# Questions tagged [differential-equations]

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

1,285
questions

**0**

votes

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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 $g(y)=0$,...

**1**

vote

**0**answers

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**

votes

**1**answer

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\\
...

**4**

votes

**0**answers

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**

votes

**1**answer

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})....

**2**

votes

**1**answer

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 $n\leq 2m+1$. ...

**2**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**3**answers

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(...

**0**

votes

**0**answers

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}
...

**1**

vote

**0**answers

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 $\ell = \{(x,y)\mid x \cos \theta + y \sin \theta = p,p \ge 0\}$...

**-2**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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}^...

**3**

votes

**0**answers

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, ...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 $x\in S^2$ ...

**0**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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{...

**-1**

votes

**1**answer

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**

votes

**2**answers

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 ...

**0**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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(...

**4**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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} \...

**18**

votes

**3**answers

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**

vote

**0**answers

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 $H(t=0)=H_0$ ($...

**1**

vote

**1**answer

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**

votes

**1**answer

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,\...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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) \...

**4**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**2**answers

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(\...

**1**

vote

**1**answer

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**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 \...