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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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 ...
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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:
$$dX_t^\varepsilon = b(X_t) dt + \varepsilon^t dW_t,\...
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Solving a ordinary differential equation Problem [closed]
i totally can't figure out how to solve it. can anyone help how to induce the process to answer ?
problem down below
Find the condition of $k$ such that
$y'=\sqrt{4-xy^2}, \quad y(2)=k$
has ...
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Derivatives and ODEs on Lie groups
I'll keep the question specific to the scenario I'm working with, which is the Lie group SO(3) and its Lie algebra so(3).
Consider two rotation matrices $R_0$ and $R_1$, and define $R_t = \exp_{R_1} ((...
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Differential-vertex-deletion equation for graph functions $f(x_1,...,x_n;G)$ on $n$ vertices
I encountered a function $f$ defined over a graph $G$ in my research which does not satisfy a deletion–contraction recurrence but an equation of the form
$$\partial_k f(x_1,...,x_k,..x_n;G)=g(x_k, N_{...
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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 ...
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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 $f(x)\ge 0$, $f(0)\ne 0$. ...
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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 ...
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Solving (or approximating) a certain delay differential equation
I'm interested in finding the (unique?) solution to the set of delay differential equations
$$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$
$$f_x(w,x) = wf(w,w^2x)$$
With the initial condition $f(1,x) = e^...
CommunityBot
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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
$$P(f '(x)) = Q(f(x))$$
...
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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:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
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Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
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Is it possible that a system of differential equation has a solution in time domain but not in Fourier domain? If so, why does it happen?
I have to solve
\begin{align}
&\frac{\partial h'_{1,1m}(t,r)}{\partial t} + \frac{2}{r} h_{1,1m}(t,r) = 0 \label{beta_0_1}\\
&\frac{\partial^2 h_{1,1m}(t,r)}{\partial t^2} = 0. \label{...
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Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane
Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...
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Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold
Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...
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The local global principle for differential equations
Are there any good reference to tackle the problem below?
Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...
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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem
This is the equation given ($n\geq2$)
$$
\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{...
CommunityBot
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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) = ...
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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 ...
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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
$$
-\...
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Random variables in conservative differential equations
I have a system of differential equations in which a variable converts to several other ones. I'd like to add randomness to the parameters, but I am not sure the best approach.
A minimal example (the ...
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Is there any work on distributional vector fields?
I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we start with an integral equation, ...
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Stability of rigid bodies spinning around $z$-axis under gravity
Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
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Differential structures on unit-root Frobenius modules
Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n$, $a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...
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The stability of the equilibria of a non-linear ODE system
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = -c f + \gamma p f,\\
\\
\dfrac{dT}{dt} = \...
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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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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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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\...
- 2,758
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Floquet coefficients under time change
Let's consider two ODEs $\tag{1}\label{1}\frac{du}{dt}=\gamma(u(t))\ F(u(t))$ and $\tag{2}\label{2}\frac{dv}{d\tau}=F(v(\tau))$ where $f\in C^\infty(\mathbb R^n,\mathbb R^n)$ and $\gamma\in C^\infty(\...
CommunityBot
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Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
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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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Allocation of $\mathbb R^2$
Let $z_1,\ldots,z_n$ be $n\ge 1$ distinct points of $\mathbb R^2$. Define the potential function $U: \mathbb R^2 \to\mathbb R$ by
$$U(x):=\sum_{1\le i\le n} \log(|x-z_i|),$$
where $|\cdot|$ denotes ...
CommunityBot
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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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A variant to the Stokes system and Navier-Stokes equation
The linearization of the Navier-Stokes equation (in a smooth bounded domain in dimensions 2 or 3) is the following non-stationary Stokes system
$$v_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$
whose $W_p^...
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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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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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Green's function on sphere
Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
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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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What kind of Lagrangians can we have?
In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-...
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Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
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The perturbation of non-Hamiltonian algebraic vector fields
In this question, we are interested in the number of limit cycles which appears in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\...
CommunityBot
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1
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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 $u:[0,1]\to\mathbb{R}$ ...
3
votes
1
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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 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 ...
CommunityBot
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Does Peano's theorem apply to spaces with infinite dimension?
Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?
Here, Peano's theorem is:
Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re \...
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1
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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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0
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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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