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

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6
votes
1answer
307 views

Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation: $y=y''.(1+y'^{2})^{-3/2}$ This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...
-5
votes
0answers
40 views

Difficult derivative of the log of a function [on hold]

Can someone help me figure out how the derivative of log p(x) wrt. theta becomes to solution below ? $ -\frac{\delta logp(x)}{\delta \theta} = \frac{\delta F(x)}{\delta \theta } - \sum p(x) ...
-1
votes
0answers
38 views

Floquet solution to Mathieu equation in terms of Mathieu sin and cos [closed]

In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution I want to know how the Floquet solution is plotted. One way I am thinking is to write Floquet solution in terms of the ...
2
votes
1answer
77 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
1
vote
0answers
26 views

Differential inequalities for a strictly diagonal dominant system of linear ODEs

Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
7
votes
0answers
231 views

Partial differential equations outside of academia [closed]

I've seen a number of career/jobs questions on mathoverflow before, so I thought I would ask. Please excuse me if this isn't the best place for this specific question. Lately I've been really ...
5
votes
1answer
233 views

Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

Main Question: What Is the correpondence between flows and vector fields in algebraic geometry? Here is a more precise statement could be an answer If it was true (I have no idea it is): ...
7
votes
0answers
819 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 ...
-3
votes
0answers
24 views

unable to solve this differential equation [migrated]

$\frac{dy}{dt}=(1-y)(1+6y)$ How can I solve this, please help. I tried using Classical Runge-Kutta , but the results are not satisfying. Can anyone suggest some other method?
4
votes
0answers
187 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 ...
3
votes
0answers
72 views

Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions $$ \nabla^2-x^2, $$ where, $$ x^2=\vec{x}.\vec{x} $$ It seems quit simple and one would think there should already be solutions ...
0
votes
0answers
147 views

Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
4
votes
0answers
68 views

Periodicity of KdV equation in relation to zero-curvature equation

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix ...
0
votes
0answers
31 views

Reparametrisation of a PDE with arclength

Suppose I have the following PDE: $\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$ I notice ...
0
votes
1answer
53 views

Uniqueness of Newton (modulo a constant) series on a compact set

Good morning everybody. My question is as follows: Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$. Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It ...
1
vote
0answers
55 views

Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?

Let $(M, \nabla)$ be a manifold together with a connection on $TM$ and let $T$ be a tensor field on $M$. Suppose the pseudogroup $\Gamma$ of locally defined smooth maps, that simultanously preserve ...
1
vote
1answer
45 views

Is this non-linear system of differential equations tractable by other means than numeric approximation and dynamic analysis?

Is there any way to solve the following system of non-linear differential equations exactly? $x'(t) = x\times(y - \frac{1}{3(t + C)})$ $y'(t) = -\frac{1}{3}x^2 - \frac{y}{t + C}$ Here $x$ and $y$ ...
0
votes
0answers
57 views

Linearization of specific plane vector field

I have a vector field $v = (f(x,y), \alpha y)$, such that $f(0, 0) = 0$ and $df (0, 0) = (1, 0)$. When is smooth linearization possible and when is it not? I only see obstacles in the form like ...
2
votes
0answers
100 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 ...
4
votes
1answer
266 views

Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$. Then we know that the eigenvalues of $-\Delta$ form an ...
-4
votes
1answer
111 views

Existence and uniqueness of solutions for a system of first order PDEs [closed]

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
0
votes
1answer
64 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 ...
1
vote
1answer
112 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
vote
0answers
57 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
votes
1answer
179 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 $. ...
4
votes
0answers
86 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
votes
1answer
195 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, ...
1
vote
2answers
917 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 ...
1
vote
1answer
55 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
votes
1answer
79 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 $v$.
0
votes
1answer
53 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 ...
0
votes
0answers
28 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 ...
4
votes
0answers
255 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
votes
1answer
137 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
votes
0answers
42 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 ...
2
votes
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 ...
1
vote
0answers
36 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
votes
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 ...
1
vote
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 ...
8
votes
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 $L^{\infty}$ ...
36
votes
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 ...
3
votes
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
votes
0answers
74 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
127 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
votes
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} &= ...
6
votes
0answers
203 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
votes
1answer
257 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 ...
1
vote
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
vote
1answer
105 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 ...
1
vote
1answer
78 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 $$ ...