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

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1answer
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
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0answers
157 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
3answers
316 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
1answer
936 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
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1answer
399 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
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1answer
680 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: $r'^2+r^2=(kt)^2$, ...
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3answers
355 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
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2answers
220 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
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2answers
355 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 $T>0$. ...
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1answer
246 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 ...
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0answers
91 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 $\Omega=\mathbb{R}^3-B_1(0)$, ...
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1answer
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 ...
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votes
2answers
336 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
1answer
206 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 = ...
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0answers
397 views

Does the following differential equation have a non-trivial solution:(f'')^4+(f)^4=2(f')^4 ? [closed]

Hello! Currently,I meet the following differential equation: ${f''}^4+f^4=2f'^4$ we can easily find the trivial solutions:$a e^{bx}$,where $a$ is an arbitrary constant,$b=\pm i$ or $b=\pm 1$,does ...
5
votes
1answer
534 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
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0answers
139 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
1answer
506 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
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4answers
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
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1answer
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?
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2answers
83 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
1answer
331 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
0answers
93 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
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1answer
126 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 ...
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2answers
445 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 ...
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1answer
272 views
1
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1answer
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 ...
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0answers
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( ...
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0answers
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
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2answers
217 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
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1answer
510 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
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1answer
175 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 ...
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5answers
859 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
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1answer
335 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
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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
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0answers
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} ...
1
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1answer
79 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
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1answer
150 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
173 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 ...
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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 ...
6
votes
3answers
328 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 $$ ...
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3answers
745 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
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0answers
78 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 $\phi_0$ ...
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0answers
149 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
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1answer
173 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
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1answer
229 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
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0answers
158 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
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1answer
178 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. ...
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2answers
266 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
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1answer
340 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 ...