Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

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6
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93 views

Closed form solution to an ordinary differential equaiton

How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where $C_1, C_2, C_3, C_4\in \mathbb{R}$ are all constants. It looks ...
5
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0answers
436 views

The heat kernel as a fundamental solution

From my undergraduate studies I know that a fundamental solution to a partial differential operator $P$ is a distribution $u$ such that $Pu= \delta$ (no reference to any boundary or initial condition)....
3
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0answers
167 views

fundamental solution of Fokker Plank equation for the Langevin equation of motion

I am studying the original paper by Ornstein and Uhlenbeck on the theory of Brownian Motion. After constructing the Fokker-Planck equation for the Langevin equation of motion, the authors arrive at ...
3
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1answer
462 views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
3
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0answers
445 views

Green function for a second-order elliptic PDE

Let $L = L^*$ be a second-order elliptic PDE with smooth and bounded coefficients in some bounded domain $\Omega \subset \mathbb R^d$, $d \geq 3$, with smooth boundary. Let $G(x,y)$ be a Green ...
2
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0answers
28 views

Analytical Solution to a linear PDE with delta function like source term

I have a time-independent Vlasov equation with source term in $R^2$: $$v\cdot\partial{f}/\partial{x}+\partial{f}/\partial{v}=s(x,v)$$ where $s(0,0)=500$, else $s(x,v)=0$. And I was wondering how to ...
2
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0answers
46 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 ...
2
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0answers
88 views

A problem about fundamental solution of Laplace equation

I would like to derive a solution to the following equation in $R^{3}$: $$-\Delta u(x) + e^{u(x)} - e^{-u(x)} = \delta(x)$$where $\Delta$ is the Laplace Operator and $\delta(x)$ is the Dirac's delta ...
2
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0answers
76 views

calculating first and fundamental form coefficients with surface normal and Gaussian curvature

I have some points (point cloud), I need first and second fundamental form coefficients at each point. before I used some polynomial surface fitting algorithm then these coefficients were calculated....
2
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1answer
64 views

What is $\frac {\partial \Gamma} {\partial \nu}$ on $\partial B_{\rho} (y)$?

I am studying Laplace's equation from the book "Elliptic partial differential equations of second order" written by Gilbarg and Trudinger. Here I am struggling to grasp a concept regarding the ...
2
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0answers
195 views

Fundamental solution for 1D nonhomogeneous wave equation

I want to get the fundamental solution for the following 1D nonhomogeneous wave equation:\begin{align}\left\{ \begin{aligned} &\frac{\partial^2u}{\partial^2t}-\frac{\partial^2u}{\partial^2x}-au=0,...
2
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362 views

Fundamental solution of Laplacian in 2d doesn't agree with 3d version physically?

The fundamental solution of the Laplacian, with $(x, y) \in \mathbb{R}^3$, is $$ \Phi(x, y) = -\frac{1}{4\pi|x - y|} $$ So, say, the further away $x$ is from some point source $y$ the lesser the ...
2
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0answers
73 views

Backwards PDE with Gaussian Kernel

I have to solve a backwards diffusion PDE of the following form: $\partial_t u-\frac{1}{2}(\partial_x u)^2+\frac{\sigma^2}{2}\partial_{xx}u=0;$ $u(x=0,t)=c(t).$ Once I use the Cole-Hopf ...
2
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0answers
46 views

How to determine a fundamental solution when only given the eigenvalues of the coefficient matrix?

Consider a homogeneous linear system of differential equations $x' = Ax$ with $A$ a real $7 \times 7$ matrix where we only know the eigenvalues and their multiplicities: $\lambda_1 = 2$ with ...
2
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0answers
185 views

Fundamental system of solutions

Theorem Let $x_i^{(k)}(t), i, k=1, \dots, n$ be a fundamental system of solutions of $x'=Ax$. Then any solution of this system can be written as a linear combination of $x_i^{(k)}(t), i,k=1, \dots, n$...
1
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0answers
49 views

fundamental solution of an ODE of second order

Let us consider the following homogenous ODE of second order: $$x''(t)+a_1(t)x'(t)+a_2(t)x(t)=0$$ where $a_1$ and $a_2$ are continuous functions. Are there conditions on these functions such that one ...
1
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0answers
32 views

Integrating log |z-w| over a disk in the complex plane.

