Questions tagged [differential-equations]

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

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25 views

A mistake in the solution of First-order system of linear differential equations [closed]

My solution: enter image description here enter image description here Wolfram Alpha solution: enter image description here Please help me to find a mistake.
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Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation: $(\omega_0 +i \partial \bar \partial \varphi)^...
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Polynomial approximations of the vector field and distance between generated flows

Let $\textbf{h} = (h_1,...,h_n)$ be a $C^1$ system of ODEs defined everywhere on on some compact subspace $\mathbb{X} \subset \mathbb{R}^n$. Suppose we have a polynomial approximation $\textbf{p} = (...
2
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1answer
86 views

Dynamical system described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations: \begin{equation} \begin{split} ...
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0answers
30 views

Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation \begin{equation} \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
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3answers
2k views

What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE $$ \dot x(t) = A(t) x(t) \...
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1answer
74 views

Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...
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0answers
73 views

Solutions of the differential equation $f'=(f^{-1})^{[n]}$

For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation. After reading these two posts (here and here)...
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What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
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1answer
140 views

Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...
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1answer
32 views

Does a scalar LTV system with odd-periodic coefficients and even-periodic inputs have no periodic solutions?

Problem Setup Suppose we have the following scalar, linear time-varying (LTV) system with parameter $\mu \in [0,\pi[$: \begin{cases} \dot{x_1}(t,\mu) = a(t,\mu)x_1(t,\mu) + b(t,\mu) \\ x_1(0,\mu) = ...
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1answer
70 views

A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$. Consider the problem of finding $u:\Omega\times[0,T]\to\...
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27 views

Floquet theory and Poincaré theorem on the continuation of periodic orbit

I read about the Floquet theory and a theorem that it named Poincaré's theorem of the continuation of periodic orbit. Poincaré's Theorem: Consider a dynamical system depending on the parameter $\...
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0answers
61 views

how to show that the operator $\mathcal {L}$ has only one negative eigenvalue?

Consider the operator $\mathcal {L}: H^2_{per}([0,L])\subset L^2_{per}([0,L]) \longrightarrow L^2_{per}([0,L])$ given by $$\mathcal{L}(y)=w\cdot y''+(3\varphi-1)y, \; \forall \; H^2_{per}([0,L]),$$ ...
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0answers
45 views

Asymptotic behaviour of solutions to system of ODEs

Let $Y:(0,+\infty)\to\mathbb{R}^n$ be a solution to the system of ODEs $$ L[Y]=0, $$ where $L$ is a linear operator which behaves, in a neighbourhood of 0, as $$ L[Y](r)\simeq-Y''(r)-\frac{1}{r}Y'(r)...
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1answer
124 views

How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
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0answers
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Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map $$ ...
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0answers
240 views

Can a polynomial family of curves be identified by a polynomial satisfied by its derivatives?

Suppose we have a 1-parameter family of curves that foliate a subset of the plane, defined by a function that is a polynomial in the coordinates $(x,y)$ and a parameter that identifies the individual ...
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0answers
66 views

Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by $$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$ So, we are looking for solutions in the form $...
2
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1answer
199 views

A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It ...
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1answer
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Analysis of solutions to a nonlinear ODE

Consider the following ODEs: $\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$. Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only ...
4
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0answers
102 views

Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...
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0answers
75 views

Solving numerically a linear ODE

I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes. Motivated by some problems in digital signal processing, I ...
3
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2answers
176 views

Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations

In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...
20
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2answers
1k views

Intercept the missile

A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal . ...
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0answers
61 views

Solution existence for two-dimensional parabolic PDEs

I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system $$ f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
4
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1answer
99 views

Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?

