Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
594
questions with no upvoted or accepted answers
2
votes
0
answers
52
views
A problem involving a highly non linear system of PDE
Is there a unique smooth function $u:\mathbb{R}^3:\rightarrow \mathbb{R}^3$ such that
$$
\begin{eqnarray}
\dfrac{\partial}{\partial t}\left(u_1^2+u_2^2+u_3^2\right) & = & \mu \left(u_1\dfrac{\...
- 205
2
votes
0
answers
104
views
How to understand the constant rank theorem for semilinear elliptic equations
Let $u$ be a solution to the equation $$\Delta u=f(u,\nabla u)$$ where $f>0$ is a smooth function with $f f_{uu} \le 2f_u^2$. The seminal constant rank theorem states that if $D^2 u$ is positive ...
- 1,320
2
votes
0
answers
100
views
What is known about gradient descent on quadratic models (not loss functions!)
Let $\mathcal X$ be any set, and $f:\mathcal X\times\mathbb R^n\to\mathbb R$ be a differentiable model, meaning that for any fixed first argument, $f$ is differentiable in its second argument. Then we ...
- 633
2
votes
0
answers
182
views
Boundedness of solutions to second order ODE
Let $q(x)$ be a probability density function over $[0,1]$. Let $\lambda > 0$ and $f: [0,1] \to \mathbb{R}$ be any solution to the following ODE:
$$
\lambda f''(x) + q(x) f(x) = 0, \text{for all }x \...
- 505
2
votes
0
answers
57
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,089
2
votes
0
answers
67
views
Optimal/Stable/...? discretization of second (higher) order ODEs
I will phrase the question only for second order ODE's although the same question might come up for higher orders.
ODEs of the form
$$
f(t, x,\dot{x}) = \ddot{x}
$$
are usually converted to first ...
- 357
2
votes
0
answers
104
views
Free energy of topological recursion
In arXiv:1805.10945, arXiv:1810.02946, arXiv:2005.08957 the free energy of topological recursion is shown to satisfy differential/difference equations for some specially chosen curves.
Are there other ...
2
votes
0
answers
73
views
Uniform bound on a certain family of hypergeometric functions
We have the following problem, which we can't solve.
Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
- 5,492
2
votes
0
answers
81
views
Gronwall-type bound for a mix-effect inequality?
This popped up in my research: we have the following mix-effect inequality that $\forall T \geq 1$
\begin{equation}\tag{*}
Y(T) - \frac{1}{100T^2}\int_1^T[\alpha^2 + e^{-(T - t)}]Y(t)dt
\lesssim \...
- 719
2
votes
0
answers
59
views
How to constrain the integral of the control function to a fixed value?
In the following, I am referring to the general "Hamiltonian control theory" using the conventions defined here.
I am working on a very simple S($x_1$)I($x_2$)R($x_3$) model for infectious ...
2
votes
0
answers
87
views
Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...
- 4,575
2
votes
0
answers
130
views
Support of a fundamental solution of wave equation
The solution of the wave equation
$$
\Box E = \delta
$$
is
$$
E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{...
- 21
2
votes
0
answers
51
views
References for generalized confluent hypergeometric differential equation
According to Wolfram, a generalization of the confluent hypergeometric differential equation is given by:
$$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{...
user161091
2
votes
0
answers
211
views
Show that the manifold interior is invariant under this flow
Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...
- 161
2
votes
1
answer
447
views
Flow induced by differentiable velocity field is differentiable
Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
- 161
2
votes
0
answers
42
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 $u:\mathbb R^3\to\mathbb R$, ...
- 1,610
2
votes
0
answers
72
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((...
- 4,089
2
votes
0
answers
47
views
Initial value problem with heterogeneous initial values
In all the references I checked the standard initial value problem for an ODE is stated as:
\begin{equation}
\begin{cases}
y'=F(y,t)\\
y(t_0)=y_0
\end{cases}
\end{equation}
for some $F:\mathbb{R}^{n+...
- 43
2
votes
0
answers
207
views
Lift the relative Frobenius automorphism to zero characteristic
Let $X$ be a algebraic variety of finite type over $\mathbb{Z}$. Let $\mathcal{F}$ be a foliation in codimension one over $X$. Let $X_p$ and $\mathcal{F}_p$ be the reductions modulo $p$ of $X$ and $\...
- 527
2
votes
0
answers
102
views
Energy functional continuous with respect of time $t$
I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation
\begin{...
- 265
2
votes
0
answers
43
views
Understand the condition of transcritical bifurcation (Crandall-Robinowitz) geometrically
Consider the dynamical system $\dot{x}=F(x,\lambda), x\in\mathbb{R^n}$, and let $F(0,\lambda)=0$ for some neighborhood of $\lambda_{0}$, the transcritical bifurcation arises if we have $w\frac{\...
