**0**

votes

**0**answers

24 views

### If $f(x) = Ax$, show that for all $t \in \mathbb{R}$, the extreme $x_n = x_n(t)$ of polygon converges to $e^{At} x_0$ [on hold]

Let $f$ is a vector field in $\mathbb{R}^n$, $x_0 \in \mathbb{R}^n$ and $x_{k+1} = x_k + f(x_k)\Delta t$, $k= 0,1,...,n-1$, where $\Delta t = \frac{t}{n}$. A polygon whose points are the $x_i$ ...

**6**

votes

**3**answers

147 views

### On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...

**0**

votes

**0**answers

22 views

### Finding a particular solution to the non-homogenous system [on hold]

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$
Step 1) Find the Eigenvalues ...

**-3**

votes

**0**answers

34 views

### A non-liner second order differentice equation with two parameters [on hold]

Can we analysis the effect of the two parameters on the equation? And how can we determine the range of parameters to obtain the solutions for the equation? And
it is much gratitude for you help even ...

**1**

vote

**0**answers

102 views

### Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the
Navier Stokes equations can be written as a semilinear symmetric first
order system
$$
u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u)
$$
for a 9 by 1 vector $u$ containing the ...

**-2**

votes

**1**answer

125 views

### This theorem is true or false in infinite dimensions?

Consider a vector space $E$ with finite dimension and linear map $A: E \rightarrow E$. The following statements are equivalent:
$x'(t)=A \circ x(t)$ defines an attractor.
All eigenvalues of $A$ have ...

**-2**

votes

**0**answers

38 views

### Show that there is a matris only $ A $ such that $ \varphi (t) = e ^ {tA} $. [on hold]

Let $ \varphi(t)$ of a matrix $n \times n$ functions $C^1$. If $\varphi(0)=I$ (identity) and $\varphi(t + s) = \varphi (t) + \varphi (s)$ for all $ t, s \in \Re $, show that there is a matris only $ A ...

**-1**

votes

**0**answers

21 views

### Solution of ODE where A has not eigenvalue [on hold]

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.

**14**

votes

**3**answers

788 views

### I would like to have a counter example that Peano's theorem does not 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 ...

**2**

votes

**1**answer

129 views

### Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...

**0**

votes

**0**answers

61 views

### Reference request: Heat kernel regularity near the boundary

Let $D$ be a domain in $\mathbb{R^d}$ and $p(t,x,y)$ be the heat kernel of $D$ (for the Dirichlet problem). I was told that if the boundary of $D$ is real-analytic, then the function $y\mapsto ...

**0**

votes

**1**answer

75 views

### Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...

**1**

vote

**0**answers

56 views

### About the integral of the plurisubharmonic functions on the whole manifold

Let $(M,\omega)$ be a Kahler manifold with positive first Chern class. $\omega\in c_1(M)$ is a Kahler current. $D$ is a smooth simple divisor in $-K_M$. $s$ is a determining section of $D$. The ...

**0**

votes

**0**answers

48 views

### Existence of BVP solutions for non-linear Bessel-like equation

The basic problem: we have the differential equation
$$
g''(r) + \frac{2}{r} g'(r) - \frac{2}{r^2} g(r) = F(g(r))
$$
in which $F(g)$ satisfies $F(0)=F(1)= 0$ and $F(x) > 0$ for $0<x<1$. ...

**-1**

votes

**0**answers

26 views

### Is there a program for convenient working with equations and coefficients? [migrated]

I perform some calculations with one differential equation. Then I got a huge expression depending on $x$ and its degrees/powers. E.g.
$$\alpha x+(4-x+\sqrt[3]{x})^2-(\beta\sqrt{x}+\frac12(x^3+1))^3 + ...

**1**

vote

**1**answer

56 views

### Solving Bessel-like equation by using Bessel Kernel [closed]

Consider the drifted-Bessel equation as follows.
\begin{equation}
x^2\ddot y + x\dot y + (x^2-n^2)y=f,
\end{equation}
where $n$ is an integer and $f$ is a known function. If $f\equiv 0$, the solution ...

**7**

votes

**1**answer

467 views

### Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$:
$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...

**5**

votes

**0**answers

76 views

### Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...

