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Questions tagged [differential-equations]

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

3
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145 views

Distributional PDE solutions as topological linear duals of PDE solutions

Let $$ P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast) $$ be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
-1
votes
1answer
128 views

An elementary question about integration by parts! [closed]

Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
1
vote
1answer
67 views

diffusion coefficient derived from simple random walk in a 1D semi-infinite domain

Suppose we have a 1D domain $x\in[0,\infty)$ and particles released at $x=0$ are doing simple random walks along the domain with reflecting boundary conditions at x=0. Then we can write down the ...
1
vote
0answers
67 views

Holonomic modules and Holonomic functions

Let $$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}\prod_{i=1}^{d}G((i-k)h) . $$ I have proved that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}\in \mathbb{C}(h)[[x]]$ is holonomic and arrive at a ...
2
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0answers
68 views

Stochastic Approximation Algorithms Converging to Local Equilibriums

Consider the stochastic iterative updates \begin{align} \theta_{t+1} \leftarrow \theta_t + \alpha_t \cdot \left [ h(\theta_t) + M_t \right ], \end{align} where $\theta_t \in \mathrm{R}^d$, $h \colon ...
0
votes
1answer
114 views

Asymptotic holonomic

Let $$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}e^{d(k-\frac{d+1}{2})h} . $$ We claim that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}$ is not holonomic??? I want to prove that above thing. Which ...
11
votes
2answers
1k views

Derivative of the flow for ODEs on manifolds

Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its flow, i.e. the unique solution to the ...
3
votes
2answers
172 views

General formula for integrating factor of an homogeneous differential 1 form

This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
5
votes
2answers
110 views

General term formulas for nonlinear recurrence sequences

It seems to be a well known question: in which cases will there be general term formulas for sequences like $p_n=a p_{n-1} ^2 +b p_{n-1} +c$ where $a, b, c$ are real or complex numbers and n is ...
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0answers
28 views

Regularity requirements of a second order parabolic PDE

I am not too familiar with the theory of regularity of parabolic PDEs, but I am wondering whether the following is true: Let $b:[0,T] \times \mathbb{R}^d \to \mathbb{R} ^d$ be in $C([0,T], C^1_b)$. ...
5
votes
1answer
256 views

Green's function for fourth order equation

I know the D'Alembert operator ${\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{3D}}$ has a well-known Green's function $\frac{\delta(t-\frac{r}{c})}{4 \pi r}$. This is very useful for studying 3D ...
7
votes
2answers
198 views

Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period?

I have heard that differential equations on $\mathbb{S}^1$ of the form \begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \...
2
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0answers
44 views

Factorization of linear ordinary differential operators

I was looking for references that give a detailed survey of techniques of factorization of linear ordinary differential operators. Specifically if there are references that do a complexity analysis of ...
2
votes
0answers
103 views

Solve 4th order ODE with variable coefficients

I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam: $u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...
1
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0answers
76 views

Differential equations with infinite-dimensional Lie groups

I am no expert in solving DEs by symmetry methods, but from pure interest - is it possible for a differential equation to have an infinite-dimesional Lie group as a symmetry group?
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0answers
129 views

Continuous Dependence of ode solution on parameters [closed]

Let $f:V\rightarrow \mathbb{R}^n$ be locally Lipschitz ($V$ is a subset of $\mathbb{R}\times\mathbb{R}^m\times \mathbb{R}^n$). Suppose we have a function $x:[t_0,\beta[\times W\rightarrow \mathbb{R}^n$...
0
votes
1answer
219 views

Center-localized oscillating modes with exponential decay tails, solved from coupled ODE

Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$: $$ -a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) U(r)+ B(r) (\partial_r-...
5
votes
1answer
166 views

Convergence of dynamical system on the sphere

Let $A(x)$ be a symmetric negative semi-definite matrix which depends continuously on the parameter $x\in\mathbb{R}^{d}$. We consider the differential equation $$\dot{x} = (I-xx^*)A(x)x$$ on the unit ...
4
votes
2answers
354 views

