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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Zero-curvature formulation of the Camassa-Holm hierarchy

In the book of Gesztesy and Holden (see the following article of the same authors), they state that the (stationary) Camassa-Holm hierarchy may be cast as a zero-curvature equation \begin{align} -V_{n,...
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An ODE is linear if and only if the maximal solutions are a linear space?

Let $I$ be a non trivial interval of $\mathbb R$, let $f : I \times \mathbb R^n \to \mathbb R^n$ and consider the following ordinary differential equation (ODE): \begin{equation}\tag{$\mathscr E$}\...
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Do you recognize these numbers related to the higher Airy equations?

I'm studying the higher Airy equations $$\left[\big({-}\tfrac{\partial}{\partial y}\big)^{n-1} - y\right] \psi = 0$$ under a coordinate transformation. The interesting coefficients $c_n^{(1)}, \ldots, ...
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Asymptotic behavior of system of differential equations

Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
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Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?

Let \begin{equation*} \zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}} \end{equation*} be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation \begin{...
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Regularity of Volterra convolution integral

I have a question about the regularity of the convolution integral. Let $f: [0,\infty) \to \mathcal{L}(\mathbb{R}^n)$ given by $f(t) = e^{tA}$. Let $g: (0,T) \to \mathbb{R}^n$ such that $g \in L^2(0,T;...
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Identity principle of solutions of SL-problems with matching values on open set

Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
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$f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? It was found ...
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6 votes
2 answers
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Exercise 8.13 - Brezis

Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set $$ D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0 $$ Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
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$V_{\lambda}$ is invariant under $A$ [closed]

Let $A \in M(n)$, let $\lambda \in \mathbb{R}$, let $V_{\lambda} := \ker(\lambda I - A)$ and let $x:\mathbb{R} \to \mathbb{R}^{n}$ be a solution of $\dot x= Ax$ such that $x(t_0) \in V_{\lambda}$ for ...
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Quasilinear second order parabolic equation

For the following parabolic equation \begin{equation*} \begin{split} u_t&=\frac{u_{xx}}{1+u_x^2}-\frac{1}{u}\\ &u(x,0)=\cosh x+1. \end{split} \end{equation*} How to show that $u(0,t)\sim \sqrt{...
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Eigenvalues of Sturm–Liouville operator

Can we calculate the eigenvalues and eigenfunctions of the following operator in $W^{1,2}(\mathbb{R})$? $$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
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Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds

Consider the general semilinear elliptic second-order PDE $$ u_t-\mathcal L u=f\left(t,x,u,\nabla u\right) $$ where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...
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Sufficient condition to be increasing, following a vector field

Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
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Can a holomorphic vector field have an attractor homoclinic loop?

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post Orbits space of real-analytic planar foliations One can ...
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Why was the region split in the following way?

Consider the paper Li, Yang, and Zhou - Global stability of an epidemic model with latent stage and vaccination. Reading the entire paper isn't required. At the top of page 2168 in the journal, the ...
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Kernel of the Laplacian + a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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Restore initial condition for distributions

I am a bit confused by the following question and I hope someone could help me out. Let $u$ be the solution of the following initial value problem $$ u''(t) = g(t) \; \text{ in } (0,\infty), \quad\...
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1 answer
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Existence of periodic solutions to scalar Riccati equations

Consider the periodic Riccati equation $y'(x)=y(x)^2+q(x)$ on the real line $\mathbb{R}$, where $q\in C^\infty(\mathbb{R})$ is a periodic function with period $T=1$. Suppose $q(x)$ can take both ...
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About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation $$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$ it has a solution given by the Gauss's Hypergeometric function $$_2F_1(a,b;c;z)...
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Conjugate and disconjugate self adjoint equations

I can't understand the difference between a disconjugate and a conjugate differential self adjoint equations? Any help please!
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Reference for unique classical solution to quasilinear uniformly parabolic PDEs

In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
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The ODE modeling for gradient descent with decreasing step sizes

The gradient descent (GD) with constant stepsize $\alpha^{k}=\alpha$ takes the form $$x^{k+1} = x^{k} -\alpha\nabla f(x^{k}).$$ Then, by constructing a continuous-time version of GD iterates ...
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1 answer
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Change of variable for differential equations

This question was previously posted on MSE at Change of variable for differential equations. Given the following differential equation \begin{equation} -y(\zeta) \left(\frac{d^2 y(\zeta)^{-1}}{d \zeta^...
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3 votes
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Banach's fixed point theorem for quasilinear parabolic PDEs

I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$ \begin{cases} \partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
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On an assertion in Payne's paper concerning the exterior Steklov eigenvalue problem

In the paper "New isoperimetric inequalities for eigenvalues and other physical quantities" by L. E. Payne [Communications on Pure and Applied Mathematics, 1956, 9(3): 531--542. https://doi....
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316 views

A singular stochastic differential equation

We consider the following SDE: $$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$ where $(B_t, \, t \ge 0)$ is linear Brownian motion. Let $\tau: = \inf\{t >0: X_t = 0\}$ be ...
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Why were these constants picked in this Lyapunov function and how did the author arrive at the final form of the Lyapunov function?

