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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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Is every complex linear algebraic group a differential Galois group?

Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication. Does there always ...
Ian Gershon Teixeira's user avatar
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0 answers
45 views

A method for solving certain difference equations

I am faced with difference equations that I need to solve (I need the main term in the asymptotic developpment). I present here the order "2" case and I would like to know if my arguments ...
 Babar's user avatar
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-1 votes
0 answers
31 views

Smoothness of solutions to the classical Dirichlet problem

Let $\Omega\subset\mathbb{R}^2$ be a domain bounded by an ellipse $\Gamma$ and $f$ a function of class $C^k(Γ).$ Will it be true that the solution of the classical Dirichlet problem is also of class $...
user526214's user avatar
1 vote
1 answer
159 views

Derivatives and ODEs on Lie groups

I'll keep the question specific to the scenario I'm working with, which is the Lie group SO(3) and its Lie algebra so(3). Consider two rotation matrices $R_0$ and $R_1$, and define $R_t = \exp_{R_1} ((...
CComp's user avatar
  • 123
0 votes
0 answers
19 views

Differential-vertex-deletion equation for graph functions $f(x_1,...,x_n;G)$ on $n$ vertices

I encountered a function $f$ defined over a graph $G$ in my research which does not satisfy a deletion–contraction recurrence but an equation of the form $$\partial_k f(x_1,...,x_k,..x_n;G)=g(x_k, N_{...
Jens Fischer's user avatar
0 votes
0 answers
44 views

Improving a condition such that the function is bounded

I'm working on a problem about differential equations and I came across the following question. Suppose $f(x)$ is a continuous function such that $f(x)\ge 0$ and $f(0)\ne 0$. Let $g(x)$ be an ...
Clario's user avatar
  • 315
1 vote
1 answer
130 views

Finding a constant to bound a function

I'm currently working on a problem about differential equations and I came across the following problem. Let $f$ be a continuous function defined on $[0,\infty)$ such that $f(x)\ge 0$, $f(0)\ne 0$. ...
Clario's user avatar
  • 315
4 votes
0 answers
136 views

Estimating $p$th moment bound of error between small noise SDE and ODE

For a $d$-dimensional standard Brownian motion $W$, and a locally Lipschitz function $b: \mathbb{R}^d \rightarrow \mathbb{R}^d$, consider an SDE: $$dX_t^\varepsilon = b(X_t) dt + \varepsilon^t dW_t,\...
ehdus113's user avatar
2 votes
0 answers
76 views

Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting. Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
George's user avatar
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3 votes
1 answer
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When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?

$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation $$P(f '(x)) = Q(f(x))$$ ...
mick's user avatar
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0 answers
80 views

Generator of an analytic semigroup

Perhaps I have a naive question. My question is as follows: When we consider a Cauchy proposition of the following form: $$ \begin{cases} x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\ x(0)=...
Jaouad's user avatar
  • 31
2 votes
1 answer
406 views

Solving $\psi_{xxx} + (u(x) - (ik)^3))\psi = 0$ for $x > 0$, $k \in \mathbb C$, and $u(x)$ smooth

I was reading Initial-Boundary Value Problems for Linear PDEs with Variable Coefficients by P. Treharne and A. S. Fokas, when I came across the following ODE formulated as part of a Lax pair for a ...
Talmsmen's user avatar
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0 answers
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Is it possible that a system of differential equation has a solution in time domain but not in Fourier domain? If so, why does it happen?

