Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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6
votes
1answer
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
1answer
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
5answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
3answers
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
3answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
3answers
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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 ...