Questions tagged [greens-function]
This tag is for questions about a Green's function which is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions.
641
questions
2
votes
0
answers
21
views
evaluate integral with the kernel of Green function (fundamental solution of Poisson equation) of infinite domain using conformal map
I must find the following definite integral
$$\phi(x,y) = \int_{0}^{\alpha} f(\theta)\frac{-1}{2\pi} \ln \sqrt{(x-\rho\cos\theta)^2+(y-\rho\sin\theta)^2} \rho d\theta$$
for a given $f(\theta)$ which ...
1
vote
1
answer
64
views
Find the Green's function of the differential operator for periodic boundaries.
On the space of functions $f(x)$ where $x \in (-\pi,\pi]$ satisfying $\int^{\pi}_{-\pi} f(x)dx = 0$ and periodic boundary conditions, find the Green's function for the operator $-\frac{d^2}{dx^2}$.
I ...
- 43
0
votes
0
answers
31
views
Green's Function for Parabolic PDE with Time-dependent Coefficients
I'm trying to work through Polyanin's solution in Handbook of Linear Partial Differential Equations for Engineers and Scientists for the following PDE:
$$
\frac{\partial C}{\partial t} = \frac{1}{\...
- 283
0
votes
1
answer
28
views
Dirac delta doublet function in simple harmonic oscillation. Conditions imposed?
I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative of dirac delta) at t=0.
$$f = \delta'(t)$$
I've already considered the case for a dirac ...
- 33
0
votes
1
answer
63
views
Solving a differential equation using Green's function where the Forcing term is a single square wave. Did I do this right?
I have the Differential Equation:
$$u_{xx} = \cases{1; \ -1\le x \le1\\ 0;-\infty<x<-1,1<x<\infty} \\ u(-1) = -1,u(1) =1$$The forcing term on the right hand side is a square wave from x = -...
- 187
3
votes
0
answers
83
views
Help with finding N-point functions with some properties
I'm a physicist and I'm working in a problem in general relativity where I end up with N-point functions that have to satisfy equations like this
\begin{equation}
\Big[\nabla_\mu \nabla^\mu +V(x_1)\...
- 187
0
votes
1
answer
22
views
Smoothness of heat kernel on Lipschitz and polygon (cornered) domain
I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general ...
- 81
1
vote
0
answers
59
views
How to show that the value of the heat kernel decreases as we move away from the heat source for a bounded domain?
The free-space heat kernel is given by
$K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}$, with $x,y \in \mathbb{R}^d$ and $t>0$.
This expression shows that the heat kernel decreases as the ...
- 49
2
votes
0
answers
36
views
A limit about a harmonic function
For $x = (x_1, x_2) \in \mathbb{R}^2$, let
$$
E(x) = \frac{1}{2\pi} \log(|x|),
$$
where $|x| = \sqrt{x_1^2 + x_2^2}$. In fact $E$ is the fundamental solution for the Laplace equation.
Let $\Gamma$ be ...
- 31
0
votes
0
answers
20
views
Solution to a dirichlet problem
Let there be a Green function $G(x,y,x_0,y_0)$ for $D={(x,y)\in{\Re}^2|x^2+2y^2<7}$ and let $(x_0,y_0)$ be any point within $D$. Calculate the integral:
$$u(x,y) = \oint_{\partial D} G_{\vec{n}}(x,...
0
votes
1
answer
61
views
Solving two coupled ODEs with delta function sources
I am interested in solving two coupled ODEs for two functions $f(r)$ and $g(r)$ of the following form:
$$r(1-g(r))+(r^2+1)\frac{f'(r)}{2f(r)}=E\delta(r-r_0)$$
$$rf(r)[2r(g(r)-1)g(r)+(r^2+1)g'(r)]=E\...
- 35
0
votes
0
answers
26
views
I need a Boundary Value Problem to test for the wikipedia table of Green's functions
I just want a specific example (preferably in 2-D) of a PDE (preferably steady-state) that I can test which can be solved using one of the Green's functions from HERE. I'm mostly asking as a sanity ...