**-2**

votes

**0**answers

35 views

### Is there an explicit ODE solution for this system?

y'(t)=A(t)*y(t)+g(t),
A(t)=[-t^2, t;
sin(t) cos(t)]
g(t)=(1+t/2;
-1)
y(0)=(3;
1)
I actually do not know how to solve an ODE with variable constants, but when I use the ...

**2**

votes

**1**answer

82 views

### Global Solutions of Ordinary Differential Equations

Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...

**2**

votes

**0**answers

150 views

### J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle ...

**2**

votes

**3**answers

231 views

### Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...

**1**

vote

**0**answers

67 views

### Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...

**1**

vote

**1**answer

141 views

### Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...

**0**

votes

**1**answer

322 views

### Teaching profession:Differential Equations and Mean Value Theorems

Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...

**0**

votes

**0**answers

58 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**18**

votes

**0**answers

826 views

### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...

**4**

votes

**2**answers

177 views

### Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...

**3**

votes

**1**answer

616 views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**2**

votes

**1**answer

173 views

### Solution to Seiberg-Witten monopole equation

To understand some physics problem, I want to know if there is (non-$L^2$ or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$
\begin{equation}
{D}_A \psi = 0\\
F_A^+ = ...

**1**

vote

**0**answers

48 views

### Implementing boundary conditions to an ODE (involving elliptic integrals)

I am trying to solve the following differential equation:
$$
\frac{\mathrm{d} f}{\mathrm{d} x} = \frac{x^2-2 a}{\sqrt{4k^2-(x^2-2 a)^2}},
$$
where $a$ and $k$ are constants ($k$ is known and $a$ is ...

**9**

votes

**1**answer

269 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

**3**

votes

**0**answers

90 views

### Nonexistence of Limit Cycle

Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where ...

**7**

votes

**1**answer

351 views

### Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...

**0**

votes

**1**answer

303 views

### phase portrait of system of differential equations

Is there a full classification of the phase portraits of the following systems of differential equations
\begin{equation}
\cases{
\dot x=a_{11}x+a_{12}y+a_{13}z \\
\dot y=a_{2 1}x+a_{22}y+a_{23}z\\
...

**4**

votes

**2**answers

195 views

### Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...

**2**

votes

**2**answers

352 views

### Does a (smooth, constant-rank, integrable) distribution have a (local) basis of divergence-free vector fields?

In coordinate-free language, my question is as follows. Let $M$ be an $n$-dimensional manifold with volume form, and let $\mathcal D$ be a smooth (integrable, if necessary) distribution with constant ...

**0**

votes

**0**answers

47 views

### ODE system with a very large number of equations

Are there any theory about a ode system (linear or nonlinear) with a very large number of equations, e.g. more than $10^5$ equations ?

**2**

votes

**1**answer

123 views

### how to solve this system of nonlinear differential equations

I want to solve this system
\begin{align*}\tag{*}
x'(s)=x^2(s)+y(s),
y'(s)=x(s)y(s)
\end{align*}
with initial conditions $$x(0)=t, y(0)=t,$$
where $t\not=0.$
With the help of Maple, the solution is
...

**15**

votes

**6**answers

1k views

### Catenary curve under non-uniform gravitational field

The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ...

**2**

votes

**0**answers

134 views

### Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear SchrÃ¶dinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...

**0**

votes

**0**answers

43 views

### Solvability and uniqueness of Fokker-Planck BVP

I have been searching for solvability for the following BVP, I believe that this is a Fokker Planck's equation but I can't find any comprehensible text on existence and uniqueness of the solutions of ...

**0**

votes

**0**answers

72 views

### Wave operator for focusing NLS

Consider the NLS equation
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u+u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where ...

**0**

votes

**0**answers

29 views

### Green function and translational symmetry

I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ...

**5**

votes

**1**answer

192 views

### Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...

**2**

votes

**0**answers

227 views

### A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
...

**0**

votes

**1**answer

111 views

### Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation :
$\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...

**5**

votes

**3**answers

144 views

### Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...

**0**

votes

**0**answers

41 views

### Mathieu equation

I have come across different versions of Mathieu equation. I am trying to solve for the stability of an inverted pendulum with oscillating base. If there any correspondence between different version ...

**1**

vote

**3**answers

166 views

### General systems of linear differential equations with variable coefficients

I am wondering what can be said in general about the fundamental matrix of a system of linear differential equations. For simplicity, let $A(t)\in\mathbb{C}^{n\times n}$ be a time dependent matrix, ...