Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
24 views

Using Green's Theorem to calculate the area of a surface

I am asked to calculate the area that is bounded by the curve r(𝑡) = (𝑡 − cos𝑡, 1 − cos𝑡), 0 ≤ 𝑡 ≤ 2𝜋, and the x-axis in the xy-plane. I calculated the line integral but the answer I get is -2𝜋....
0
votes
0answers
10 views

Green's function for homogeneous PDE

I was looking for some Green's function method to solve a homogeneous PDE with nonhomogeneous boundary conditions (i.e., $Lu=0$ in $D$ with $u=f(\mathbf{x})$ in $\partial D$), but most of the ...
1
vote
1answer
20 views

Green's function for 1D modified Helmoltz' equation

My equation is $-k^2 \phi + \frac{\partial^2{\phi}}{\partial{z^2}} = -2 \delta(z)$ with $\phi =0 $ on $z=\pm a$ How do I show the Green's function is $\phi = \frac{\sinh(k(z+a))}{k\cosh(ka)}$ if $z&...
0
votes
0answers
5 views

Continuity in derivatives of the Green's function in RHB

I am currently reading chapter 15 of Riley, Hobson and Bence's "Mathematical Methods for Physics and Engineering" ($3^{\text{rd}}$ edition), which discusses $n^{\text{th}}$ order linear ordinary ...
1
vote
0answers
16 views

Green's function in cylindrical coordinates, a small question

Here is a nice derivation for Green's function of a Laplacian in cylindrical coordinates. For the $r$ coordinate, the equation looks like this: $$\frac{1}{r}\,\frac{d}{dr}\!\left(r\,\frac{dg_m}{dr}\...
0
votes
0answers
25 views

Solution to Green's function second degree differential equation

I'm given a differential equation in the form of $$y''+P(x)y'+Q(x)y=R(x)$$ with the specified boundaries $y(a)=y(b)=0$ I've already shown that the Green function of this ODE is $$G(x,z)=\begin{...
0
votes
0answers
18 views

Green's Equation for $\Delta u=0$ with specified conditions.

I need to find the Green's function for $$\Delta u=0$$ in the upper half plane subject to $$G(x,0;x_0,y_0)=0=\lim_{y\rightarrow\infty} G.$$ I know that $\Delta G=0$ and I have tried solving this with ...
0
votes
2answers
59 views

Finding the fundamental solution of an ODE

I want to find the fundamental solution of this ODE: $$- u^{\prime\prime} + k^2 u=0, -\infty<x<\infty, k\neq 0$$ I know that it is: $$\Gamma(x,\epsilon) = \frac{e^{k|x-\epsilon|}}{2k}$$, but I ...
3
votes
1answer
43 views

Green's Function for Differential Equation $t\dfrac{d^2}{dt^2} - \dfrac{d}{dt}$

I have a Second Order Differential Equation $tu'' - u' = 1-t^2$. How do you find the Green's function $G_a(t)$ for any $a\in(-1,1)$? We define the Green's function $G_a(t)$ as a solution to $tG_a'' - ...
0
votes
0answers
16 views

Gaussian diffusion propagator with reflecting boundary conditions?

In their 1993 paper, Mitra et al introduce (Equation 23) the three-dimensional diffusion propagator with reflecting boundary conditions at $z=0$ in the following form, $$ G_0( \textbf{r}',\textbf{...
0
votes
0answers
17 views

Greens Function where $G'(0)=0, G \in L^2[0,\infty)$

I have the following problem: $-\frac{d^2G}{dx^2} - \mu G = \delta(x-\xi)$ where $G'(0)=0, G \in L^2[0,\infty)$ The general soln is $Acos(\sqrt{\mu} x) + Bsin(\sqrt{\mu} x)$ i.e. $Ae^{i\sqrt{\mu}x} + ...
0
votes
1answer
29 views

Continuity and compactness of this operator?