How to find the symmetry group of a differential equation

If one is given a differential equation, e. g. the KdV equation $\ u_t + u_{xxx} + uu_x = 0$, how can he find all of the symmetries of the differential equation? Is there also a method that works for ...
3
votes
2answers
114 views

Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$

The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function). $$(x^2y')'-x^2y=\lambda \;y$$ Now for a higher-degree ...
4
votes
2answers
140 views

Polynomial vector field tangent to a given analytic simple closed curve

Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin. Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, ...
0
votes
1answer
161 views

One side Harnack inequality for Subharmonic function

It is well known that for any non negative Harmonic function w ($\Delta w=0$, $w\geq 0$) in a ball, $B_1(0)$, $\exists$, C>0 such that $\forall y\in B_{1/2}(0)$ $$ Cw(0)\leq w (y) $$ It is a clear ...
3
votes
0answers
101 views

First order linear ODE with some decay condition

In Kronheimer [1, p.183], a certain statement is made of which I extract the following special case. Let $\alpha:\mathbb{R}\to \mathrm{Mat}(n\times n,\mathbb{C})$ be smooth and suppose that there ...
1
vote
1answer
80 views

a question about complex Hessians on complex tori

Suppose we have a real-valued smooth function on a complex torus: $$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$ i.e., this $f$ is a real-valued smooth function on $\...
6
votes
2answers
299 views

Solution to at least one ODE in a family of ODE's

In my research I have stumbled across the following 1st order complex differential equation for smooth functions $\eta:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}-\lbrace0\rbrace$ defined on the circle, $$...
1
vote
1answer
60 views

The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$

What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of ...
0
votes
1answer
118 views

Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function

Edit : According to the comments of Michael Renardy and Christian Remling I revise the question as follows: Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such ...
3
votes
2answers
119 views

ODEs whose finite-time solutions are not L^2 on their interval of definition

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be analytic and consider the ODE $$x'(t)=f(x(t)).$$ It is well-known that if $(t_{min},t_{max})$ is the maximal domain of a solution $x$ and $t_{max}<\infty$, ...
1
vote
0answers
28 views

How to find maximize the variable in the given scenario?

Consider an aloha like wireless communication algorithm with $n$ nodes, with each node transmitting with an exponential distribution with rate $\lambda_o$, with $\tau$ be the packet transmission time. ...
1
vote
0answers
47 views

Ratio dependent predator prey model

In the article on Global qualitative analysis of a ratio-dependent predator–prey system- Kuang, 1998 The system is where a, K, c, m, f, d are positive constants that stand for prey intrinsic ...
2
votes
1answer
45 views

The number of limit cycles of a quadratic vector field with a unique singularity

Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
11
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2answers
424 views

Solving ODE via contact geometry

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential ...
5
votes
2answers
220 views

Reference needed: $C^r$ convergence of Euler's method

Let $U\subset R^n$ be open, $F\colon U\to \mathbb{R}^n$ a $C^\infty$ vector field, and $x(t)$ the solution of $$x’ = F(x)$$ with initial condition $x(0) = y$, which we assume defined at least for $t\...
2
votes
1answer
179 views

Isochronization of quadratic vector fields with center

What is a classification of all quadratic vector fields $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}\qquad (V)$$ with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...
3
votes
1answer
153 views

An explicit formula for a flat metric compatible to certain polynomial vector field with center

Let $X$ be the following vector field on the plane: $$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$ The vector field $ (X)$ has a non isochronous center at the origin.The ...
-1
votes
1answer
84 views

transforming a Ricatti equation into a generalised Ricatti equation [closed]

C̶o̶n̶s̶i̶d̶e̶r̶ ̶a̶ ̶R̶i̶c̶a̶t̶t̶i̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶r̶m̶ $$ y' + y^2 = S(x), \qquad \qquad \qquad (1)$$ w̶h̶e̶r̶e̶ ̶$̶S̶(̶x̶)̶$̶ ̶i̶s̶ ̶a̶ ̶m̶e̶r̶o̶m̶o̶r̶p̶h̶i̶c̶ ...
3
votes
1answer
195 views

