Questions tagged [green-function]
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101 questions
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Expression for the (1+1)-dimensional retarded Dirac propagator in position space
Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone?
In particular, is it ...
5
votes
1
answer
1k
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Green's Function for 3D Relativistic Heat Equation
On the Wikipedia page here , it states that the Green's function for 3D relativistic heat conduction (with $c=1$)
$$[\partial_t^2 + 2\gamma\partial_t -\Delta_{3D}] u(t,x) = \delta(t,x) = \delta(t)\...
- 131
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vote
0
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246
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Diffusion equation solution using Laplace transform [closed]
Consider the operator
$$
L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}
$$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
- 111
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178
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Green's Function for Fractional Laplacian on the Union of Two Balls
I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:
$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
- 93
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3
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views
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...
- 1,353
2
votes
0
answers
314
views
Fundamental solution of parabolic PDE with variable coefficients
Let us consider the parabolic operator
$$
\mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x)
$$
over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...
- 21
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0
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89
views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
- 53
3
votes
1
answer
297
views
Reconstructing the Green's function of an initial-value problem of partial differential equation
Consider a partial differential equation that is of the following form:
\begin{equation}
(-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t)
\end{equation}
where $g(x)$ is a real function. Suppose that $f(...
- 197
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128
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Paving property
In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...
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0
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110
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Parabolic (heat) PDE Green's function spatial asymptote at infinity
Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...
- 2,363
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442
views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
- 61
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138
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Green's third identity potential massive object
Consider a massive object occupying a volume $U$ with boundary $\partial U$. Let the gravitational potential inside be $V_{in}$ and outside $V_{out}$
Normally the gravitational field of a massive ...
- 583
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Limit of Green's function as metric changes (S^2 -> R^2)
The Laplace-Beltrami operator is invertible on the space of 1-forms on $S^2$ (since $S^2$ has zero first betti number). Therefore it has an inverse, the Green's function. Now let the radius $r$ of $S^...
- 73
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When are Green's functions causal convolution kernels
Let $L$ be a linear differential operarator acting on distributions over $\mathbb{R}$ and $G(t, s)$ be a Green's function, i.e., a solution to $LG(t, s) =\delta(t-s)$.
$G$ is said to be causal if $G(...
- 547
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The study of dynamics of a polynomial vector field via Green's function methods
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 ...
- 330
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Green's function for fourth order equation
I know the D'Alembert operator ${\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{3D}}$ has a well-known Green's function $\frac{\delta(t-\frac{r}{c})}{4 \pi r}$. This is very useful for studying 3D ...
- 51
2
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1
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779
views
hyperbolic "Green function" on a product of upper half-planes
Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...
- 7,811
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1
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How to determine the spectrum from the diagonal Green's function
Let $L: L^2(\mathbb{R}) \supseteq Dom(L) \rightarrow L^2(\mathbb{R})$ be a densely defined closed operator. Assume that the resolvent admits an integral kernel (Greens function) $G$, i.e. for $z\in \...
- 1,127
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vote
0
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133
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singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
- 4,189
10
votes
1
answer
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views
Variation of the Green function with respect to the metric
Consider a (closed) Riemann surface and let $G(x,y)$ be the Green function of the Laplace-Beltrami operator. We can informally identify $G$ with the two-point correlation function for the Gaussian ...
- 731
3
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1
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Singularity of the heat kernel
The heat kernel in one dimension for the real line is given by the usual gaussian density function:
$$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$
In particular, by differentiating ...
- 630
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votes
1
answer
390
views
Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?
I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...
- 125
4
votes
0
answers
555
views
Convexity of the electrostatic energy on a Riemann surface
Let $M$ be a compact Riemann surface.
Let $\Lambda$ be a differentiable real $2$-form of integral one.
Let $G$ be the Green function associated to $\Lambda$, i.e.
$G: M \times M \to \mathbb R \cup \{...
- 41
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vote
0
answers
430
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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)....
- 185
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vote
0
answers
101
views
Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions
I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...
- 151
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1
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281
views
Modified mean value property
Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant.
I met this modified mean value property of a solution $u$ of $Lu=0$ as
$$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4\...
- 1,360
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0
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165
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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 ...
- 151
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0
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1k
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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 ...
- 111
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0
answers
182
views
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 ...
2
votes
1
answer
146
views
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,...
user50806
3
votes
0
answers
521
views
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, ...
- 151
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1
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887
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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 ...
- 51
12
votes
1
answer
2k
views
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 $d\geq 2$ ...
12
votes
5
answers
1k
views
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?
- 3,920
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vote
0
answers
386
views
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 ...
- 41
3
votes
1
answer
535
views
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 ...
- 31
1
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1
answer
522
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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 ...
- 11
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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 ...
- 1,729
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votes
1
answer
394
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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 ...
- 327
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0
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239
views
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 ...
- 1,729
3
votes
1
answer
559
views
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 ...
- 31
1
vote
1
answer
2k
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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 ...
- 3,644
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1
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Existence of Green's function and the Dirichlet problem
Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \...
- 31
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0
answers
2k
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Nonlinear PDE and Green functions
This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
\partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real ...
- 1,729
1
vote
2
answers
917
views
Estimates for Green's function
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 < \...
- 13
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votes
2
answers
877
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 ...
- 59
3
votes
3
answers
955
views
Hyperbolic Riemann Surface
Let $X$ be a compact Riemann surface and $x\in X$.
Is $X - \overline{D(x,r_x)}$ hyperbolic?
- 59
0
votes
0
answers
899
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 $g(x)$,
$f' + af + bg = S_1(f,g) + S_2(f,g)$
...
- 3,631
2
votes
1
answer
259
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_{...
- 41
3
votes
1
answer
631
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Math background needed for Stakgold's Boundary Value Problems & Green's Functions Book
I saw a reference in Jackson's "Classical Electrodynamics" book for Stakgold's book on "Boundary Value Problems and Green's Functions" as a reference for Green's functions. The text is sort of clear, ...
