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

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16 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
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1answer
80 views

Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot ...
1
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0answers
46 views

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...
5
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1answer
166 views

Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $

I am looking for a reference to the following problem: Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices $A(t)$ in $\mathbb{R}$ with hyperolic limits at $\pm \infty $. ...
3
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0answers
58 views

Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul ...
3
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1answer
178 views

How to learn concepts of Functional Analysis which are common in PDE

I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
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50 views

Are these two difference scheme of the same differential equation equivalent?

I have been doing numerical simulation of the ODE below for days: $$i\frac{dc_n}{dz} = -\sigma(c_{n+1}+c_{n-1})+(-1)^n\delta c_n$$ I tried two different difference scheme of this ODE, which in my ...
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2answers
912 views

Solution of Heat equation with Neumann BC in an arbitrary domain

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary. Is this true: Any solution ...
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0answers
14 views

First Order Linear Differential Equation [migrated]

I'm having trouble solving this first order linear differential equation. I know the answer (thanks Mathematica!), but I'm having trouble with the steps. The equation is, $ \begin{align*} y' + ...
1
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1answer
53 views

Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko. I can not solve the problem 11.6 that says Consider ...
-1
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1answer
63 views

PDE with harmonic function [closed]

I'm looking for the solution of the following equation $$2cE^3=E_u^2+ E_v^2$$ where $c$ is a constant and $E$ is an harmonic function w.r. to the variables $u$ and $v$.
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1answer
50 views

What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for $$ u''+u'-i\lambda V(x)u=0, \, x\in [0,1], $$ What is the ...
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0answers
26 views

Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation: $$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with $$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...
7
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748 views

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
4
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0answers
249 views

Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
2
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1answer
136 views

Differential equations → predicate logic mapping

I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark): I was gonna say, there was a book I ...
2
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0answers
41 views

1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...
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0answers
19 views

converse lyapunov theorems for differential inclusion

Let $A$ be a global attractor of a differential inclusion $$\dot{x}(t) \in h(x(t))$$. I am interested in knowing converse lyapunov theorems under this condition. Thanks
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0answers
16 views

existence and smoothness of separable PDE

Suppose a PDE can be separable into ODEs by separation of variables method, can those ODEs be used to prove existence and smoothness of the original PDE. If so, how to do it?
2
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0answers
27 views

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let $f$ be ...
3
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95 views

Models used for the Zika virus?

I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ...
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0answers
35 views

Monotonocity of non-trivial solution of ODE [closed]

Consider the differential equation $$y^{''}+q(x)y=0,q(x)<0$$ where $q(x)$ is a continuous function. Let $y$ be a non-trivial solution of ODE. How to prove that $y,y^{'}$ are strictly monotone ...
7
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1answer
6k views

A particularly hard nonlinear first order ordinary differential equation

I hope this is an appropriate place to post this question, however, I am stuck on solving an apparently simple ODE. I have checked numerous texts, references, software packages and colleagues before ...
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0answers
80 views

A specific type of first-order nonlinear ordinary differential equation

I am trying to divide $\mathbb{R}^+\times \mathbb{R}^+$ into some curves so that the integration of the function $ h(x)h(y) $(where $h(x)$ is a $C^1$ function from $\mathbb{R}^+\to \mathbb{R}^+$ that ...
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2answers
2k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
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5answers
1k views

Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...
2
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0answers
69 views

Deriving Milne's predictor of order four from extrapolation polynomial [closed]

I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = ...
2
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0answers
71 views

How can I prove that this D-module is free?

I have the following setup, I expect that it is studied in the theory of $D$-modules, and I apologize in advance if I am wrong. First, I have an algebra $A$ of differential operators on $n$ ...
3
votes
2answers
122 views

Many-Body Green's Functions for Interacting Systems of Fermions

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition: ...
2
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1answer
224 views

Constructing an example of Hamiltonian flow

I have this Hamiltonian flow generated by $$ h(x, \xi ) = V(x) + \sqrt{\xi ^2 + 1}, \quad x, \xi \in \mathbb{R}^3, $$ so the defining equations are $$\begin{align*}\begin{cases} \frac{dx}{dt} &= ...
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198 views

Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?
2
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1answer
252 views

Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
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1answer
46 views

Is it possible to find the final state of the bilinear ODE system?

I need to find the final state (i.e. the state at $t\to+\infty$) of the following ODE system: $$\begin{eqnarray*}\frac{dA}{dt}&=&-aAX\\ \frac{dB}{dt}&=&-bBX\\ ...
1
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1answer
101 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
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1answer
70 views

On Wazewski's theorem on system of differential inequalities

According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem: Consider the system of differential inequalities given by $$ ...
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1answer
67 views

Closed formula for a homogeneous second order linear ODE [duplicate]

Let $A, B, C\geq 0$ be constants. Is there an explicit formula to a nontrivial solution to the homogeneous linear ODE $$y''(t) -(A+B\,\sin t)\,y'(t) -C\, y(t)=0$$ for $t\in(0,2\pi)$ with periodic ...
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0answers
44 views

Explicit solution of a Cauchy-type singular integral equation with regular part

I am doing research on the Riemann boundary value problem for bi-half-planes, and in a certain case I was able to reduce this problem to a linear singular integral equation of the form ...
2
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0answers
102 views

Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...
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0answers
55 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') ...
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0answers
61 views

Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous function, $ \omega $ be a modulus of continuity and assume $$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$ whenever $ x,h \in ...
2
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0answers
60 views

Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...
0
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1answer
54 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
2
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1answer
73 views

Second order linear ODE question with boundary conditions

I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't ...
0
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1answer
79 views

Explicit solution for one-dimensional Gelfand problem

I wonder if the ODE $y''+e^{y}=a$ can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in ...
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0answers
60 views

Scattering in (pseudo-)Riemannian spaces

I will ask my question in a broad way, leaving a lot of freedom for answers. Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
8
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1answer
140 views

How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely ...
6
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3answers
223 views

Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form: $(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} ...
1
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0answers
59 views

inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define ...
2
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0answers
82 views

Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form: $\frac{d U_t}{dt} = (A + w(t)B)U_t$ where $A,B \in \mathfrak{su}(n)$ generate the ...
17
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0answers
767 views

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...