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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17 views

Complex delay differential equation with time-dependent lag

I am trying to find a solution $g$ to the following delay differential equations (DDEs): $$ \beta(t)g^\prime(y)=g(y)-g(y-t)-t \quad (1)$$ $$ \beta(t)g^\prime(y)=g(y+t)-g(y)+t \quad (2)$$ with $g(y)=0$,...
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23 views

Convergence of the solution to ``some stochastic equation''

For every $\epsilon>0$, consider \begin{equation*} X^{\epsilon}_t = Z + W_t - 1 + \mathbb E\left[\exp\left(-\frac{1}{\epsilon} \int_0^t \max\left(-X_s^{\epsilon},0\right)ds\right)\right],\quad \...
5
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1answer
183 views

Initial conditions in the Klein-Gordon equation

I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$) \begin{equation}\label{kg} \left\lbrace \begin{array}{ll} (\square+m^2)F(x)=0\\ ...
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48 views

A continuity argument for a dispersive $gKdV$ estimate

I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The ...
5
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1answer
107 views

Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?

Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor. Let $T_n = J(x_{n-1})J(x_{n-2})....
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1answer
88 views

Finding a local normal form regarding distribution rank properties

I am working in geometry control field, fall last week on this exercice and I can't figure it out. I have a distribution $\mathscr{D}$ with $rank(\mathscr{D})=m+1$ in $\mathbb{R}^n$ with $n\leq 2m+1$. ...
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1answer
78 views

References Request: A paper Tanno's equation

I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet ...
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21 views

Series solution of an ODE with nonpolynomial coefficients

Basically, I have a second-order differential equation for $g(y)$ and I want to obtain a series solution at $y=\infty$ where $g(y)$ should vanish. That would be easy if the ODE contains polynomial ...
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87 views

A Kazhdan-Warner type problem

Let $X$ be a compact Riemannian manifold and I am interested in the following set of equations: \begin{align*} \Delta f+u\cdot e^{f+\lambda}=c\\ \lambda-2f=g \end{align*} where $u,g$ are given real ...
7
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3answers
269 views

Planar flow with bounded orbits and a single equilibrium point

Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x, $$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$ $$\lim_{t\rightarrow -\infty}\varphi_t(...
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36 views

Feynman Kac representation for nonlinear heat equation

Consider the following Cauchy problem \begin{align} \begin{cases} \partial_t u=\sigma(t)\partial_{xx} u+ b(u),\; (t,x)\in[0,T]\times \mathbb R\\ u(0,x)=u_0(x)=Ce^{-x^2/2}, \end{cases} \end{align} ...
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59 views

'Partial Integral' Equation

Given the following 'partial integral' equation of a measurable function $u$: $$\int_\ell u(x,y) \, d\mathcal{H}^1=F(\bar{\ell})$$ where $\ell = \{(x,y)\mid x \cos \theta + y \sin \theta = p,p \ge 0\}$...
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1answer
103 views

Hypergeometric function with changed argument [closed]

I have the hypergeometric function $_2F_1 (a, b,c, p\cdot z)$, where $p$ is a parameter and $z$ is the independent variable. I would like to know how the former function is related to the standard ...
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28 views

Solve differential system with a parameter

Consider the following system: $$ \begin{cases} \frac{x_1}4 + \frac{3 x_3}4 = a, \\ f(x_1) + 9 f(x_3) \ge 8 f(a), \\ f'(x_1) = 3 f'(x_3). \end{cases} $$ I want to find all functions (or at least learn ...
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0answers
92 views

Parabolic regularization for the Navier-Stokes equations

I'm looking for some references about a result on the Navier-Stokes equations which seems to be folklore but for which I didn't manage to find a proof. The setting is the following : Let $Q=\mathbb{R}^...
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0answers
81 views

Solving a integro-differential equation

I am trying to solve an integro-differential equation: $$ {\frac{d}{dt}} f(t)=\int_0^t k(t-\tau)S(\tau)f(\tau) d\tau $$ with initial condition $f(0)=1$ If $k(t)=c\delta(t)$ with $c$ being constant, ...
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0answers
74 views

Clarifications about a proof of (the measurable Riemann) mapping theorem in Hubbard's book on Teichmuller theory,

On page 151 of Hubbard's book, the author is proving the following theorem( Prop.4.6.2 ): Suppose $\mu$ is a real analytic function on a domain $U$ of $\mathbb{C}$. Then every $z \in U$ has a ...
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0answers
55 views

Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?

Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties? The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
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1answer
76 views

Nonlinear differential equations with zero initial conditions

Suppose that we have $n$ differential equations of the form: $\dot{x}_i(t) = f_i(g(x_1(t)), \ldots, g(x_n(t))) \qquad \qquad$ ($i=1,\ldots,n$). where $f_i$ are linear functions, and $g$ is an ...
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1answer
121 views

Solve differential system of equations

Consider the following system: $$ \begin{cases} x_1 + 3 x_3 = 4a, \\ f(x_1) + 3 f(x_3) = 8 f(a), \\ f'(x_1) = 3 f'(x_3). \end{cases} $$ I want to find all functions (or at least learn some properties ...
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0answers
87 views

Is there a concentric map from the disk onto the ellipse with constant sum of singular values?