I have to solve \begin{align} &\frac{\partial h'_{1,1m}(t,r)}{\partial t} + \frac{2}{r} h_{1,1m}(t,r) = 0 \label{beta_0_1}\\ &\frac{\partial^2 h_{1,1m}(t,r)}{\partial t^2} = 0. \label{...
AleNekro97's user avatar
2 votes
1 answer
132 views

Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane

Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral: $$I(v) := \int_0^1 \det(v(t),v'(t))dt$$ tells us about $v$, where $\det(v(t)...
stupid_question_bot's user avatar
0 votes
0 answers
55 views

Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold

Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...
wsz_fantasy's user avatar
0 votes
0 answers
46 views

Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem

This is the equation given ($n\geq2$) $$ \begin{cases} u_{tt}=a^{2}\left(\Delta u\right), \\ \left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\ \left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) . \end{...
Zydragon's user avatar
3 votes
0 answers
321 views

The local global principle for differential equations

Are there any good reference to tackle the problem below? Or, are there any know result? Problem Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
George's user avatar
  • 103
3 votes
1 answer
114 views

Converting an algebraic equation into a ODE

I'm working on a method to solve algebraic equations by converting them into ordinary differential equations (ODEs) and then integrating these ODEs over time. Given an algebraic equation $f(x(t), t) = ...
Joe's user avatar
  • 31
3 votes
0 answers
161 views

What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
Alexander Chervov's user avatar
3 votes
3 answers
313 views

Existence and uniqueness of solutions to a distributional ordinary differential equation

Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation $$ \dot{x}(t) = v(x(t)) $$ which I might interpret as an integral equation $$ -\...
cheshircat's user avatar
2 votes
2 answers
231 views

Is there any work on distributional vector fields?

I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we start with an integral equation, ...
cheshircat's user avatar
1 vote
0 answers
95 views

Stability of rigid bodies spinning around $z$-axis under gravity

Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
Zhang Yuhan's user avatar
4 votes
1 answer
175 views

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: $$ \begin{cases} \dfrac{dp}{dt} = -\gamma p f,\\ \\ \dfrac{df}{dt} = -c f + \gamma p f,\\ \\ \dfrac{dT}{dt} = \...
A novice's user avatar
  • 143
1 vote
0 answers
103 views

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 ...
Yifan's user avatar
  • 73
2 votes
1 answer
184 views

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 ...
George's user avatar
  • 103
3 votes
3 answers
338 views

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 $t$...
Math2024's user avatar
  • 109
0 votes
0 answers
26 views

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 ...
user1598's user avatar
  • 177
5 votes
0 answers
551 views

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? ...
Math2024's user avatar
  • 109
2 votes
1 answer
369 views

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 ...
mathoverflowUser's user avatar
5 votes
2 answers
3k views

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 ...
Moritz Firsching's user avatar
0 votes
0 answers
102 views

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
33 votes
8 answers
3k views

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&...
AlpinistKitten's user avatar
1 vote
1 answer
90 views

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 $u:[0,1]\to\mathbb{R}$ ...
gmvh's user avatar
  • 2,758
3 votes
1 answer
103 views

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 $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
Ervin's user avatar
  • 395
1 vote
1 answer
127 views

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 ...
JMK's user avatar
  • 299
0 votes
0 answers
27 views

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 ...
Josh Burby's user avatar
0 votes
1 answer
72 views

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 ...
NicAG's user avatar
  • 247
0 votes
0 answers
50 views

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 ...
GJC20's user avatar
  • 1,230
1 vote
0 answers
64 views

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 ...
Eli Bartlett's user avatar
0 votes
0 answers
76 views

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 $\mathbb R_+$...
GJC20's user avatar
  • 1,230
0 votes
0 answers
113 views

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) &...
yarchik's user avatar
  • 482
7 votes
1 answer
313 views

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 |\...
Xin Qian's user avatar
  • 125
2 votes
0 answers
62 views

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
1 vote
0 answers
133 views

$L^{p}$ 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\...
Xin Qian's user avatar
  • 125
5 votes
1 answer
373 views

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 ...
yarchik's user avatar
  • 482
0 votes
0 answers
37 views

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 \: ...
Hollis Williams's user avatar
0 votes
0 answers
45 views

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
0 votes
0 answers
133 views

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(\...
MrPie 's user avatar
  • 305
2 votes
2 answers
137 views

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 ...
lming2's user avatar
  • 45
6 votes
0 answers
211 views

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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