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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The stability of the equilibria of a non-linear ODE system

I have the following coupled non-linear ODE system, which describes a biological system: $$\frac{dp}{dt} = -\gamma p f,$$ $$\frac{df}{dt} = \gamma p f,$$ $$\frac{dT}{dt} = \left( 1 - \frac{k}{T} \...
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$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?

Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$. \begin{align} &f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
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Frobenius theorem and the size of integral manifold

Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and $[X,Y]:=XY-YX=0$. Then by ...
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Generalized Fuchsian-type PDE?

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
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Neumann vs Stefan conditions in Free Boundary Problems

Suppose I have a free boundary problem of the form in which, in an interval $(\alpha(t),\beta(t))$, we have $u_t(x,t) = \mathcal{L}u(x,t)$, and $u(x,t)=0$ for $x\notin(\alpha(t),\beta(t))$, for some ...
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A 4th-order linear PDE

I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $x^3 f_{xxxt}+ f =0$ Does anyone know if this type of PDE already appeared in the literature? ...
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The Fourier transform of the Liouville function?

The Liouville function in number theory is defined as: $$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$ Taking the discrete time Fourier transform and then taking the ...
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Analysis of an autonomous plane system

I am studying a plane autonomous system described by the differential equations: $$\frac{dx}{dt} = f_\delta(x,y), \quad \frac{dy}{dt} = g(x,y), $$ where $f_\delta,g$ are rational functions in $x,y$ (...
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Nicer expression for 2.1369288...?

In Drift Analysis and Evolutionary Algorithms Revisited by Johannes Lengler and Angelika Steger in Theorem 10, there is mention of a constant "$2.2$", and in the proof it becomes apparent ...
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Who first gave a result stronger-or-equal to this one on ODEs

After some thinking I've come to the following conclusion. Consider the initial value problem $$\text{(P)}\begin{cases}x'(t)=f(t,x(t)),\quad t\geq t_0\\x(t_0)=x_0 \end{cases}$$ where $f:D\subset\...
Luca T. Castrillón's user avatar
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Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
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Bound on $L^1$ norm of solution of two-point boundary value problem

This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
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Fréchet-valued symbols

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
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Existence of solution to nonlinear first order PDE with C^1 bounds

I'm looking for general existence of a PDE of the form $$ f: U \times [0, \delta) \to \mathbb{R}$$ $$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$ where $f(p,0)$ is prescribed and $F$ is non-linear ...
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Tableau and its first prolongation for linear Pfaffian systems

This question concerns characterization of tableau associated with an exterior differential system (EDS). On the one hand, we have prop 4.2 in the EDS book by Bryant et al.: Given an EDS on a manifold ...
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Same occupation measure $\Rightarrow$ same trajectory

Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ The occupation ...
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Question on the modelling of (viscous) fluid in a bag with holes

Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture) What is the corresponding PDE to model the ...
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Looking for examples of 3rd-order contact transformations

In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated: As a final ...
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What is the PDE corresponding to this weak formulation?

Consider a flow $(\mu_t)_{t\ge 0}$ such that every $\mu_t$ is a probability on $\mathbb R_+$; $\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
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One solution is known, how to find another one

This question is driven by pure curiosity as for all practical purposes I can generate numerical or series solutions of the given ODE \begin{align} p(a,u) [a'(u)]^2+q(a,u)a'(u)+a(u) &= 0,\\ a(0) &...
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Estimates of $\Delta|\nabla u|$ for harmonic function $u$

The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$, $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
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On improving the regularity of solutions to nonlinear parabolic pde

There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
Agustín Oyarce's user avatar
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$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$

I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
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Asymptotic solution of a system of ODEs

I have asked this question on math.stackexchange, however, have not got any answer. Therefore, I suspect that this system of ordinary differential equations cannot be solved analytically. But I still ...
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System of equations with one integral equation

Start with a system of three equations such that two of the equations are ordinary or partial differential equations, but one of them is an integral equation as follows: $C = \int_{0}^{\infty} X \: ...
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Numerically finding periodic solution to Riccati equation with a scalar unknown parameter

We have a Riccati equation with an unknown scalar parameter $a$. It is proven to have unique real solution pair $\{y(x), a\}$ exists so that $y$ is periodic: $$ y'=ap_{0}(x)+q_{0}(x)+q_{1}(x)y+q_{2}(x)...
That Frank Guy's user avatar
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Relative bounds for vorticity

Write the vorticity equation as \begin{equation}\label{Eq20} \begin{split} \dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
free_lancer's user avatar
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2 answers
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Upper bound estimation for second-order variable-coefficient ODE

I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of $y(t)$ s.t $|y(t)|\le M$. The equation in question ...
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Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
Brian Hepler's user avatar
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How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?

Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
boundary's user avatar
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Complex integration in a separable EDO for an eigenvalue problem

I'm trying to prove a result from Leon Cohen's book "Time-frequency Analysis", Chapter 18. Namely, I want to verify the solution of the eigenvalue problem $$\mathcal{C}s(t) = c s(t)$$ for ...
Bernardo Boechat's user avatar
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A self-consistent equation that turns into a differential equation

Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation \begin{align*} & f(x,y) = \frac{1}{1-y} + ...
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How to force my differential equations give a bounded solution?

I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation): $$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...
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On the decay of a supersolution of a Sturm Liouville eigenvalue problem

I am not really familiar with Sturm-Liouville theory, so possibly the answer to my question is rather trivial. Consider the SL problem \begin{equation*} L \varphi (x) : = \frac{d}{dx}\Big[ (x^2+1)^\...
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Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
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Can a solution to this parameterized ODE converge to zero?

Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$? $$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$ We are also given y(0) = 2/3, $...
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory

Let $M$ be a compact three-manifold. I am interested in the following equation on $M$: $ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$ together with the following condition:...
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3 votes
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Distance between solutions of differential inclusions

Suppose that we have two differential inclusions $$\frac{dY^1}{dt}(t)\in b_1(Y^1,t)$$ with $Y^1(0)\in Y_0^1$ and $$\frac{dY^2}{dt}(t)\in b_2(Y^2,t)$$ with $Y^2(0)\in Y_0^2$. Can we then control $d(Y^1(...
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Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
11 votes
4 answers
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Interesting examples of systems of linear differential equations with constant coefficients

In this paper, Gian-Carlo Rota wrote: A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal processing, even in ...
Michael Hardy's user avatar
1 vote
1 answer
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Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system? that is, does ...
li ang Duan's user avatar
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Nonnegative eigenvector form the point of view of variational characterization

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$ $(N\geq 3)$. We denote by $G(x, y)$ the Green function for the boundary value problem $$ -\Delta_x G(x, y)=c_n \delta(x-y) \quad \text { in } \Omega, ...
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Sufficient conditions for chain recurrent set equal to set of non wandering points

Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
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Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?

The special form of Riccati equation $$ \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} $$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$. The sufficiency is ...
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Is the global solution to this ODE bounded?

Consider $$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$ where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
tony's user avatar
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6 votes
1 answer
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Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
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5 answers
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PDEs and algebraic varieties

Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
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Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
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Coefficients in the series expansion of a central manifold are all zero

I have a system of 4 ODEs, which linearized around the origin gives $$ \begin{align} &\dot{q_1}=a\, q_1\\ &\dot{q_2}=b\,q_2\\ &\dot{q_3}=0\\ &\dot{q_4}=c\,q_4 \end{align} $$ with $a$, $...
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