Consider the following paper: "A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004 The methodology is understood in this paper ...
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Bounded derivative of diffeomorphism from implicit function theorem

Let us define $f$ and a set of functions $\{g_n\}$ as follows: $$f:[0,T]\mapsto\mathbb{R}\textrm{ s.t. } f(t)>0,\forall t \in[0,T]$$ $$\{g_n:\mathbb{R}\mapsto\mathbb{R}^+\}\textrm{ a set of ...
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Time regularity of traces

I have a question about the time regularity of the traces in one dimension. Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
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1 vote
1 answer
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What are semipositone functions? [closed]

I am reading a paper on multiple solutions for boundary value problems of fourth-order differential systems. In the paper, there is a nonlinear term $f\in C\left[(0,1)\times \mathbb{R}^+\times \mathbb{...
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2 votes
1 answer
87 views

Decay rate for a small perturbation of a simple linear ODE

MOTIVATION. Let $f:[0,+\infty)\to \mathbb{R}$. Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$. This property is preserved if we apply an ...
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3 votes
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A foliation version of S.Husseini counter example in fixed point theory

In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977 "The Products of Manifolds with the f.p.p. Need Not have the f.p.p" who gave an example of two ...
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10 votes
1 answer
219 views

Non-integrability of Abel's equation

I frequently encounter in the literature the statement that Abel's equation $$\frac{dy}{dx}=x+y^3$$ is not integrable. This is always stated without reference. My questions are a) What is the precise ...
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1 vote
1 answer
181 views

How to solve a nonlinear PDE?

I want to solve the problem : $$\frac{\partial u}{\partial t}=\frac{1}{\left\vert \nabla u \right\vert^p} \operatorname{div} \left( \frac{\nabla u}{\left\vert \nabla u\right\vert^p}\right) $$ We ...
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1 answer
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How to solve this delay differential equation?

How to solve this DDE: \begin{align} \frac{1}{N_t} \frac{dN_t}{dt}=r\left(1-\frac{N_{t-\tau}}{K}\right) \end{align} where $N_0,K,r,\tau$ are constant? This differential equation is based on logistic ...
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Amplitude and phase of trajectories of non-linear dynamical system

Assume we have a 2D non-linear dynamical system with 2x2 Jacobain matrix (at the steady state). We also assume that we have periodic solutions, i.e. we have complex eigenvalues of the form $\pm i \phi$...
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3 votes
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Does there exist a framework for determining if a power series is "differentially algebraic"

It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
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1 answer
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Why is the largest invariant set the following?

Consider this paper: Hai-Feng Huo and Li-Xiang Feng, "Global stability for an HIV/AIDS epidemic model with different latent stages and treatment", Applied Mathematical Modelling, Volume 37, ...
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Existence of solutions to $\alpha(s)=\mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt$

Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process $$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$ where $Y_0>0$ has a ...
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0 votes
1 answer
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Existence of a global solution to a differential inclusion that does not blow up

Let $\dot{x}(t) \in F(x(t))$ be a differential inclusion, with $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ an uppersemicontinuous, convex and compact valued set-valued map. On Wikipedia it is ...
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1 vote
1 answer
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Second order matrix differential equation in the space of symmetric positive definite matrices

In the construction of interpolations in the space of Gaussian measures, I encountered a second order matrix differential equation in the set of symmetric positive definite matrices $\mathbb{S}_+^d\...
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3 votes
2 answers
368 views

Is the function $F(x) = \exp(x) + \exp(\exp(x))x$ a hypertranscendental function?

The function $F(x) = \exp(x) + \exp(\exp(x))x$ plays a role in the formulation of the Lagarias inequality: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ If we put $x = \log(H_n)$, then this inequality ...
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1 vote
0 answers
59 views

Fitting a model

I have a function expressed as the ratio of two exponential series with certain parameters $$\frac{\sum\limits_{j=1}^{i-1} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i-1} (b^j-b^l)}}{\sum\limits_{j=1}^{i}...
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2 votes
0 answers
75 views

A complicated equation of integro-differential type

Consider the following equation of $\beta$ : $\beta(0)=2$ and $$\dot\beta(t)\beta(t)^2 = \frac{1}{2}\int_0^t\frac{\dot\beta(s)(t-s)}{A_{\beta}(s,t)^{3/2}}f\left(\frac{t-s}{\sqrt{A_{\beta}(s,t)}}\right)...
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1 vote
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Show that an integral operator with Bessel function kernel is bounded on $L^2(0,\infty)$

Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2), $$ J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}. $$ Put $u_0(r) = r^{1/...
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3 votes
1 answer
92 views

Factoring higher-order differential operators

I have been researching various methods for solving differential equations. In particular, I want to better understand the factoring approach. For example, if we want to solve a general second order ...
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2 votes
0 answers
44 views

Harmonic function over a square with linear Neumann boundary conditions

For a rectangle with height 1 and length 2, here is the unique numerical solution (showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum) to ...
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1 vote
1 answer
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Why should we model infectious diseases with fractional differential equations?

With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
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0 answers
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Error estimates for orthogonal polynomial approximation

tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials? There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
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