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.

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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
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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.
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$N$-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
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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
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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
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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 simple but difficult to solve.
LCFactorization
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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
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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
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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
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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
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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.
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Second order linear ODE with polynomial coefficients

I am currently stuck at solving an ODE of the form \begin{equation} 0=\psi''(x)+(\varepsilon-(\alpha x^2-\beta)^2)\psi(x) \end{equation} where $\alpha,\beta$ and $\varepsilon$ are real parameters. Anticipating $\psi\sim\mathrm{e}^{-\alpha x^3/3}$…
ciabatta
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assumption such that the explicit formula of heat equation holds

In Evans PDE book, Section 2.3, Theorem 2. It says that if $f\in C^2_1(\mathbb{R}^n\times [0,\infty))$ and $f$ has compact support, then $$ u(x, t)=\int_{0}^{t} \int_{\mathbb{R}^{n}} \Phi(x-y, t-s) f(y, s) d y d s $$ is the solution of $u_t-\Delta…
Q-Y
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Green's function on unit ball

In Evans' book on PDE it is stated (p.50 in my edition) that a Green's function for solving Laplace's equation on $B_1(0)$ is given by $$G(x,y)=\Phi(y-x)-\Phi(|x|\cdot (y-\tilde x)),$$ where $x,y\in B_1(0)$, $x\not= y$, $\tilde x:=\frac{x}{|x|^2},…
JohnSmith
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The Green's function of the beam deflection equation

This is a problem in a textbook used in my class: Suppose we have an infinite elastic beam, where the deflection $u(x)$ satisfies the differential equation $$\frac{d^4 u}{dx^4}+k^4 u = > f(x),$$ where $k^4$ is a positive constant regarded as…
Olorun
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