# Questions tagged [green-function]

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### References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
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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 $u(x,0)=g(x), \forall x\in \Bbb R$...
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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 $f$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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{...
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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 ...
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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 ...
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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 ...
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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\... 0answers 133 views ### 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 ... 0answers 658 views ### 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 ... 0answers 125 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 ... 1answer 119 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,... 0answers 307 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, ... 1answer 523 views ### 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 ... 2answers 1k 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. $$Is there a known closed form expression of the Green's function of L on \mathbb R^d (for d\geq 2 ... 5answers 889 views ### Reference request for a treatment of Schwinger–Dyson equations Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog? 0answers 311 views ### What's Known About the Green's Function to the 1D Diffusion Equation with Position-dependent Diffusion Coefficient? Consider a one-dimensional diffusion equation$$ C(x) \partial_t \Phi(t,x) = \partial_x^2 \Phi(t,x), $$on the interval [0,1]. The function C(x) has a pole of order 1 at x=0 and a pole of finite ... 1answer 232 views ### What can we say about the left inverse of the Green's function? Let \mathbb{D} be an self-adjoint elliptic operator of a compact manifold and G(x,y) the Green's function of \mathbb{D}. By definition G(x,y) is the right inverse of \mathbb{D} in the sense ... 1answer 332 views ### Is Green's function of an elliptic operator always symmetric? Let D be an elliptic operator of a compact Riemannian manifold and G(x_0,x_1) the Green's function of D. Is G always symmetric in variables x_0 and x_1, i.e. G(x_0,x_1)=G(x_1,x_0)? If ... 0answers 200 views ### Green function and translational symmetry I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ... 1answer 318 views ### Green's function for *GJMS* operator Consider a Riemannian manifold (M^n, g) of dimension n with a metric g. We assume M to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ... 0answers 217 views ### Solvable PDEs and their Green's functions I have a class of PDEs of the form$$ -\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 ...
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 ...
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 ...