(x,s)/2= \delta ( \frac{x}{s}-1) $i do not know , since it is a first odrder differntial operator, the formal solution i've found would be$ G(x,s)= \sum_{n} \frac{u_{...
Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some constant \$\rho \in [-...