Set $X=C([a,b],\mathbb{R})$ with the uniform norm. I want to prove that $T:X\rightarrow X$ given by $$T(x)(t)=\int_a^b K(t,s)f(s,x(s))ds $$ is continuous and compact, where $K:[a,b]\times [a,b]\...
2
votes
1answer
59 views

Green's function for $\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}$

Compute the Green's function for the Laplacian, for the region $$\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}.$$ My approach is to use a reflection argument similar to the one used for ...
1
vote
0answers
21 views

How to evaluate the imaginary part of this one-sided fourier transform?

So, I came across the following integral $\tag{1}\Gamma(\omega) = \int_{0}^{\infty}dse^{i\omega s}G^{+}(s)$ where $G^{+}(s) = \langle \phi(t)\phi(t - s)\rangle = \left[-16\pi\alpha^2\sinh^2(\frac{s}...
0
votes
1answer
37 views

Obtaining the Fundamental Solution of $1-\partial_x^2$

How does one obtain the fundamental solution of the differential operator $(1-\partial_x^2)$? There does not seem to be any easily accesible literature specifically describing how this is done, except ...
1
vote
0answers
39 views

Fourier Transform of Dyson's Equation

I'm attempting to show that the Fourier transform of Dyson's equation for a constant potential V, \begin{equation} G(\mathbf{r},t,\mathbf{r}_0,t_0) = G^0(\mathbf{r},t,\...
2
votes
0answers
33 views

Modified Helmholtz for imaginary constant

The Green's function for the free space modified Helmholtz equation in two dimensions is \begin{equation} \alpha^{2}u - \Delta u = 0, \end{equation} with $\alpha^{2}$ real, is $K_{0}$. However, I am ...
3
votes
0answers
90 views

Converting between Solution forms using Green's Functions in Linear Differential Equation

EDIT: Bounty is over tomorrow so I tried to clean up the question a bit, and put the additional work below as optional to read. I summarized the current results and the solution form I am trying to ...
0
votes
1answer
54 views

Heat equation PDE (nonhomogeneous); Green's function; Dirac delta

(Sorry for the messy title, trying to include the keypoints of the problem.) I am new to the theory on how to solve this kind of PDE problem which is presented below; I am unsure on which method to ...
0
votes
0answers
7 views

Axial Green’s function

Which is the solution of the Poisson equation in axial symmetric coordinates (r,phi,z)? I would like to know the potential generated in the point (r_f,phi_f,z_f) by a circular loop (the source) ...
0
votes
0answers
28 views

Green's function of a differential operator

I am looking for pointers on how to find the Green's function of the following differential operator: $V = V_0(x) + V_1(x)\frac{d}{dx}+V_2(x)\frac{d^2}{dx^2}$ In my particular problem, I have $V_0$, ...
0
votes
0answers
15 views

Green's function modified Helmholtz operator in square periodic domain

My question is motivated by point vortex dynamics. The motion of $N$ point vortices, each of circulation $\Gamma_i$ and located at $\mathbf{x}_i$, is induced for a given elliptic operator (e.g. ...
0
votes
0answers
26 views

Green function region of convergence nonlinear differential equation

Given a second order nonlinear differential equation with the boundary values: $y'' + \frac{1}{x}y' + y^2 = 0$, with $y'(0)= 0, y(1)=1$ How would one estimate the region of convergence given by: $$...
7
votes
3answers
148 views

How to solve a second order partial differential equation involving a delta Dirac function?

In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function: $$ a \, \frac{\partial^2 w}{\partial x^2} + b \, \frac{\partial^2 w}{\...
2
votes
1answer
61 views

Green function for solving ODE: continuity and discontinuity of derivatives

I am studying Green functions and I want to find the Green function $G(x,x_0)$ of the linear operator \begin{equation} \mathcal{L}=\left(\frac{d^n}{dx^n}+a_1(x)\frac{d^{n-1}}{dx^{n-1}}+\dots+a_n(x) \...
0
votes
0answers
43 views

Boundedness of the L2 norm of the boundary derivative of the Greens function.

