# Questions tagged [differential-equations]

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

1,656
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 ...

0
votes

0
answers

45
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### 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 ...

-1
votes

0
answers

31
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### 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 $...

1
vote

1
answer

159
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### 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} ((...

0
votes

0
answers

19
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### 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_{...

0
votes

0
answers

44
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### 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 ...

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 $f(x)\ge 0$, $f(0)\ne 0$. ...

4
votes

0
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136
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### 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,\...

2
votes

0
answers

76
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### 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 ...

3
votes

1
answer

145
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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
$$P(f '(x)) = Q(f(x))$$
...

0
votes

0
answers

80
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### 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)=...

2
votes

1
answer

406
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### 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 ...

0
votes

0
answers

54
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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{...

2
votes

1
answer

132
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### 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)...

0
votes

0
answers

55
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### 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. ...

0
votes

0
answers

46
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### 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{...

3
votes

0
answers

321
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### 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 ...

3
votes

1
answer

114
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### 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) = ...

3
votes

0
answers

161
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### 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 ...

3
votes

3
answers

313
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### 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
$$
-\...

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, ...

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 ...

4
votes

1
answer

175
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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:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = -c f + \gamma p f,\\
\\
\dfrac{dT}{dt} = \...

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 ...

2
votes

1
answer

184
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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 ...

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 $t$...

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 ...

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? ...

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 ...

5
votes

2
answers

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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 ...

0
votes

0
answers

102
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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\...

33
votes

8
answers

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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&...

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 $u:[0,1]\to\mathbb{R}$ ...

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 $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...

1
vote

1
answer

127
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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 ...

0
votes

0
answers

27
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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 ...

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 ...

0
votes

0
answers

50
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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 ...

1
vote

0
answers

64
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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 ...

0
votes

0
answers

76
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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 $\mathbb R_+$...

0
votes

0
answers

113
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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) &...

7
votes

1
answer

313
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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 |\...

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 ...

1
vote

0
answers

133
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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\...

5
votes

1
answer

373
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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 ...

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 \: ...

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)...

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(\...

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 ...

6
votes

0
answers

211
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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 ...