For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...
2
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1answer
171 views

Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -is+\alpha r \\ i' &= i s - \gamma i\\ r' &= \gamma i-\alpha r \end{align} \...
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0answers
39 views

How to solve a system of second order ODE from time t = T to t = 0

I have a system of second-order ODEs $$ \mathbf{M\ddot{x} + C\dot{x} + Kx = f} $$ I want to know some good numerical methods to solve this system of the equation given the initial conditions at time $...
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0answers
39 views

Obtaining generating function for multivariate recurrence with non-constant coefficients

Consider a second order recurrence of the form below for some fixed $n$ \begin{align} (n+s-1) A_{n}(k,s) &= (n-k+1) A_{n}(k-1,s-1) + (k+s-3) A_{n}(k,s-1) \\ A_n(k,s) &= 0 \tag{when $s < k$ ...
5
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2answers
273 views

Name for a standard trick to construct a diffeomorphism

The following construction is standard, and it deserves a name. Suppose we need to construct a diffeomorphism from a manifold $M$ to itself with some additional properties. Observe that the flow $\...
4
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1answer
89 views

Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

There are some known criteria for the Sturm-Liouville Problem \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \...
4
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0answers
73 views

Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem. Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup of $\mathcal{B}^{-1}$ (the group of ...
24
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1answer
768 views

Solving a delay-differential equation related to epidemiology

For some inexplicable reason, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the following system of first-order ...
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0answers
66 views

ODE operator splitting with second order time discretization not possible?

I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach: $\frac{\partial f}{\partial t} = A(f) + B(f)$ Let's assume that $A$ and $B$ are very simple: $\...
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1answer
94 views

restricted three body problem equations of motion using particle distances and one angle variable

If we solve numerically a three (or $N$) body planar problem, it's easy to calculate the distances of the bodies as function of time. Conversely if we know the interparticle distances as functions of ...
0
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1answer
79 views

A solution of a system of equations that involve directional derivatives

[Edited on 29-March-2020 to make the question clearer] Let $f, g : [0,1]^2 \to \mathbb{R}$ be two smooth functions, which are strictly increasing and concave in each coordinate. That is, for every $0 ...
3
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0answers
96 views

Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
2
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0answers
57 views

Convergence rate of Toda/Morse flow

Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow \begin{align} \frac{dA}{dt} &= \left [ C\circ A , A \right ] \\ A(0) &= A_0 \ . \end{align} ...
2
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0answers
35 views

When does the PDE $F\cdot \nabla u+Gu+H=0$ admit a global solution $u:\mathbb R^3\to\mathbb R$?

Let $F:\mathbb R^3\to \mathbb R^3$ and $G,H:\mathbb R^3\to\mathbb R$ be some given smooth maps. In order for the PDE $F\cdot \nabla u+Gu+H=0$ to admit a global solution $u:\mathbb R^3\to\mathbb R$, ...
4
votes
1answer
106 views

Obstruction to the existence of a globally defined integrating factor

Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The Frobenius Theorem implies that, near each point of $U$, $\omega$ may be written as $g\,{...
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0answers
63 views

Intuition from Hopf lemma (boundary point lemma )

Consider the classical boundary point lemma: Let $L$ be an elliptic operator. Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
3
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1answer
194 views

Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
1
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0answers
43 views

wave equation with non-smooth coefficients

Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
1
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2answers
255 views

Simple bound on $\log(x)/x$

I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...
7
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0answers
152 views

No intermediate denominators growth for holonomic functions?

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form $$ a(n+k) = \sum_{i=0}^{k-1} p_i(...
13
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2answers
639 views

Solution to differential equation $f^2(x) f''(x) = -x$ on [0,1]

I'd like to solve a differential equation $$ f^2(x) f''(x)=-x $$ where $f(x)$ is defined on $[0,1]$ and has a boundary condition $f(0)=f(1)=0$. I somehow found out that the solution is fairly close ...
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1answer
57 views

Lorenz ODEs with negative parameters

Consider the Lorenz system $$\dot{x}(t) = \sigma(y-x) \, ,$$ $$\dot{y}(t) = x(\rho-z) - y \, ,$$ $$\dot{z}(t) = xy-\beta z \, .$$ Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be ...
1
vote
0answers
61 views

Fixing constants of a series solution of a fourth-order PDE

The following is the PDE I want to solve, $$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}...
0
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1answer
128 views

Conditions to determine sign of real roots

From a delay system, I obtain the following as part of a characteristic equation: $$f(\lambda) = \lambda - a + be^{-c\lambda},$$ where $a, b,$ and $c$ are positive number and $a<b, ac<1$. My ...

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