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

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Existence and Uniqueness of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
-2
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2answers
88 views

Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [on hold]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
0
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0answers
30 views

existence of solution of volterra integral equation for the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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0answers
90 views

Calogero-Moser eigenfunction

The folllowing function \begin{equation} J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...
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0answers
46 views

Differential equations [closed]

For k > 0,x = x(t),y = y(t) Solve this system: \begin{array}{l} x\frac{{{d^2}x}}{{d{t^2}}} = k\frac{{dy}}{{dt}}\\ x\frac{{{d^2}y}}{{d{t^2}}} = - k\frac{{dx}}{{dt}} \end{array}
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98 views
+50

A derivational approach to the Poincare Bendixson Theorem

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a smooth vector field on the plane. Assume that $K\subset \mathbb{R}^{2}$ is a compact subset (not necessarily invariant under ...
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0answers
29 views

numerical solution of volterra integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$ \int_{a}^{x}f(s)g(s,x)ds=h(x) $$ ...
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0answers
26 views

How to solve integral equation? [duplicate]

I have an integral equation such that $$ \int_t^T f(s)g(s,t)ds= h(t) $$ where g and h is given. we want to know function f explicitly. As i know, this type of question is about the integral equation. ...
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1answer
71 views

Non Linear PDE's [closed]

I want to solve two systems of questions being $$\frac{C(r,y)''}{C(r,y)} + \frac{C(r,y)'^2}{C(r,y)^2}=0$$ and $$A(r,y)'' + 2A(r,y)' \frac{C(r,y)'}{C(r,y)}=0$$ where ' is differentiating with respect ...
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1answer
102 views

Global version of the Picard-Lindelöf theorem [closed]

Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) intervall and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ ...
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0answers
27 views

Backlund transformation related to two NL differential equations

I'm looking for a Backlund transformation linking the following two nonlinear differential equations for real $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ ...
5
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0answers
103 views

The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...
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1answer
159 views

Quadratic PDE Systems

(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.) I have a problem that leads me to the following quadratic system of PDEs:- $ c_1 ...
3
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0answers
70 views

Is there any weighted sobolev embedding with non-decaying weight

Is there any weighted sobolev embedding like $$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$? Here $s>0$, and for some appropriate $p, q$.
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0answers
64 views

if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity?

if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? And in dimension $n$? I know there is singularity when $n=2$.
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0answers
70 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
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0answers
56 views

Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
3
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1answer
89 views

Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude). Given $g \in (0,\infty), k \in C^1( [0, ...
2
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0answers
74 views

What transformations preserve the von Mises distribution?

The von Mises distribution is entirely defined on the circle with a density given by $$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$ where $x$ is in an arbitrary real interval of ...
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0answers
187 views

The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
2
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1answer
61 views

How to find the general solution of Mathieu differential equation?

Two solutions of the equation are the Mathieu sine and cosine and they form an orthogonal basis. The equation has arisen from the wave equation for an infinite string with a sinusoidal cross-section. ...
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0answers
91 views

How to choose appropriate norms in the analysis of PDEs?

In many papers on the analysis of nonlinear PDEs, the authors could always choose appropriate norms to do their estimates. Sometimes these norms look very odd, which I don't know how these authors ...
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1answer
71 views

A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs $$ Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t) , $$ where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...
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3answers
119 views

Harmonic Function with special property

I would appreciate any help with the following problem: Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...
4
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1answer
70 views

Explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one. But when I tried to add spin into account in this scheme, it ...
2
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1answer
61 views

First order ODE and possible periodic solutions

I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions: $\partial_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$. This equation roughly ...
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94 views

projecting Laplacian onto tangent and normal bundles of submanifold

If I have a simple linear differential equation involving covariant derivatives such as $\nabla^2 g_{\mu\nu}+ 2g_{\mu\nu}=0$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) ...
0
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1answer
109 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( ...
7
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1answer
137 views

Differential analogue of the newton polygon

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows: Let us recall the ...
3
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1answer
76 views

Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...
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1answer
95 views

first order differential equation

Is there any solution formula for a differential equation like this $$ y'(x)=f(x)y(x)+g(x)y(ax)\qquad \text{where}\qquad a\gt1. $$ I have a differential equation a little more complicated than this. ...
2
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1answer
78 views

General four-term recurrence relations

I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form ...
3
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2answers
202 views

Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that ...
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0answers
63 views

Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of ...
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46 views

how can i solve this nonlinear problem?

let be the differential equation in dimension $ n > 3 $ or $ n =3 $ $$ -\Delta u =|grau|^{2}u $$ (1) with the constraint that the vector 'u' $ |u|=1 $ then how could i prove that the unitary ...
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0answers
78 views

G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...
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0answers
49 views

Single parameter bifurcations caused by a simple additive term

Note: I asked this question on Math.SE over two months ago, and it has not received any answers. Motivation: A practical dynamical system is often described by an ODE that has a parameter that ...
0
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1answer
191 views

Evolution equation on Banach space

Let $X$ be a Banach space. We consider the evolution equation: $$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$ where $A$ is a bounded operator. I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...
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0answers
46 views

Allen Cahn Equation with Dirichlet Data

consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions ...
0
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1answer
148 views

Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
4
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1answer
395 views

Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange. My question is where the following differential equation arises naturally and where it ...
2
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0answers
78 views

Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
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0answers
61 views

Constrained optimization problem with system of differential equations [closed]

Thanks in advance for any help. My question relates to a homework problem. I have been given a system of generalized differential equations with the goal of creating an algorithm that will solve ...
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0answers
42 views

Special case of Forced Harmonic Oscillator

$$ \frac{d^2l}{dt^2} + c \frac{dl}{dt} +k= F_0\sqrt{1-l^2}$$ Either a closed form or something like given initial conditions, the max value of $l$. For small $l$, this boils down to simple step ...
6
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3answers
216 views

On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...
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1answer
144 views

This theorem is true or false in infinite dimensions?

Consider a vector space $E$ with finite dimension and linear map $A: E \rightarrow E$. The following statements are equivalent: $x'(t)=A \circ x(t)$ defines an attractor. All eigenvalues of $A$ have ...
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0answers
118 views

Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the Navier Stokes equations can be written as a semilinear symmetric first order system $$ u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u) $$ for a 9 by 1 vector $u$ containing the ...
15
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3answers
849 views

I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension

Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample? Here, Peano's theorem is: Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re ...
1
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1answer
113 views

Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...
5
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0answers
89 views

Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...