I wish to prove that an integral is bounded. For $\delta \in (0,1)$ and $\textbf{c} \in (\delta - 1, 1-\delta)^2$ we define our domain to be $\Omega := (-1,1)^2 \setminus B_\delta (\textbf{c})$. Let $\...
1
vote
0answers
18 views

Contour Integral over Heaviside Function

I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression $$G^+(\mathbf{0})=\frac{2m}{(2\pi)^2}\int d\...
0
votes
0answers
24 views

Rotationally invariant Green's functions for the three-variable Laplace equation in all known coordinate systems

Green's function for the three-variable Laplace equation in Cartesian coordinates is $$\frac{1}{|\mathbf{r}-\mathbf{r'}|} = \frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$ It may be written in ...
0
votes
0answers
72 views

How does one use Green's function of the operator to get the solution of the arbitrary boundary value problem?

Assume I've been given an operator $L$ and its Green's function $G(s, s')$. This is the function that solves the following: $$ L G(s, s') = \delta(s-s'), G(a,s') = G(b, s') = 0 $$ I know how to get a ...
0
votes
0answers
38 views

How does one obtain Green's function on practice? Is it used besides than a solution representation nowadays?

If it is used, than how to numerically compute it. It seems to me that it would take a great deal of time to get it with a reasonable precision. And as the function is used as accumulator, the ...
1
vote
1answer
73 views

A Triple integral

While trying to obtain a Green's function for a PDE,I stumbled upon this integral $\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{e^{i(s_{x}x+s_{y}y+s_{z}z)}}{...
1
vote
0answers
29 views

How to find the Green's function for a non-homogeneous PDE?

Very general question on how to find the Green's function that satisfies $$y(x) = \int_a^b G(x,s)f(s)ds$$ For a non-homogeneous PDE problem of the form: $$L[y(x)] = f(x)$$ In the domain $[a,b]$. ...
1
vote
0answers
31 views

Stationary phase for retarded potentials in electromagnetism

I want to apply something like a stationary phase approximation to the following expression $\int_V d^3x' \frac{B(x')}{|x-x'|}e^{ik|x-x'|}$ with $x\in \mathbb{R}^3$, $k\rightarrow \infty$ and $B$ is ...
0
votes
1answer
56 views

Constructing a green function

Construct Green's function to solve the boundary value problem $$ \frac{d^2y}{dx^2} + 4y = f(x), \qquad y(0)=0, \qquad y(\pi/4)=0. $$ Now consider this equation on the interval $[0,L]$ with the ...
0
votes
0answers
11 views

General solution to inhomogeneous recurrence equation/ Analogon Greens function

I have a recursive equation of the form $u_{n} = u_{n-1} - a_{n}$ For finite $n\leq k$ with $u_{k+1}=u_{1}$Where $a_{n}$ is an inhomogenity that I do not presently know. The goal is find a ...
0
votes
0answers
27 views

Find the fundamental set of solutions for $-y''+k^2y=f$ with the boundary conditions $2y(0)-y'(0)=0$ and $y(1)=0$

I am reading a chapter from a textbook about Green's functions, and the example in the text asks to find the Green's function for $-y''+k^2y=f$ with the boundary conditions $2y(0)-y'(0)=0$ and $y(1)=0$...
0
votes
1answer
34 views

How to solve this Complex Integral using poles?

I want to find the green's function of a free particle, which depends of the integral: $$ I = \frac{1}{4\pi ²ir} \int^{+\infty}_{-\infty} \frac{ke^{ikr}}{E-\frac{\hbar²k²}{2m}+i\eta} dk\,. $$ Then, ...
2
votes
0answers
66 views

Forced wave equation using Green's functions and Laplace transforms

Solve $$\boxed{\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=F(x,t),0<x<\infty} $$ with the boundary conditions $\frac{\partial u}{\partial x}=0$ at $x=0$ and $u\to 0$ as ...
0
votes
0answers
34 views

how to find the fundamental solution of $-\Delta u + e^u - e^{-u} = \delta(\vec{x})$ in 2D?

