# Questions tagged [differential-equations]

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

**0**

votes

**0**answers

188 views

### Canard limit cycle for certain singularly perturbed system(Is there a contradictory situation?)

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system
$$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) ...

**1**

vote

**0**answers

46 views

### How to prove a set of delay differential equations never converge (the delay is not constant)

Two functions $x(t)$ and $y(t)$ are coupled via:
$$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$
where $a<0$, $b\neq 0$ is some constant.
I am mostly confused with the second equation. What is the ...

**2**

votes

**1**answer

67 views

### The regularity of ODE with Zygmund coefficients

A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...

**2**

votes

**0**answers

84 views

### Limit circle/point of an ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...

**44**

votes

**4**answers

4k views

### Could the Riemann zeta function be a solution for a known differential equation?

Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...

**-4**

votes

**0**answers

96 views

### Le produit de série entier [on hold]

Salam. J’aurais une question concernant les séries entiers, j’aurais besoin de votre aide pour ceci résulta de cette multiplication:
$$t(∑_{n=0}^{∞}a_{n}t^n)(∑_{n=0}^{∞}(a_{n}t^n)-a)(∑_{n=0}^{∞}(a_{n}...

**0**

votes

**0**answers

77 views

### On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...

**-1**

votes

**0**answers

39 views

### How to solve this kind of second order equations with variable coefficients?

Let $f(x)$ be a function satisfying the functional equation
$$
c_1(x)f(x)^2 + c_2(x) f(x) + c_3(x) f(x-1) + c_4(x) f(x+1) = 0,
$$
where $c_1, \ldots, c_4$ are known functions. What can be said about $...

**3**

votes

**2**answers

254 views

### Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...

**-1**

votes

**0**answers

45 views

### Fractional differential equation and inverse Laplace transform

I am writing my bachelor thesis to prove that a certain linear fractional differential equation of order $(n,q)$ has $N$ linearly independent solutions, where $N$ is the smallest possible integer ...

**13**

votes

**1**answer

288 views

### Projective-invariant differential operator

This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...

**1**

vote

**1**answer

68 views

### The difference between the nonlocal and local conditions problems

In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness ...

**4**

votes

**0**answers

112 views

### Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...

**0**

votes

**0**answers

67 views

### Biharmonic equation

Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation
\begin{equation}\label{\star}
\Delta^2u+V(x)u=g(x, u)+K(x)u,
\end{equation}
where $|g(x,s)|\leq \varepsilon|...

**2**

votes

**1**answer

161 views

### Functional equation $\int_z^{2z} [f(x)-f(z)] dx = 0$

Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$,
$$\int_z^{2z} [f(x)-f(z)] dx = 0.$$
It is clear that a constant ...

**-4**

votes

**0**answers

127 views

### Finding a paper [closed]

Could you please help me to find the paper "Rüssmann, Helmut Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. (German) Nachr. Akad. Wiss. Göttingen Math.-Phys. ...

**4**

votes

**1**answer

378 views

### Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$:
\begin{equation}
\frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...

**7**

votes

**1**answer

610 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

**4**

votes

**0**answers

66 views

### Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel

Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...

**1**

vote

**0**answers

130 views

### Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...

**2**

votes

**1**answer

179 views

### Isochronization of quadratic vector fields with center

What is a classification of all quadratic vector fields
$$\begin{cases}
x'=P(x,y)\\
y'=Q(x,y)
\end{cases}\qquad (V)$$
with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...

**2**

votes

**1**answer

285 views

### Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I ...

**2**

votes

**1**answer

73 views

### convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture).
Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...

**3**

votes

**0**answers

120 views

### Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$,
and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree).
I wish to prove or find a counterexample to the following claim:
If ...

**2**

votes

**0**answers

167 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**1**

vote

**0**answers

83 views

### Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:
Let $X$ be a compact $k$-...

**1**

vote

**0**answers

82 views

### Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...

**0**

votes

**1**answer

100 views

### Delay equations

In an effort to solve a delay partial differential equation
$$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$
with
$$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$
Where $\alpha$ is the delay ( a real ...

**0**

votes

**1**answer

219 views

### Center-localized oscillating modes with exponential decay tails, solved from coupled ODE

Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$:
$$
-a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) U(r)+
B(r) (\partial_r-...

**2**

votes

**1**answer

94 views

### The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...

**1**

vote

**0**answers

42 views

### Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...

**0**

votes

**0**answers

49 views

### Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...

**0**

votes

**0**answers

54 views

### Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$.
Which vector fields $F$ are so that ${\rm div} (F)= f$ ?

**41**

votes

**8**answers

3k views

### Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...

**2**

votes

**3**answers

282 views

### Non-linear Basis for PDE's

Asked this on stack exchange and got no response, so I'll try here.
An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...

**1**

vote

**0**answers

38 views

### Solutions of nonlinear equations with multiple parameters

In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...

**1**

vote

**0**answers

71 views

### One-point partition

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$.
$$
\mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...

**52**

votes

**2**answers

10k views

### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...

**-1**

votes

**0**answers

70 views

### Solutions to the Bond Pricing Equation

Consider a spot rate of the form:
$dr = (\eta - \gamma r) dt + \sqrt{\alpha r + \beta} dW$
where all parameters are constants.
Lets look for a solution of the form $Z(r; t) = e^{A(t;T) - r B(r; T)...

**33**

votes

**4**answers

7k views

### Function satisfying $f^{-1} =f'$

How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?

**1**

vote

**1**answer

57 views

### Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...

**16**

votes

**2**answers

782 views

### The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...

**7**

votes

**1**answer

494 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} f(t,x)=\frac{f(t,...

**2**

votes

**0**answers

39 views

### Floquet stochastic process

Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...

**0**

votes

**0**answers

45 views

### The limit of infinite ODE solver iteration with zero time step

Suppose I am trying to find a solution of an ordinary differential equation:
\begin{equation}
\begin{aligned}
y'(x) &= f(y(x))\\
y(0) &= y_0
\end{aligned}
\end{equation}
on ...

**0**

votes

**0**answers

18 views

### Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...

**5**

votes

**1**answer

209 views

### Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...

**1**

vote

**1**answer

96 views

### Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form.
We have a type I singularity if
$$
\max_{p \in M} |A(p,...

**4**

votes

**0**answers

262 views

### Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...

**11**

votes

**2**answers

1k views

### Derivative of the flow for ODEs on manifolds

Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its flow, i.e. the unique solution to the ...