Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

An ordinary differential equation of order n will in general be satisfied by n linearly independent functions known as fundamental or elementary solutions. A general solution to the differential equation may be constructed as a linear combination of the fundamental solutions, whose coefficients are constants of integration.

232 questions
16
votes
2 answers

Fundamental solution for Helmholtz equation in higher dimensions

The fundamental solution for Helmholtz equation $(\Delta + k^2) u = -\delta$ is $e^{i k r}/r$ in 3d and $H_0^1(kr)$ in 2d (up to normalization constants). Is there an explicit expression (eventually in terms of special functions) for the fundamental…
user71623
  • 297
8
votes
2 answers

Must any nth order homogeneous ODE have n solutions?

I am quite confused about ordinary differential equations and the number of solutions they have. In particular, it seems that an nth order homogeneous differential equation has n solutions, not more or less. I cannot figure out why this would be…
Meep
  • 3,267
7
votes
1 answer

$N$-dimensional Heat equation + BC's

Problem I have to solve the nonhomogeneous classic problem $$\left(P_{1}\right)\;\,\left\{ \begin{aligned} u_{t}\;-\; \Delta u\; &= \;f& &\textrm{on}\;\;\; \Omega\times\left(0,\,\infty\right) &\\ u\; &=\; 0 & &\textrm{on}\;\;\;…
user588775
6
votes
0 answers

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). Now, while reading about the heat equation I…
Alex M.
  • 35,927
6
votes
1 answer

Connection Between Fundamental Solution and Symmetries of PDE

The typical derivation of the fundamental solution of Laplace's equation is to look for a radially symmetric solution because the Laplace equation has radial symmetry, and a similar heuristic can be used to derive the fundamental solution of the…
JLA
  • 6,932
6
votes
2 answers

The Method Of Frobenius

The ODE $xy'' + y = 0$ has a real degeneracy. Use The Method Of Frobenius to find a fundamental set of solutions. Here is the procedure, as I understand it: 1) Plug the guess $y = x^s \sum_{n = 0}^\infty a_n x^n$ into the ODE and do the…
user10478
  • 2,118
6
votes
1 answer

Fundamental solution of heat equation with zero initial condition

Consider the following problem: $u_t = ku_{xx}$ on the semi-infinite strip $S = [0,\infty)\times [0,\infty)$, with the $zero$ initial condition $u(x,0) = 0,\, u(0,t)=g(t).$ I tried both the separation of variables and the fundamental solution…
dezdichado
  • 14,761
  • 2
  • 26
  • 56
6
votes
0 answers

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 simple but difficult to solve.
LCFactorization
  • 2,102
  • 18
  • 23
5
votes
0 answers

What is a distinct feature of an ambiguous result

This question comes from my experience in radar signal processing. As I am going more deep into the theory of sampling, statistical signal processing and estimation theory in general, I have a very silly but important mathematical question that I…
CfourPiO
  • 119
  • 9
5
votes
2 answers

What is the solution to this non-linear second order differential equation?

I'm trying to solve the following non-linear second order differential equation: $$\tag{1} \frac{d\, }{dx} \Bigl( \frac{1}{y^2} \, \frac{dy}{dx} \Bigr) = -\, \frac{2}{y^3}, $$ where $y(x)$ is an unknown function on the real axis. I already know the…
Cham
  • 322
5
votes
1 answer

Is there a Green function for the p-Laplacian?

The Green's function is defined for a linear differential operator $L$ as the solution of the equation $LG = \delta$, where $\delta$ is Dirac's delta function. A direct consequence of the definition of $G$ is that the solution of the problem $Lu =…
korina
  • 203
5
votes
2 answers

Method of solving no-homogeneous recurrence equation

I need to obtain a closed form of $M(t)$, satisfying the following recurrence equation: $$M(t+1)=a+bM(t)+\frac{c}{t+1}\sum_{t'=0}^tM(t')+df(t)$$ Where $f(t)$ is a known function and $a$, $b$, $c$ and $d$ are known constants. And with the initial…
Ana S. H.
  • 953
5
votes
2 answers

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 function (fundamental solution). In a distributional…
jinawee
  • 2,667
4
votes
1 answer

Green's function for $\frac{\partial}{\partial\bar{z}}$

I've read some complex analysis texts and often there is some appeals to Green's theorems when proving facts about contour integrals of holomorphic functions yet there seems to be a lack of appeals to Green's functions (at least explicitly). We know…
Cameron L. Williams
  • 30,485
4
votes
1 answer

Exactly one homogeneous differential equation of second order to given fundamental solution

I am working on: Let $\phi_1,\phi_2$, so that $\phi_1(x)\phi_2'(x)-\phi_1'(x)\phi_2(x)\neq 0.$ for all $x\in\mathbb{R}$. Then there exists exactly on homogeneous differential equation of second order $$y''(x)=f(x)y'(x)+g(x)y(x)$$ so that the…
john_psl1298
  • 387
1
2 3
…
15 16