Let $ E=\frac{1}{\pi} \log |z|^{2} $ and $ \chi_{r}=\frac{1}{\pi r^{2}} \chi_{\Delta(0, r)} $ where $\chi_{\Delta(0, r)}$ is the characteristic function of the disk with radius $r$ centered at $0$, ...
1
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0answers
32 views

Transition densities for (elliptic) SDEs [or: Fundamental solutions for (elliptic) PDEs]

General question: When are the transition kernels corresponding to the SDE $$ dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \quad t\in[0,1], \tag{$\heartsuit$}$$ absolutely continuous w.r.t. Lebesgue's ...
1
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0answers
71 views

The fundamental solution for Laplace's equation in cylindrical coordinates

I am working my way through the excellent textbook of Garabedian on partial differential equations and have two questions related to the topic of the fundamental solution in chapter 5: Garabedian ...
1
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1answer
44 views

Find differential equation general solution

The task is to find $u_{xy}+yu_x+xu_y+xyu=0$ this equation general solutions. First of all I wrote it as $\frac{d}{dx}(u_y+yu)+x(u_y+yu)=0$ Then marked $u_y+yu=t$ and got equation $t_x+xt=0$ And ...
1
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0answers
68 views

Difference of fundamental solutions

I am reading a set of notes in which the author claims that the difference between any two fundamental solutions (to the Laplacian) is harmonic. That is, if $E_1$ and $E_2$ are fundamental solutions ...
1
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1answer
205 views

A question regarding the fundamental solution of a 1D Laplace equation

Disclaimer: My mathematical background is mathematical physics so this question most likely lacks rigor. I am trying to understand the nature of the singularity of the fundamental solution (aka Green ...
1
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0answers
60 views

Nonvanishing Gradient of a harmonic polynomial

Let $e$ be a unit vector in $\mathbb R^d$, $d\geq 3$. Consider the function $\Gamma(x):=\frac {1}{|x-e|^{d-2}}$, which is the (translated) fundamental solution to the Laplacian on $\mathbb R^d$. Let $...
1
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0answers
182 views

Deriving the fundamental solution for the Helmholtz operator in $\mathbb{R}$

I want to derive $G\in\mathcal{S}'(\mathbb{R})$ satisfying $$\left(\Delta+\frac{\omega^{2}}{c^{2}}\right)G=\delta,\qquad(\star)$$ where $\Delta$ is the Laplace operator, $\omega\in\mathbb{R}$ and $c&...
1
vote
1answer
116 views

Linear ODE with one periodic coefficient

I'm interested in the following equation $$\ddot{f} - \cos^2(\omega t) \dot{f} - f = 0$$ with an initial condition $f(0) = f_0$. I don't think there is a closed-form solution, but do you know if ...
1
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0answers
42 views

Uniqueness of solution in a scalar equation.

Show that $\det \Phi(t)=\det \Phi(t_0) e^{\int_{t_0}^{t} \sum_{j=1}^{n} a_{jj}(s)ds}$ is the unique solution of the scalar equation: $$y'=\big(\sum_{k=1}^{n} a_{kk}(t)\big)y$$ Satisfying the initial ...
1
vote
1answer
815 views

Converging differential equation solutions

Sometimes solutions to differential equations can converge to an equilibrium path as time approaches infinity. Also, if you were given a second order differential equation and two different sets of ...
1
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0answers
117 views

Potential theory: Fundamental solution.

After a brief introduction to distribution theory, I have been given the definition of fundamental solution for an operator $\mathcal{L}$ as follows: Given a domain $\Omega\subset\mathbb{R}^N$ and ...
1
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0answers
111 views

Fundamental solution matrix of a linear PDE

A vectorial function $\boldsymbol{f}(\boldsymbol{x})$, satisfies the following PDE $$ (\boldsymbol{c \cdot \nabla}) \boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{A}(\boldsymbol{x}) \boldsymbol{f}(\...
1
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0answers
185 views

A simple assumption when deriving the fundamental solution of the heat equation

When deriving the fundamental solution in his book PDE (section 2.3.1, page 46), Evans comes to the equation $$r^{n-1}w'+\frac{1}{2}r^n w=a,$$ where $w:=w(y)=w(|y|)=w(r)$. After that he assumes that ...
1
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0answers
115 views

What is the use of the Rofe-Beketov formula?

Let $y_1(x)$ be an elementary solution of the Sturm-Liouville equation, $$ \frac{d}{dx}\left( p(x)\frac{dy}{dx} \right) + q(x)y = 0 $$ It is well known that a second linearly independent solution ...
1
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0answers
344 views

Forward-time, centered space evalaution of the heat equation: numerical stability and unique solution

I have a script of code which models a planetesimal that is accreted into a planetary atmosphere. In the code, I include the physics of frictional ablation and thermal ablation. Frictional ablation is ...
1
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0answers
114 views

Analytical Solution of diffusion equation(with Laplace Beltrami operator)

The analytical solution of diffusion equation is an exponential function. How can i find the analytical solution of the diffusion equation, if we have Laplace Beltrami operator(on a sphere), instead ...
1
vote
0answers
18 views