Inverting the cumulative probability function to find roots of stochastic function

Given a function: $$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$ where $\Phi$ is the cumulative density function of the standard normal ...
2
votes
1answer
63 views

Boundedness of particle motion with time-varying force

Consider the differential equation $$ m \ddot{x} + k \dot{x} = - W_t x $$ where $m$ and $k$ are nonnegative. $x_t \in \mathbb{R}^n$ $W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...
2
votes
0answers
73 views

Positively invariant with respect to nonlinear dynamics

I have the set of nonlinear differential equations describing a system I modeled for my research (spread of epidemics or information for instance): $$\begin{array}{rl} \dot{p}(t) &= \gamma r(t)-u(...
1
vote
0answers
43 views

How to solve differential equation for cylindrical diffusion?

How the differential equation for diffusion along a hollow cylinder, $$ \frac{\partial c}{\partial t} = D \Biggl(\frac{1}{r^2}\frac{\partial^2 c}{\partial \phi^2}\ + \frac{\partial^2 c} {\partial z^2}...
1
vote
0answers
71 views

When is $|\int_a^b \exp(-izx)f(x) \, \mathrm{d}x| \leq |\int_a^b f(x) \, \mathrm{d}x|$ for a general $f$? [closed]

Obviously, if $f(x) \geq 0 $ on $(a,b)$, the above claim is trivial. We also see that if $a = -\infty, b = \infty$, above is claiming that $|\hat{f}(z)| \leq |\hat{f}(0)|$ for all real $z$, where $\...
1
vote
0answers
41 views

Two semi stable limit cycles with disjoint interior

What is a precise example of a quadratic vector field on the plane with at least $1$ semi stable limit cycles? Furthermore, is there a quadratic polynomial vector field on the plane with two ...
4
votes
1answer
123 views

Parametric ODEs - when do there exist solutions independent of the parameter?

I have a complicated 3rd-order ODE of the form $P(y, y', y'', y''') = 0$, where $P$ is a complicated polynomial (5th-order with 24 terms) and coefficients that are (unknown) functions of a parameter $\...
2
votes
1answer
53 views

propagation speed for a modified wave equation

Given a 1+1 dimensional wave equation ($c$ constant) plus a small ($\left|k\right|\ll 1$, $k$ imaginary (?)) third order derivative in $x$ term, $$ f_{tt}=c^2\ f_{xx}+k\ f_{xxx} $$ is there a ...
0
votes
1answer
103 views

Numerical stable soliton solution

It is well known that the non-linear equation $f'' + 2f(1-f^2) = 0$ admits a soliton solution $f = \tanh(x)$. Is it possible to solve this equation numerically? For example on a finite interval $[-L,...
1
vote
0answers
132 views

A cubic system with two nested limit cycles with opposite orientations(2)

The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
1
vote
0answers
34 views

Error bounds for non-autonomous systems with respect to input

The error bounds for an ordinary differential equation: \begin{equation} \dot{x}(t) = f(x(t)) \end{equation} with respect to initial conditions $x(t_0) = x_0$, $\hat{x}(t_0)=\hat{x}_0$ \begin{...
3
votes
0answers
138 views

Flat Riemannian metrics adapted to quadratic vector fields with center

Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
3
votes
0answers
78 views

A criterion for a differential equation to be realized as an Euler-Lagrange equation on the infinite dimensional space

I study PDEs that arise in fluid dynamics in an infinite dimensional Riemannian geometric perspective. For example, Ebin-Marsden(1970) showed that the group of volume preserving diffeomorphisms has an ...
1
vote
0answers
90 views

Existence theorems Volterra Equation of second kind on unbounded domains

The general Volterra Equation of the second kind in convolution form can be described by: $$ \phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a $$ Suppose we wish to ...