- 187
2
votes
0
answers
59
views
How does this proof for the solution of Dirichlet's problem work?
Our professor proved a certain theorem in our PDE class and I have troubles understanding the proof. Let me start with the relevant definitions.
Problem: Suppose $\Omega \subset \mathbb{C}$ is a ...
- 1,568
0
votes
0
answers
10
views
Did I do this right for finding the solution to a BVP using Green's Function in 2-D? I wanted to work out a finite plane.
Problem: On the upper right quarter plane $D = {(x,y):0 < x < a}, 0 < y < b; \partial D = C = (x,y):(x=0,y),(x,y=0),(x=a,y),(x,y=b)$, we have the BVP:
$$\frac{\partial^2 u(x,y)}{\partial x^...
- 187
1
vote
0
answers
36
views
What is the admissible function in the convolution integral?
I know this is a dumb question, but I'm having some trouble with the notation.
In 1-D, the solution to a BVP with some linear differential operator $$Lu(x) = f(x)$$ is of the form: $$u(x)=\int_{-\...
- 187
0
votes
0
answers
29
views
Solution to a Boundary Value Problem using Green's Function leads to a diverging integral
I set up a basic Laplace's Equation to try solving using Green's Function, and I noticed that the infinite integral diverges. The problem is as follows:
$$u_{xx} + u_{yy} = 0; \ \ \ \ \ x>0, y>0 ...
- 187
0
votes
1
answer
33
views
Level set of green function
$\textbf{Background for problem statement}$:
Let $B \subset \mathbb{C}$ be a bounded domain, and $g_{B}(z,z_0) = g$ its Green's function with pole at $z_0 \in B$, so $g$ is harmonic in $B \setminus \{...
- 2,302
0
votes
0
answers
35
views
What should I use for $f(\zeta)$ to test for the solution to Laplace's Equation obtained using Green's Function?
I'm trying to compute the solution obtained via Green's Function to test it. It's for Laplace's equation for a semi-infinite slab defined for $y>0$. Here is a snippet from the document which I have ...
- 187
0
votes
0
answers
28
views
Green's Function in some problems (2-D) causes the occurrence of ln(0) at $G(x=x_0,y = y_0)$. How does one get around this?
From the document I found here (and many others which cover this exact same problem as well), we have the following where I added some edits to help follow along:
So I went ahead and tested the ...
- 187
0
votes
1
answer
58
views
Are these two integrands the same?
$$u(x) = \int_{-\infty}^{\infty}f(s)G(x,s)ds = \int_{-\infty}^{\infty}f(x)G(x-s)ds$$
I was looking at an example where the integrand was written in the latter way and not the former and I wanted to ...
- 187
0
votes
0
answers
9
views
Confused about setup of equation for Green's Function
This is my first time ever attempting to use Green's Function and I have taken no formal classes on the subject, so my mistake might be rather entry level, so please bare with me. From the Wikipedia ...
- 187
2
votes
0
answers
46
views
Solving $\iint_{\mathscr{D}}\frac{(x-x')^2f(x',y')\,\mathrm{d}x'\mathrm{d}y'}{\left((x-x')^2+(y-y')^2\right)^{3/2}}=1$ for the unknown function $f$
While solving a fluid dynamical problem, the following integral equation arised:
$$
\iint_\mathscr{D} \frac{(x-x')^2 f(x',y') \, \mathrm{d}x' \mathrm{d}y'}{\left( (x-x')^2 + (y-y')^2 \right)^{3/2}} = ...
- 431
1
vote
2
answers
76
views
1D Heat Equation with Insulated Boundaries with Dirac Delta
This question is very similar to the question I previously asked here; however, there is a subtlety which I did not appreciate at the time. The previous question which I asked was correctly answered, ...
- 1,380
0
votes
0
answers
37
views
The divergence of gradient of an integral
Let given a $ C^2(\bar{\Omega}) $ function
$$u(r)=\int_{\Omega}(\nabla f).(\nabla g))dv$$
Then how to find $ \nabla^2 u(r)? $ Can i pass the Laplace operator inside the integral? If so how could I do ...
