**6**

votes

**1**answer

507 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

174 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

810 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

329 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

260 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

77 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

144 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

172 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

116 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

303 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

680 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

77 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

148 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

172 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

157 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

177 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

263 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
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**3**

votes

**1**answer

337 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

307 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

190 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

212 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

483 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

121 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

267 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

359 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 ...

**2**

votes

**1**answer

257 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 $\partial_{x}^{-1}(uv)$, ...

**2**

votes

**0**answers

343 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

273 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

351 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

485 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

163 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

540 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

385 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 ?

**5**

votes

**0**answers

273 views

### Rigorous results on the method of multiple scales

The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation ...

**1**

vote

**0**answers

115 views

### The existence of the solution of $u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$

Prove the existence of the solution of the Cauchy Problem:
$$u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$$
where$u_0\in C^{\infty}(R).u_0,u_{0x}\to 0,when |x|\to \infty$
AS Robert Bryant's comment,$u=0$ is ...

**1**

vote

**0**answers

102 views

### The existence of the solution of the perturbed KdV Equation(semi-group operator)

Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...

**1**

vote

**3**answers

839 views

### book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...

**3**

votes

**2**answers

174 views

### second order operator with real coefficients and not locally solvable

This question is a follow up of the following answer I posted recently:
Counterexamples in PDE
Is there a second order partial differential operator with real coefficients which are not solvable in ...

**2**

votes

**0**answers

383 views

### Optimal transport warping implementation in Matlab

I am implementing the paper "Optimal Mass Transport for Registration and Warping", my goal being to put it online as I just cannot find any eulerian mass transportation code online and this would be ...

**1**

vote

**1**answer

167 views

### Constructing an example of Hamiltonian flow

I have this Hamiltonian flow generated by
$$
h(x, \xi ) = V(x) + \sqrt{\xi ^2 + 1}, \quad x, \xi \in \mathbb{R}^3,
$$
so the defining equations are
...

**0**

votes

**1**answer

278 views

### Solution to a second order PDE

Looking for a general solution to the following second-order PDE, where the unknown is a function $f(x_1, x_2)$ of two variables:
$
0=a^2f+a^2x_1{\partial f\over \partial x_1}+b^2x_2{\partial f\over ...

**2**

votes

**1**answer

146 views

### Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients.
$$ f = \sum_I f_I \mathbf x^I $$
This power series is said to ...

**3**

votes

**1**answer

278 views

### Moving under the influence of a vector field

I have a continuously varying vector field $v(p)$ on $\mathbb{R}^2$, and a particle
at point $p$ in the plane that can move in a direction $u(p)$ as long as $u(p)$ is
turned at most $\pi/2$ left of ...

**3**

votes

**2**answers

265 views

### delay differential equation

I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE):
$ y_x(x) = A y(x-1)^a $
where $ 0 < a < 1 $ and $ A > 0 $ are given ...

**3**

votes

**1**answer

190 views

### ODE in symmetric definite positive matrices

It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt ...

**2**

votes

**1**answer

92 views

### Are the real components of s-roots subharmonic?

Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...

**1**

vote

**0**answers

79 views

### Semi implicit DAE integration using an implicit Runge Kutta scheme

I'm looking for some references regarding integration of DAEs in the form
$M(t) \frac{dy}{dt} + G(y(t),t) + f(t) = 0, \quad y \in R^n, M(t) \in R^{nxn}$
using a high order implicit Runge Kutta ...