- 125
2
votes
0
answers
121
views
GUE, tau-function of Painlevé II, and an article of Forrester-Witte
Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a ...
- 549
2
votes
0
answers
385
views
Lax Milgram for non coercive problem?
I obtained the variational form of my problem. and the bilinear form is below.
Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have
$$a(u,v)=\int_\Omega u(t)...
- 55
2
votes
0
answers
36
views
Is there an extension of the Kovacic algorithm to handle algebraic coefficients?
Kovacic's algorithm solves second-order linear homogeneous differential equations with rational function coefficients.
I'm wondering if anybody has extended this algorithm to handle algebraic ...
- 415
2
votes
0
answers
66
views
Annihilator of the of the generating function not holonomic
The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
- 685
2
votes
0
answers
96
views
Limits of the wave equation with piecewise constant propagation speed
This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty.
Consider a wave equation
$$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...
- 201
2
votes
0
answers
522
views
Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
- 611
2
votes
0
answers
109
views
Solution of equation on vector field
I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
2
votes
0
answers
103
views
State of art of hyperfunction theory in solving partial differential equations
What are the advantages of 'representing distribution(or more generalized functions) as boundary value of holomorphic functions', and their use in solving pde?
- 39
2
votes
0
answers
139
views
Global solution of second order ODE defined on riemannian manifold
Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
- 345
2
votes
0
answers
149
views
Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?
Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
2
votes
0
answers
211
views
The regularity of ODE with Zygmund coefficients
A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...
- 896
2
votes
0
answers
149
views
Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
2
votes
0
answers
105
views
Bifurcations due to a nonlinearity parameter
Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...
- 243
2
votes
0
answers
72
views
Floquet stochastic process
Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
- 205
2
votes
0
answers
161
views
A question about whether an operator can be lipschitz or not
Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...
- 361
2
votes
0
answers
188
views
Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems
I have a query regarding two equalities in the lemma in the book.
But first I'll provide two definitions that one needs for this lemma.
Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...
- 1,524
2
votes
0
answers
59
views
Stability of ODEs with exponentials in the vector field
What is known about fine stability properties of ODEs of the following kind?
$$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$
where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...
- 225
2
votes
0
answers
55
views
Region of attraction of simple ODE with perturbation
Consider the following simplest example:
$$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{equation}
\begin{aligned}
&\dot{x} = x(x-1)(x+1)\\
&...
- 345
2
votes
0
answers
108
views
Does a smooth dynamical system always come with a metric
Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me.
Question: Consider a smooth dynamical system $...
- 21
2
votes
0
answers
168
views
Stochastic Approximation in Reproducing Kernel Hilbert Space
Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
- 1,127
2
votes
0
answers
173
views
Location of the endpoints of two parametric curves
I have two curves, $C_1$ and $C_2$ parametrized by $\theta$, the angle of the outward normal with the X-axis.
$C_1$ is given by the following equations (say $r = 0.2$):
\begin{align*}
\frac{dx}{d\...
- 543
2
votes
0
answers
141
views
When are Green's functions causal convolution kernels
Let $L$ be a linear differential operarator acting on distributions over $\mathbb{R}$ and $G(t, s)$ be a Green's function, i.e., a solution to $LG(t, s) =\delta(t-s)$.
$G$ is said to be causal if $G(...
- 547
2
votes
0
answers
112
views
Holonomic modules and Holonomic functions
Let
$$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}\prod_{i=1}^{d}G((i-k)h) . $$
I have proved that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}\in \mathbb{C}(h)[[x]]$ is holonomic and arrive at a ...
- 685
2
votes
0
answers
82
views
Stochastic Approximation Algorithms Converging to Local Equilibriums
Consider the stochastic iterative updates
\begin{align}
\theta_{t+1} \leftarrow \theta_t + \alpha_t \cdot \left [ h(\theta_t) + M_t \right ],
\end{align}
where $\theta_t \in \mathrm{R}^d$, $h \colon ...
- 1,127
2
votes
0
answers
47
views
Factorization of linear ordinary differential operators
I was looking for references that give a detailed survey of techniques of factorization of linear ordinary differential operators. Specifically if there are references that do a complexity analysis of ...
- 131
2
votes
0
answers
163
views
Solve 4th order ODE with variable coefficients
I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam:
$u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...
- 21
2
votes
0
answers
173
views
Positively invariant with respect to nonlinear dynamics
I have the set of nonlinear differential equations describing a system I modeled for my research (spread of epidemics or information for instance):
$$\begin{array}{rl} \dot{p}(t) &= \gamma r(t)-u(...
- 21
2
votes
0
answers
234
views
A cubic system with two nested limit cycles with opposite orientations(2)
The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
- 312