# Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

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### 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 ...
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)....
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
46 views

### Modified Helmholtz for imaginary constant

The Green's function for the free space modified Helmholtz equation in two dimensions is $$\alpha^{2}u - \Delta u = 0,$$ with $\alpha^{2}$ real, is $K_{0}$. However, I am ...
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### 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 ...
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....
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### 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 ...
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,...
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 ...
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### 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 ...
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### 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 ...
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### 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 $x_i^{(k)}(t), i,k=1, \dots, n$...
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### 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 ...
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### 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 $0$, ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...
111 views

A vectorial function $\boldsymbol{f}(\boldsymbol{x})$, satisfies the following PDE $$(\boldsymbol{c \cdot \nabla}) \boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{A}(\boldsymbol{x}) \boldsymbol{f}(\... 0answers 185 views ### A simple assumption when deriving the fundamental solution of the heat equation When deriving the fundamental solution in his book PDE (section 2.3.1, page 46), Evans comes to the equation$$r^{n-1}w'+\frac{1}{2}r^n w=a,$$where w:=w(y)=w(|y|)=w(r). After that he assumes that ... 0answers 115 views ### What is the use of the Rofe-Beketov formula? Let y_1(x) be an elementary solution of the Sturm-Liouville equation,$$ \frac{d}{dx}\left( p(x)\frac{dy}{dx} \right) + q(x)y = 0 $$It is well known that a second linearly independent solution ... 0answers 344 views ### Forward-time, centered space evalaution of the heat equation: numerical stability and unique solution I have a script of code which models a planetesimal that is accreted into a planetary atmosphere. In the code, I include the physics of frictional ablation and thermal ablation. Frictional ablation is ... 0answers 114 views ### Analytical Solution of diffusion equation(with Laplace Beltrami operator) The analytical solution of diffusion equation is an exponential function. How can i find the analytical solution of the diffusion equation, if we have Laplace Beltrami operator(on a sphere), instead ... 0answers 18 views ### Fundamental solution of the frozen opearator Let L be some differential operator of the form$$ Ly = y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) $$with all a_k(x) being smooth. Let also M be the frozen at x=0 operator L... 0answers 67 views ### Solution to a PDE that requires fundamental Laplace solutions I would like to derive the spherical symmetric solutions (or in another words make use of the fundamental solutions) of the following PDE: \Delta u(x)+\lambda u(x)=0, \quad \lambda&... 0answers 71 views ### uniform absolute convergence for function series Consider the Theta function \theta(x,t)=\sum\limits_{m=-\infty}^{\infty}K(x+2m,t), where K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}. The \theta function is seen in the solution of the ... 0answers 171 views ### From fundamental solution to differential equation. There are various techniques to find the fundamental solutions for a given linear ordinary differential equation (ode). I am interested in reverse engineering; to find a differential equation from a ... 0answers 77 views ### Existence of fundamental solutions of Heston's PDE The Heston's PDE is a variable coefficient linear PDE. I know the existence of fundamental solution of constant coefficient linear PDE has been proofed by Bernard Malgrange and Leon Ehrenpreis in 1954 ... 0answers 36 views ### Increasing Powers of Independent Variable As Factor in Linear ODE Solution Terms According to my understanding, whenever would-be terms of a general solution to a linear ODE with constant coefficients are linearly dependent, inserting increasing powers of the independent variable (... 0answers 82 views ### Fourier transform of the Hankel function of first kind and zero order Given the Hankel function of first kind and zero order$$ 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} ...
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}{\...