Fundamental solution of the frozen opearator

Let $L$ be some differential operator of the form $$ Ly = y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) $$ with all $a_k(x)$ being smooth. Let also $M$ be the frozen at $x=0$ operator $L$...
1
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0answers
67 views

Solution to a PDE that requires fundamental Laplace solutions

I would like to derive the spherical symmetric solutions (or in another words make use of the fundamental solutions) of the following PDE: \begin{equation} \Delta u(x)+\lambda u(x)=0, \quad \lambda&...
1
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0answers
71 views

uniform absolute convergence for function series

Consider the Theta function $\theta(x,t)=\sum\limits_{m=-\infty}^{\infty}K(x+2m,t)$, where $K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}$. The $\theta$ function is seen in the solution of the ...
1
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0answers
171 views

From fundamental solution to differential equation.

There are various techniques to find the fundamental solutions for a given linear ordinary differential equation (ode). I am interested in reverse engineering; to find a differential equation from a ...
1
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0answers
77 views

Existence of fundamental solutions of Heston's PDE

The Heston's PDE is a variable coefficient linear PDE. I know the existence of fundamental solution of constant coefficient linear PDE has been proofed by Bernard Malgrange and Leon Ehrenpreis in 1954 ...
0
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0answers
36 views

Increasing Powers of Independent Variable As Factor in Linear ODE Solution Terms

According to my understanding, whenever would-be terms of a general solution to a linear ODE with constant coefficients are linearly dependent, inserting increasing powers of the independent variable (...
0
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0answers
82 views

Fourier transform of the Hankel function of first kind and zero order

Given the Hankel function of first kind and zero order $$ H_0^{(1)}(\omega|x-y|), $$ with $|x-y| \geq C >0$, I would like to calculate its Fourier transform, that is $$ \int_{-\infty}^{\infty} ...
0
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0answers
39 views

I have a question about Elzaki variational iteration method.

How to get from the first line to the second line? enter image description here $$u_{n+1}(x,t) = u_n (x,t) - \int_0^t \left\{ (u_n)_s(x,s) - \frac{\partial}{\partial s} G(x,s) + \frac{\partial}{\...
0
votes
0answers
49 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{...
0
votes
0answers
43 views

Hipoelliptic altered wave operator?

We know that the wave operator in $\mathbb{R}^{2}$: $L=\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$ Can I say that $L-\lambda$, $\lambda(x,y) \in C^\infty(\mathbb{R}^2)$ is not ...
0
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0answers
76 views

Fundamental Matrix of OED 3x3 Constant matrix

Given the constant matrix A= \begin{bmatrix}4&-1&0\\3&1&-1\\1&0&1\end{bmatrix} find the fundamental matrix for the system $y'=Ay$. After attempting to solve the system, I ...
0
votes
0answers
66 views

Finding Eigenvector from a 3x3 matrix

Given the constant matrix A = \begin{bmatrix}4&-1&0\\3&1&-1\\1&0&1\end{bmatrix} Find the fundamental matrix. After finding the eigenvalue $\lambda$=2 with multiplicity 3, ...
0
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0answers
22 views

ODE, free-space Green's Function

In my PDE study, I learned about $\textit{free-space Green's Functions}$ and I thought if there was also for ODE ($\frac{d}{dx^2}$ or Sturm-Liouville). I looked for free-space green's functions for ...
0
votes
1answer
185 views

bessel function of second kind

I'm working through "Elementary Differential Equations", 10th Ed (Boyce/DiPrima) on my own -- 120 miles from the nearest easily accessible prof or grad student -- and I've a question on a technique in ...
0
votes
0answers
145 views

Estimates for the derivatives of the solution of heat equation

Let $ u_0\in L^\infty (\mathbb{R^d})\cap C^\alpha(\mathbb{R}^d), \alpha \in (0,1), \Phi $ the fundamental solution of the heat equation and $u(t,\cdot ):= \Phi (t,\cdot )*u_0(\cdot)$ for $t>0$. Let ...
0
votes
0answers
121 views

Finding the Fundamental Solution of the Operator $\frac{\partial}{\partial x}+c$

I am trying to find the Fundamental Solution for the Operator $\frac{\partial}{\partial x}+c$. I approach it by considering a radial function $v(r)$, and considering the ODE $v'(r)+c v=0$. This ODE ...
0
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1answer
20 views

CDA Fundamentals of Comp.S - Reduce an equation

how can I reduce: $co = ci'ab+cia'b+ciab'+ciab$ to: $co = ab + cia + cib$ I don't know how they did that on the book... This is for Adders, working with circuits. Thank you very much