- 7
0
votes
0
answers
33
views
Laplace equation's solution as a "convex combination" of the Dirichlet data
I was playing around with numerical solutions of the Laplace equation with mixed boundary conditions:
\begin{alignat}{3}
\Delta u(x) &= 0, &\quad &x \in \Omega, \\\\
u(x) &= g(x), &...
- 1,843
0
votes
0
answers
37
views
Solution of backward heat equation
I am attempting to solve the following PDE: for $y >0$ and $t>0$,
$$V_t(y,t) + a V_{yy}(y,t) - b V_y(y,t) = f(y,t), \ V(y,0) = 0 . $$
By some changes of variables, I covert the above Euler-type ...
- 389
2
votes
1
answer
110
views
1D Heat Equation with Insulated Boundary Conditions; Green's Function
I would like to determine the solution to the 1D heat equation where the initial condition is a Delta function at the boundary
$$ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$$
...
- 1,380
0
votes
0
answers
25
views
Associating a non-local differential operator to its integral representation
It is known that, in $\mathbb{R}^2$, we can define the non-local operator $\frac{1}{\Delta}$ with the Green function of the Laplace operator $\Delta$. This provides the non-local operator with an ...
- 11
0
votes
0
answers
17
views
Dirichlet Green's function outside the sphere
I am trying to find the Dirichlet Green's function for the spherical region centred at 0 and of radius a. I have found the Dirichlet Green's function inside the sphere by method of images, but what is ...
- 11
0
votes
0
answers
58
views
Deriving green function for Biharmonic equation and more in general polyharmonic.
As I am not an expert on biharmonic and more in general polyharmonic equation. Is there a procedure similar to the one highlighted here to derive the green function?
Some people however don't seem to ...
- 5,126
3
votes
0
answers
116
views
Evans Partial Differential Equations - Derivation of Green function
I am reading through PDE book by Evans to get a better understanding of what a Green function is.
In my understanding the Green function enables an explicit representation of the solution of certain ...
- 5,126
0
votes
1
answer
91
views
On the existence of Green's functions by Peter Lax
I'm reading Peter Lax's paper On the existence of Green's functions. He showed that the regular part of Green's functions is continuous on the boundary. My question is : to have the normal derivative ...
- 1
0
votes
1
answer
158
views
How to find the maximum of this oscillating function?
Given $n \in\mathbb{N}$, $a_1, \dots, a_n \in\mathbb{C}$, $k \in \mathbb{R}$ and $x_1, \dots x_n \in \mathbb{R}^3$, let $\Phi : \mathbb{R}^3 \to \Bbb C$ be defined by
$$\Phi (y) := \sum_{i=1}^{n} {a_i ...
- 45
0
votes
0
answers
45
views
Integral representation for the operator $\frac{\log{\Delta}}{\Delta}$ in two dimensions
Recently, I have stumbled upon the following non-local operator in $\mathbb{R}^{2}$
$$
\frac{\log{\Delta}}{\Delta} \ ,
$$
where $\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^...
- 11
0
votes
0
answers
62
views
Fundamental Solution of a heat equation with a drift term
Let consider the following equation on $[0,T]\times \mathbb{R}$:
$$
\partial_t u + \beta \partial_x u =
\alpha \partial_{xx} u.
$$
Question: Is it possible to construct a fundamental solution for ...
- 75
0
votes
0
answers
41
views
Integrating product of Heaviside step function and Delta functions
I want to calculate the following integral:
$$I_1 = \int_{\eta_0}^0d\eta'\int d^3x'~a^4(\eta')~G(x,x')~\frac{\lambda~\delta^3\left(\vec{x}'\right)}{a^3(\eta')}~,$$
where $$G(x,x') = -\frac{\theta(\...
- 49
1
vote
0
answers
59
views
Finding the Green's function of a given PDE
I'm currently studying this article because I'm interested in the technique for finding the Green's function $G$ of the following PDE:
$$\frac{\partial B}{\partial t}+\frac{\sigma^2}{2}\frac{\partial^...
