**0**

votes

**0**answers

292 views

### Green's function of coupled ODEs

For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$,
$f' + af + bg = S_1(f,g) + S_2(f,g)$
...

**1**

vote

**0**answers

268 views

### How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...

**2**

votes

**1**answer

296 views

### Strongly parabolic PDE vs weakly parabolic PDE

In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...

**3**

votes

**1**answer

349 views

### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...

**15**

votes

**1**answer

458 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 ...

**5**

votes

**2**answers

187 views

### Constructing a linear ODE for a product of two holonomic functions without introducing additional singularities

A function $f$ is called holonomic if it satisfies some linear differential equation with polynomial coefficients $$p_n(x) f^{(n)}(x)+\dots+p_1(x)f'(x)+p_0(x)f(x)=0.$$ Now if $f,g$ are holonomic then ...

**1**

vote

**0**answers

79 views

### differential equation with delay

Suppose, i have an differential equation with a fixed $\tau$ delay $\frac{dx}{dt}=f(x(t)+\alpha x(t-\tau))$. Here $\alpha$ is a small parameter. When $\alpha=0$ its a simple Self-oscillatory ...

**0**

votes

**0**answers

137 views

### Time-delay differential equation

Is it possible that the system \begin{equation}
\begin{cases}
2\dot{q}(t) + \dot{q}(t-1) + \dot{q}(t+1) = c & \text{if} \hspace{5mm} 0 \le t \le 2 ,
\dot{q}(t) + \dot{q}(t-1) = k & ...

**17**

votes

**0**answers

854 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 ...

**1**

vote

**0**answers

227 views

### Whether does the following equation have only one finite zero?

Dear MOs,
Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ...

**0**

votes

**2**answers

503 views

### Solution to the fractional differential equation

What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this ...

**2**

votes

**1**answer

241 views

### symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...

**8**

votes

**2**answers

813 views

### Reference for a nice proof of “undetermined coefficients”

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...

**1**

vote

**0**answers

280 views

### a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...

**1**

vote

**1**answer

422 views

### Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...

**1**

vote

**1**answer

140 views

### finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation
$det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$
you can find the definition of effective forms here

**1**

vote

**0**answers

158 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**2**

votes

**3**answers

334 views

### Laurent series expansion for ODE.

OK, then I read Frobenius method in mathworld (I learned when I took ODE 2):
http://mathworld.wolfram.com/FrobeniusMethod.html
My question is:
Are there any ODEs where the solution is given by full ...

**2**

votes

**1**answer

985 views

### First Order PDE Solution Method Issues

Greetings, I have a rather pedantic problem that's honestly been holding me back for over a month & is something neither stackoverflow nor some of my professors nor some graduate students have ...

**2**

votes

**1**answer

413 views

### closed form solution of an ODEs

I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help ...

**4**

votes

**1**answer

684 views

### What is symmetry group of non-linear equation?

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, ...

**-1**

votes

**3**answers

363 views

### problem related to Airy functions [closed]

I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems:
What are the normalization ...

**5**

votes

**2**answers

221 views

### When is a solution to an ODE determined by its average value?

Fix a smooth vector field $\vec v$ on $\mathbb R^n$. It is well-known that any trajectory $\vec x : [0,1]\to \mathbb R^n$ solving the ODE $\dot x(t) = \vec v(\vec x(t))$ is determined by its ...

**3**

votes

**2**answers

365 views

### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. ...

**1**

vote

**1**answer

247 views

### A 'conjecture' on critical elliptic pde

I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define
$$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$
$E_{\mathbb{R}^3}$ is defined ...

**1**

vote

**0**answers

92 views

### A critical elliptic PDE

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...

**1**

vote

**1**answer

281 views

### Solution of a PDE and its uniqueness

Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
...

**0**

votes

**2**answers

347 views

### Solution to differential equation

a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y_0>0$ and $y'(s)=y_1$, $s<0$,
$$y''+(2-n)\coth(t) ...

**3**

votes

**1**answer

209 views

### A partial differential equation on $\mathbb{CP}^1$

Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation
$\frac{\partial}{\partial z} f = ...

**5**

votes

**1**answer

553 views

### Numerical calculation of Arnold tongue

Hello.
I am working on investigation of family of dynamical systems on the torus
$$\dot{x}=\cos(x)+b\cos(t)+a$$
$$\dot{t}=1$$
and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$
I need to find ...

**2**

votes

**0**answers

140 views

### Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general
solution $g(t)$ to a following linear functional differential equation
$$
\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)
$$
with ...

**3**

votes

**1**answer

528 views

### Are (Frobenius) integrability conditions covariant?

Starting with a system of partial differential equations for functions on $\mathbb{R}^n$, Frobenius' theorem gives a bunch of integrability conditions (on e.g. functions which are 'data' for the ...

**14**

votes

**4**answers

1k views

### ODE's without a Lipschitz condition

When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is ...

**0**

votes

**1**answer

66 views

### Self-inhibitions are diagonal matrix

In all the discontinuous neural networks models, the self-inhibitions are diagonal matrix, what is the reason for this assumption?

**1**

vote

**2**answers

85 views

### Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm ...

**5**

votes

**1**answer

345 views

### total variation distance between two solutions of SDE

Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...

**5**

votes

**0**answers

94 views

### Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...

**4**

votes

**1**answer

131 views

### Dependence of the blow-up time of existence of an ODE with respect to initial condition.

Let $V$ be a smooth vector field on $\mathbb{R}^n$. Assume that the maximal solution to the Cauchy problem $x'=V(x), x(0)=x_0$ exist only for $t\in [0,T)$, where $T$ is finite, denote this time by ...

**7**

votes

**2**answers

455 views

### What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...

**1**

vote

**1**answer

290 views

### Books on Numerical Methods for Partial Differential Equations

Any good references for undergraduates?

**1**

vote

**1**answer

147 views

### Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.
Here ...

**0**

votes

**0**answers

156 views

### Harmonic Function?

Hi,
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...

**1**

vote

**0**answers

149 views

### two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...

**1**

vote

**2**answers

224 views

### Invariant set of Lotka-Volterra equation

I have the Lotka-Volterra equation
$\dot{x}=x(1-y),$
$\dot{y}=y(x-1),$
where $x$ and $y$ are non-negative.
It is easy to see that the $x$- and $y$-axis are invariant sets. I can see from plots that ...

**6**

votes

**1**answer

511 views

### ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \wedge \theta^k$ for the ...

**1**

vote

**1**answer

177 views

### Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$

I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j ...

**10**

votes

**5**answers

862 views

### Is there any techniques for solving a differential equation including iterated function? f ' (x) = f( f( x ) )

I am trying to solve this differential equation but I have no idea how.
$f ' (x) = f( f( x ) ) $
Although I don't think this differential equation is solvable, I'd like to know if there is any ...

**0**

votes

**1**answer

344 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

**1**answer

1k views

### Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...

**6**

votes

**0**answers

266 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} ...