$-\Delta u + e^u - e^{-u} = \delta(\vec{x})$, where $\Delta$ is the Laplace operator and $\delta(\vec{x})$ is the Dirac's delta function and satisfies: $\delta(\vec{x}) = \begin{cases} 0, & \vec{...
2
votes
0answers
53 views

Finding the Green Function

I'm having some trouble finding the Green function of the following differential equation: $$ \frac{d[x y'(x)]}{dx} = f(x)\\ 0 \leq x \leq 1\\ y(1) = 0 $$ $y(0)$ is finite.
1
vote
0answers
54 views

Green's function for 1d heat equation on finite interval: is it non-negative?

The Green's function for the 1d Heat equation with Dirichlet BCs on the domain $(0,1)$ is $$G(x,y,t) = 2\sum_{n=1}^\infty \sin(n\pi x)\sin(n\pi y)\exp(-n^2\pi^2t)$$ Is this function non-negative? Is ...
3
votes
2answers
100 views

PDE Laplace equation. Integral representation form and Green function

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and assume that for any $y \in \Omega$ there is a function $h_{y} \in C^{2}(\overline{\Omega})$ such that \begin{equation} \label{eq8.1} \begin{cases} ...
2
votes
0answers
42 views

Finite sum involving trigonometric functions

The temperature Green's function in momentum-frequency representation of a system of free phonons (in one-dimension) is given by [1]: \begin{equation} D^{(0)}(k_n,\omega_{n'}) = -m\frac{\omega_{k_n}^...
1
vote
0answers
35 views

Solding pdf using Green's function.

Use the Green’s function obtained in problem 2 to obtain the leading term in the asymptotic expansion for $ \phi(x) $ as $ \lvert x \rvert \to \infty $, where $ \phi $ satisfies $ \triangledown^2 \phi ...
0
votes
0answers
125 views

Laplacian Green's function on the $n-$sphere

I was looking for the explicit expression for the Green's function of the Laplace operators on the Euclidean $n-$sphere, $S^n$, namely the distribution $G(x, x_0)$, with $x$, $x_0$ unit vectors in $\...
0
votes
2answers
111 views

Evaluate $\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx\text{ ,with }a>0$ [duplicate]

$$\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx,\;\;\;\;\text{with }a>0$$ How to evaluate Integral of $\ln(1+a^2-2a\cos x) dx$? where $x$ from $0$ to $2\pi$ and $a>0$, $\ln$ is the natural logarithm.
0
votes
0answers
20 views

Writing down the equation for the Green's function

I am considering diffusion on a sphere with spherical symmetry: $$\frac{\partial c}{\partial t}=\frac{D}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial c}{\partial r}\right)$$ With $c(0,...
2
votes
0answers
90 views

Prove Property of Green Function Solution to Laplace Equation in a 2D-square

Let's consider a 2D-square with 4 euqal subsquares containing different dielectrics. Inside the square domain, the unkown potential function $\Phi$ satisfies the Laplace equation: $\nabla^2\Phi=0$ ...
2
votes
1answer
181 views

Heat equation problem with initial condition in a disk

I was solving this problem Find the solution $u(x,y,t)$ of the problem $$ \begin{cases} u_t=D\nabla^2u \\[5pt] u(x,y=0,t) = 0\\[5pt] u(x,y,0) = \phi(x,y) \end{cases} $$ where $$ \phi(x,y) ...
0
votes
0answers
58 views

Anti-holomorphic derivative on 1/z

We know the fact that $\partial_{\bar z} (1/z) \propto \delta(z, \bar z)$. But I am slightly confused about the proof. I have seen a few of them, and of course they use the Green's theorem, or the ...