- 131
3
votes
1
answer
115
views
On the solution of the heat equation using distribution theory
The Green's function
$$
\tag{1}
\displaystyle K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}$$
solves the heat equation
$$
{\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\
$$
for ...
- 119
1
vote
0
answers
43
views
Green function of a Schrodinger operator
If $V:\mathbb R\to \mathbb R$ is an $L$ periodic function in $\operatorname L^{\infty}$ we can always find two independent solutions for $$\psi''(x)+V(x)\psi(x)=E\psi(x)$$
$\psi^{\pm}(x)=e^{\pm ipx}\...
- 2,665
1
vote
0
answers
97
views
Bond-pricing under the Vasicek short rate model
Crossposted at Quant SE
I'm currently studying the Vasicek model of the short interest rate
$$dr_t=a(\mu-r_t)dt+\sigma dW_t$$
I know how to solve this stochastic differential equation (SDE) and how ...
- 131
0
votes
0
answers
18
views
Green's function of Laplace equation in axissymmetric bounded region
Given a Laplace equation in an axissymmetric region, i.e., $\nabla^2 \varphi(r,z) = \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial \varphi}{\partial r}\right) + \frac{\partial^2\varphi}{\...
- 211
0
votes
0
answers
55
views
Use Green's Function to solve ODE
I have the following ODE to solve via Green's function:
$$
\frac{d^2y}{dx^2} - y = F(x)
$$
for $0<x<1$ with boundary conditions $y(0)=y(1)=0$.
So far I have done the following:
I try a Green's ...
- 359
1
vote
0
answers
48
views
integral over Green's function of a $d$-Ball
I'm studying for an exam on PDE's and I'm stuck trying to solve the following problem.
This is in context of the following boundary value problem:
\begin{align}
\begin{cases}
-\Delta u \equiv 0,&\...
0
votes
0
answers
50
views
A priori estimate of solution of Poisson equation
I'm studying for an exam on PDE's and came across the below problem. I found an approach for solving it, but am missing a last argument. Can you help me conclude my proof?
This is related to this ...
1
vote
1
answer
103
views
Green function of a forced undamped oscillator using two different methods don't match!
The differential equation for a forced undamped oscillator has the form $$\mathcal{L}x\equiv \frac{d^2x}{dt^2}+\omega_0^2x=f(t),$$ and the Green function $G(t,t')$ defined as $$\mathcal{L}G(t,t')\...
- 641
0
votes
0
answers
50
views
Green function on unbounded domains (subgraph domain)
Consider the equation $\Delta u = 0$ in $\Omega$ with $u = g$ on $\partial \Omega$. Usually, we can follow Evans's PDE book to construct the Green function
$$ G(x,y) = \Phi(y-x) - \phi^x(y)$$
where $\...
- 1,371
1
vote
0
answers
38
views
resolution of the identity applied to exponential functions
Context
I am working with three-dimensional Green's functions. A Green's function must satisfies
$$
1 = \lim_{\boldsymbol{\varepsilon}\to0 } \int_{\mathbf{x}^{\prime}-\boldsymbol{\varepsilon}}^{\...
- 948
2
votes
1
answer
64
views
Green function of the 2-dimensional "curl" operator
I want to find the solution to the equation:
$$
x \partial_y f - y \partial_x f = \delta^{(2)}(x,y) \,,
$$
where $f : \mathbb{R}^2 \to \mathbb{R}$ is a scalar function, and $\delta^{(2)}(x,y)$ is the ...
- 93
1
vote
0
answers
77
views
What's the exact expression of Green's function on poincare disk?
Consider the Poincare disk $\mathbb{H}^2$, that is the unit disk endowed with the following metric
$$
g_{ij}=\frac{2}{1-|x|^2}\delta_{ij}.
$$
I believe there exists a global Green's function on $\...
- 473
0
votes
0
answers
21
views
Effect of point source perturbation on lift for 2D subsonic flows past airfoils
I am trying to determine the effect of inserting a stationary point source perturbation on the lift exerted on an airfoil inmersed in an inviscid, compressible, subcritical flow in 2D (no shock waves, ...
- 1