**2**

votes

**0**answers

137 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

408 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 ...

**13**

votes

**4**answers

866 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

79 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

312 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 ...

**6**

votes

**0**answers

88 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

102 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

408 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

228 views

### Books on Numerical Methods for Partial Differential Equations

Any good references for undergraduates?

**1**

vote

**1**answer

146 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

151 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

148 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

201 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

500 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

171 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

800 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

301 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

251 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} ...

**1**

vote

**1**answer

73 views

### A linear equation related to Camassa-Holm equation

I would like to study the well-posedness of the following equation
$u_t - u_{txx} + a u + b u_x + c u_{xx} + \gamma u_{xxx} = f$
with $u(0)=0$ and $f \in H^{s-1}(\mathbb{R})$, where $a, b, c, ...

**1**

vote

**1**answer

115 views

### Bessel and Neumann functions:ordering the zeroes

I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could ...

**3**

votes

**1**answer

166 views

### nonlinear delay differential equation

Consider the delay differential equation:
$ y_x(x) = \sqrt{y(x-\bar{x})} $
where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter.
This equation does not seem to have a ...

**1**

vote

**1**answer

112 views

### Boundary condition for a non linear schrodinger equation

I'm studying an anrticle on a non linear Schrodinger equation posed on $\Theta=(x\in R^2:|x|<1)$.
I read this: "we will only consider initial data of Sobolev regularity $s<1/2$ and thus we will ...

**4**

votes

**3**answers

276 views

### ordered exponential of unbounded operators

Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...

**2**

votes

**3**answers

632 views

### Jacobi method on first order partial differential equations

Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance

**1**

vote

**0**answers

73 views

### Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency?
Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...

**1**

vote

**0**answers

146 views

### “Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...

**5**

votes

**1**answer

170 views

### Periodic Holomorphic ODE

Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.
I would like to know if there are (tractable) conditions on $V$ that ensure that the ...

**1**

vote

**1**answer

228 views

### Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies
$$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$
where $|\cdot|$ denotes the ...

**0**

votes

**0**answers

155 views

### search for a function satisfying some conditions

Hi everyone, I would like to find a function
$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$
satisfying the following conditions:
...

**1**

vote

**1**answer

172 views

### Reference to the Existence and Uniqueness of the PDE system

Hi all
I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...

**1**

vote

**2**answers

246 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**2**

votes

**1**answer

308 views

### Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...

**1**

vote

**1**answer

279 views

### Inhomogeneous Bernoulli Equation

Does anybody suggest how to face the inhomogeneous Bernoulli differential equation
$y'+P(x)y=Q(x)y^n+f(x)$
for the simple case $f=const.$ and for the generic case.
I would like to know about ...

**5**

votes

**0**answers

186 views

### Linear ODEs in a locally convex vector space

Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...

**3**

votes

**2**answers

204 views

### how can i solve a boundary value numerically on an infinite interval ??

let be the differential equation $ -y''(x)+x^{4}y(x)-E_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty) $
how could i use the shooting method or other numerical method to solve this ...

**2**

votes

**0**answers

456 views

### Problem using finite difference to solve a initial value problem

Hallo, I tried to use 'finite difference' method to solve a Initial Value Problem(IVP). For the two boundaries I used periodical condtion and for the differential operators I used 4th degree center ...

**1**

vote

**1**answer

120 views

### Quantitative deformation and lusternik schnirelman method

The lusternik-schnirelman method relates the topology of manifolds with the critical points of functionals defined on them, giving lower bounds for the number of critical points in terms ...

**0**

votes

**1**answer

261 views

### W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation

In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
http://arxiv.org/abs/1111.7207
They proved the $W^{2,1}$ estimate for standard monge-ampere equation
$detD^{2}u=f$
with $f$ bounded from below ...

**2**

votes

**2**answers

348 views

### How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...

**1**

vote

**1**answer

240 views

### Leibniz rule for Pseudo-differential operators of negative order

Does anyone know of some good references for a fractional Leibniz rule for pseudo-differential operators of negative order? As a specific example, I would like to compute $\partial_{x}^{-1}(uv)$, ...

**3**

votes

**0**answers

323 views

### L^1-convergence of convolution exponential

Consider a differential
equation
\begin{eqnarray*}
\frac{d}{d\tau}q^{\tau}\left(x\right)=Aq^{\tau}\left(x\right)+h\star q^{\tau}=Aq^{\tau}\left(x\right)+\int ...

**1**

vote

**1**answer

96 views

### Bidirectional ODE

Hello all,
Could you please help with the following problem? I have a set of two coupled ODE for $a$ and $b$ waves [wave is a general form of solution $a(z)=A(z)\exp(\imath \beta z)$]. The equations ...

**4**

votes

**1**answer

270 views

### Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$.
Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...

**4**

votes

**0**answers

342 views

### A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE):
$\frac{d}{dt}y= A(t) y$, where $A(t)$ is a ...

**3**

votes

**0**answers

478 views

### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...

**1**

vote

**2**answers

162 views

### Dealing with a Matrix ODE in integral form

Supose we have
$v'(t) = A(t)v(t)+b(t),$ with $v(0)=0$
where $v(t),b(t) \in \mathbb{R}^n $, $A$ is a $n\times n$ real matrix and $t \in [0,1]$
Now, if A and b are discontinuous but integrable ...

**0**

votes

**2**answers

488 views

### duality argument in PDE

Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references?
Occasionally I see this term appears in ...

**0**

votes

**0**answers

362 views

### What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?