**5**

votes

**1**answer

557 views

### Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?

**0**

votes

**0**answers

59 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 $0<x<1$. ...

**0**

votes

**0**answers

61 views

### How to apply initial conditions to the Mathieu differential equation?

The Mathieu equation was a result of the separation of variables method used on a wave equation for a string with sinusoidally changing cross-section with the coordinate, $S(x)=S_1-S_2 \cos{x}$. The ...

**9**

votes

**1**answer

282 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 ...

**1**

vote

**0**answers

192 views

### Question on measure zero set of initial conditions in dynamical systems

[Update] Let $S \subseteq \mathbb{R}^n$ be a closed, bounded, convex set with measure $m(S)>0$ and let an autonomous dynamical system (system of ODEs) be given by
$$\frac{dx}{dt} = f(x),$$
where ...

**5**

votes

**1**answer

393 views

### Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?

I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...

**0**

votes

**1**answer

97 views

### Product of derivatives of a convex function [closed]

Given a well-behaved function $f(x,t)$ such that $\frac{\partial f}{\partial x}<0$ and $\frac{\partial^2f}{\partial x^2}>0$, is there a way to show
\begin{equation}
...

**4**

votes

**1**answer

150 views

### Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin ...

**2**

votes

**2**answers

205 views

### Any suggestions on a rigorous stochastic differential equations book?

I have been looking through some books and they are not very rigorous. Any suggestions would be great.

**4**

votes

**1**answer

98 views

### For a linear dynamic system, what can we learn from its singluar value and rank?

Given a linear system $\frac{dx}{dt}=Mx$, what's the relationship between the dynamic's property and the singular value decomposition/rank of $M$ ?

**3**

votes

**1**answer

108 views

### Convergence of trajectories and asymptotic stability

Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are ...

**2**

votes

**2**answers

159 views

### What is the right initial domain for the Dirichlet-Laplacian on a bounded domain?

Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian.
Some books ...

**4**

votes

**1**answer

214 views

### Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form
$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$
for ...

**3**

votes

**1**answer

133 views

### Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...

**-1**

votes

**2**answers

123 views

### Does an implicit Runge Kutta scheme applied on a nonlinear ODE give a nonlinear set of equations to solve in each step?

We want to approximately solve an ODE
$$\frac{dy}{dt} = f(y,t)$$
with the Runge Kutta method
$$y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$$
$$k_i = f\left(y_n + h \sum_{j=1}^s a_{ij} k_j,\,t_n + c_i ...

**25**

votes

**5**answers

1k views

### Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...

**0**

votes

**0**answers

114 views

### Prove that origin is globally exponentially stable with Lyapunov Indirect Method

I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially ...

**1**

vote

**1**answer

171 views

### Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...

**4**

votes

**3**answers

262 views

### On discrete version of curve shortening flow

One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...

**2**

votes

**1**answer

141 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...

**1**

vote

**1**answer

123 views

### Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...

**3**

votes

**1**answer

45 views

### Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3
$$x'(t)=f(x(t),d)$$
where the dynamics f is homogeneous of degree 1 and there is exactly one
line of equilibrium points. This line is independent of the ...

**4**

votes

**2**answers

175 views

### First order Elliptic operator

Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$?
For example, is the dimension of $V$ ...

**1**

vote

**1**answer

400 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**-1**

votes

**1**answer

142 views

### How do I solve this nonlinear ODE with a fractional order term

Problem:
Let $p$ and $q$ be two integers, and $q > p>0$. Does the following ODE have a general solution on some finite time interval $[0,T]$? If yes, how can I obtain the solution?
$$
\dot ...

**2**

votes

**2**answers

141 views

### Unusual Differential Equation for CDF

Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...

**3**

votes

**1**answer

181 views

### Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation:
$$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$
for $x>1$.
If that is not possible, I would at ...

**1**

vote

**0**answers

53 views

### On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by
$$\left\{
\begin{array}{l}
\dot{\theta}_1 = \omega_1 - ...

**1**

vote

**1**answer

251 views

### solving elliptic system of first-order linear PDE's

I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which ...

**2**

votes

**3**answers

255 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 ...

**2**

votes

**1**answer

208 views

### Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...

**3**

votes

**0**answers

103 views

### Approximating solutions of non-linear differential equations

I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
...

**0**

votes

**0**answers

69 views

### How to construct a harmonic functions satisfying the following conditions?

Let $\Omega\subset R^2$ be a bounded domain with a smooth boundary $\partial \Omega$. Let $\Gamma_1\subset\partial$ such that $\partial\Omega\setminus\overline\Gamma_1\neq\emptyset$. Does there exist ...

**4**

votes

**2**answers

196 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 ...

**2**

votes

**2**answers

159 views

### Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...

**4**

votes

**1**answer

221 views

### Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...

**1**

vote

**1**answer

168 views

### Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...

**4**

votes

**1**answer

326 views

### A Differential Equation with Nested Functions

This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.
I would like to know whether there exists a differentiable function from the ...

**0**

votes

**1**answer

533 views

### Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ?
assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...

**4**

votes

**1**answer

410 views

### Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.

Suppose that a continuous function $f$ on the line and satisfies
$$
|f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
...

**1**

vote

**1**answer

123 views

### A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) ...

**1**

vote

**0**answers

90 views

### growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation
$$
A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2
$$
where
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ ...

**1**

vote

**1**answer

99 views

### whether there are some books and original papers ergodic theory approach to ODE

Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications.
People always said that most of the ideas in ...

**3**

votes

**1**answer

320 views

### Path integrals for stochastic equations

Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example
http://arxiv.org/abs/hep-ph/9912209v1
For imaginary time rigorous ...

**1**

vote

**1**answer

99 views

### Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$.
Minimal ...

**2**

votes

**1**answer

147 views

### Total variation distance between diffusion processes with different volatility coefficient

Preamble:
This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...

**6**

votes

**1**answer

288 views

### Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$.
It seems that the solution of linear ODE
$$\dot x= Ax$$
can not be written in a closed form using $\int$, $A$, $x(0)$ ...

**2**

votes

**1**answer

118 views

### Is autonomous dynamical system equivalent to one single higher-order ode?

We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a ...

**13**

votes

**5**answers

977 views

### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**2**

votes

**0**answers

117 views

### Solving ODE with negative expansion power series

I'm moving this here, as suggested from physics.stackexchange. The original is here.
So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...