$\newcommand{Vol}{\text{Vol}}$ Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...
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1answer
57 views

Solving a fully nonlinear first order PDE

given a symmetric matrix of Holder continuous functions $A(x)$ such that $$ \frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2 $$ find a vector field $\Phi$ such that $$ D \Phi(x)^t D ...
7
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2answers
695 views

Young-Fibonacci version of Nekrasov-Okounkov

This question addresses a hierarchy of linear recurrences which arise from an attempt to generalize the Nekrasov-Okounkov formula to the Young-Fibonacci setting. A related posting extensions of the ...
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1answer
88 views

Existence of solutions of a system of first order PDEs

Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset. Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions. That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
3
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1answer
220 views

Find strictly subharmonic function that vanishes at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
2
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1answer
106 views

Solving the Airy equation by Borel summation

The Airy equation is the canonical example of the Stokes phenomenon but, as of yet, I've not seen it being solved by Borel summation (which is the main way to explicitly construct examples of Stokes ...
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41 views

Solving a differential equation related to the hypergeometric differential equation

I need to solve the following equation: $x*(1 - s*x) y''[x] + y'[x] + r*y[x] =0,$ where $s$ and $r$ are two parameters. It would seem that is similar to the hypergeometric differential equation, but ...
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0answers
56 views

Is there an analytic formula (or even a name…) for a plane curve with curvature inversely proportional to x?

I'm interested in plane curves with curvature inversely proportional to distance from the axis: $$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(...
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1answer
101 views

Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion. Consider a classical mechanical system with ...
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0answers
35 views

how to linearize a set of differential equations and convert them into state-space model? [closed]

Below is a system of linear differential equations that describe the motion of a control moment gyroscope: $$ \begin{aligned} \dot{v}_1 &= - \left( \dfrac{5 \left(200 \tau_{3} \sin{\left(q_{2} \...
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3answers
3k views

Is there a general solution for the differential equation $f''(x) = f(f(x))$?

I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation $f''(x) = f(f(x))$ for the past 2 days. I can't seem to crack it but it feels like it ...
1
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0answers
64 views

symplectic Runge-Kutta for matrix differential equation

I would like to solve, for $t>0$ the following matrix differential equation: $$U'(t)=H(t)U(t)$$ with initial condition $U(t=0)=U_0$ ($2N\times2N$, symplectic and unitary matrix) and $H(t=0)=H_0$ ($...
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1answer
89 views

Qualitative analysis of the equation and symmetry (point on sphere)

A point moves on the surface of sphere ($R>0$ - radius) along the curve defined by the differential equation in spherical coordinate system: $R^2(|\dot \theta|^2 + w^2 \sin^2 \theta)=(at)^2$, ...
3
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1answer
112 views

What is the ideal form of an h-curve?

This question concerns mathematical modelling of the citation curve, well-known in the sciencemetry. The citation curve (or else the $h$-curve) of an individual researcher is the vector $(c_1,c_2,\...
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0answers
36 views

Modelling stochastic solutions for the general stochastic epidemic

I have found the deterministic solutions for the following system of differential equations: $$ s'(t) = -\beta s(t) i(t) \\ i'(t) = \beta s(t) i(t) -\gamma i(t)\\ r'(t) = \gamma i(t) $$ So the ...
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0answers
24 views

Existence theory for first order scalar discontinuous ODE

Consider the scalar i.v.p. in ${\mathbb R}$ $$ x'=f(t,x), \; t\in[0,T], \; x(0)=x_0, $$ where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R} $, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$...
3
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1answer
154 views

Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral $$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$ where $P$ is a linear ...
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108 views

Solve nonlinear, forced and damped Duffing oscillator

I am trying to solve a Duffing type equation by using Van Der Paul's method: \begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align} with $$x(t) = Re[A(t) \...
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0answers
58 views

Poincaré's Lemma in the space of tempered distributions

It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$. Question. Does the ...
3
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0answers
87 views

Regular Lagrangian flows on a domain of $\mathbb{R}^d$ with a boundary

I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space). Here, ...
2
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0answers
83 views

Regularity of the dependence of the flow on the vector field definining it

Let $M$ be a smooth compact manifold and $k \geqslant 1$. Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
5
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0answers
121 views

Overtwisted contact forms on open manifolds

I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here. It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...
2
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1answer
111 views

Second order inhomogeneous PDE

I'm trying to get an exact solution to this second order inhomogeneous PDE: $$ \frac{\partial^2}{\partial{x}^2} y(x, z) - \frac{\partial^2}{\partial z^2} y(x, z)=k^2y(x, z)-\frac{1}{3}e^{4(x-2z)}y(x, ...
6
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2answers
392 views

Non-linear hyperbolic PDE

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively $$ 2\...
7
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1answer
187 views

Spaces of solutions to algebraic linear differential equations

What is the name of the function space formed by solutions to algebraic linear differential equations? Where can I find a discussion of its properties? By an algebraic linear differential equation I ...
0
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2answers
150 views

Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$

How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with $$ \big(xJ_a'(\ell x) \big)'+\left(\...
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1answer
162 views

Beltrami equation with harmonic coefficient

I need to find solutions to the Beltrami equation $$ \frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z} $$ for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
1
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0answers
48 views

Lyapunov theory in coupled nonlinear dynamic system with input

Suppose I have the following nonlinear coupled dynamic system \begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
0
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1answer
58 views

Are two closed curves similar to each other if they have the same tangential vectors at the same position $\xi \in [0, 2\pi)$?

Given two closed curves $X,Y:[0,2\pi) \rightarrow \mathbb{R}^2$ with $X(0)=X(2\pi)$ and $Y(0)=Y(2\pi)$, if their tangential vectors are the same, which means $\mathcal{T}(X)|_\xi=\nabla_\xi X(\xi)/|\...
2
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0answers
48 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 \...

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