Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems. Are there any papers or books that ...
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13 views

reference help on regular singular points of differential equations

I'm looking for books on systems of holomorphic partial differential equations with regular singular points. I know the book 'Équations différentielles à points singuliers réguliers' by Deligne, but I ...
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25 views

Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...
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1answer
84 views

A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation $$ \frac{d x (t)}{dt} = f(x(t)) $$ with some initial condition $x(0)=x_0$ has no solution?
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0answers
97 views

Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...
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44 views

Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation $$ -\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x) $$ with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...
3
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1answer
295 views

Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$ M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0. $$ My question is to ...
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31 views

Where can I take Partial Differential Equation (Online) courses? [closed]

I need to take a PDE course with an official grade before the end of 2015. In order to fulfill a conditional offer. Does anyone know where I can take the course? (I'm in the Netherlands, so if outside ...
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55 views

Proving a differential inequality without performing iteration [migrated]

I'm seeking a better proof of the following fact: If $g$ is a non-negative bounded function, $g(0)=0$ and $g'(t)\leq \sqrt{g(t)}$ for all $t>0$, then $g(t)\leq t^2/4$. The upper bound $t^2/4$ is ...
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1answer
107 views

Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
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1answer
70 views

First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation $$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$ where $A_0, A_1, A_2, B$ are some constants, ...
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0answers
60 views

[This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question: Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as ...
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1answer
653 views

Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
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0answers
20 views

Finding incomplete geodesics [migrated]

I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic? In particular, I am trying to solve the following exercise: Consider the upper ...
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0answers
61 views

Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
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1answer
188 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
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1answer
168 views

Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...
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1answer
45 views

Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential" \begin{equation} -\frac{d^2 \psi}{dx^2} ...
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2answers
124 views

How to find an ODE with prescribed terminal values?

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
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1answer
87 views

Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE: $\frac{d U_t}{dt} = (a + w(t)b)U_t$ consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...
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0answers
41 views

Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.) Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...
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2answers
90 views

Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation: $\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$ which is essentiall in matrix notation: $\dot{\mathbf{x}} = ...
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1answer
240 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
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1answer
344 views

Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg: Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...
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116 views

Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...
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1answer
115 views

How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE $$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$ using Charpit's method. The ...
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0answers
86 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form ...
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3answers
514 views

Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
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50 views

smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as \begin{align} \mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega \end{align} where the ...
2
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1answer
131 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
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2answers
2k views

Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
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0answers
53 views

About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper: A concave–convex elliptic problem involving the fractional Laplacian - C. Brandle, E. Colorado, A. de Pablo and U. Sánchez. At the Pgs 41, 42, the ...
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50 views

regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies: \begin{equation*} b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...
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237 views

Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B. Wonderfulness of roller comes from this property ...
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1answer
89 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
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1answer
163 views

Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...
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14 views

solution to Helmholtz equation with non circular boundary

I have 2D homogeneous domain $D$ with non circular boundary $\partial D$ and I am trying to solve the Helmhotz equation $\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)$ in which $k$ ...
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0answers
87 views

Is the trivial solution the unique solution to the following initial value problem?

This question is a duplicate one asked by myself elsewhere. But there are no answers or comments so far. The initial value problem that I am considering is: $$ y'' (3y+2x)^2=3(3y'-1)(9yy' + 4xy' -y), ...
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1answer
95 views

Heat kernel for non bounded domains

consider a domain $U\subset \mathbb{R}^n$ which is not bounded and denote its boundary by $\partial U$. (The boundary I'm confronted with is actually not very irregular. Think of a smooth manifold ...
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0answers
46 views

Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
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0answers
72 views

Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
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1answer
73 views

Fundamental solution to the heat equation with zero boundary values

let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ ...
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42 views

Causal (Volterra type) differential equation with local Lipschitz condition

Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
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1answer
125 views

What are some good references on the Galois theory, factorization, or minimality of differential equations?

Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u ...
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0answers
60 views

Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
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1answer
254 views

BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better? The BDF2 method requires the ...
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0answers
85 views

Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function. When I consider price of American ...
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2answers
229 views

ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension. The first one I know is the Peano existence theorem. I ...
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1answer
153 views

Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...
4
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2answers
155 views

Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote: ...