Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,637
questions
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6
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The stability of the equilibria of a non-linear ODE system
I have the following coupled non-linear ODE system, which describes a biological system:
$$\frac{dp}{dt} = -\gamma p f,$$
$$\frac{df}{dt} = \gamma p f,$$
$$\frac{dT}{dt} = \left( 1 - \frac{k}{T} \...
0
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0
answers
36
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$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
1
vote
0
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92
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$
I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
2
votes
1
answer
167
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Frobenius theorem and the size of integral manifold
Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and
$[X,Y]:=XY-YX=0$.
Then by ...
3
votes
3
answers
281
views
Generalized Fuchsian-type PDE?
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
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19
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Neumann vs Stefan conditions in Free Boundary Problems
Suppose I have a free boundary problem of the form in which, in an interval $(\alpha(t),\beta(t))$, we have $u_t(x,t) = \mathcal{L}u(x,t)$, and $u(x,t)=0$ for $x\notin(\alpha(t),\beta(t))$, for some ...
4
votes
0
answers
384
views
A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
2
votes
1
answer
357
views
The Fourier transform of the Liouville function?
The Liouville function in number theory is defined as:
$$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$
Taking the discrete time Fourier transform and then taking the ...
-1
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answers
33
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Analysis of an autonomous plane system
I am studying a plane autonomous system described by the differential equations:
$$\frac{dx}{dt} = f_\delta(x,y), \quad \frac{dy}{dt} = g(x,y),
$$
where $f_\delta,g$ are rational functions in $x,y$ (...
5
votes
2
answers
3k
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Nicer expression for 2.1369288...?
In Drift Analysis and Evolutionary Algorithms Revisited by Johannes Lengler and Angelika Steger in Theorem 10, there is mention of a constant "$2.2$", and in the proof it becomes apparent ...
0
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99
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Who first gave a result stronger-or-equal to this one on ODEs
After some thinking I've come to the following conclusion.
Consider the initial value problem $$\text{(P)}\begin{cases}x'(t)=f(t,x(t)),\quad t\geq t_0\\x(t_0)=x_0 \end{cases}$$ where $f:D\subset\...
33
votes
8
answers
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Motivation and physical interpretation of the Laplace transform
Concerning the one-sided Laplace transform,
$$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$
what is a motivation to come up with that formula? I am particularly interested in "physical&...
1
vote
1
answer
84
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Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
3
votes
1
answer
98
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Fréchet-valued symbols
Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
1
vote
1
answer
120
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Existence of solution to nonlinear first order PDE with C^1 bounds
I'm looking for general existence of a PDE of the form
$$ f: U \times [0, \delta) \to \mathbb{R}$$
$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$
where $f(p,0)$ is prescribed and $F$ is non-linear ...
0
votes
0
answers
25
views
Tableau and its first prolongation for linear Pfaffian systems
This question concerns characterization of tableau associated with an exterior differential system (EDS).
On the one hand, we have prop 4.2 in the EDS book by Bryant et al.:
Given an EDS on a manifold ...
0
votes
1
answer
72
views
Same occupation measure $\Rightarrow$ same trajectory
Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
The occupation ...
0
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0
answers
49
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Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
1
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0
answers
63
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Looking for examples of 3rd-order contact transformations
In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated:
As a final ...
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73
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What is the PDE corresponding to this weak formulation?
Consider a flow $(\mu_t)_{t\ge 0}$ such that
every $\mu_t$ is a probability on $\mathbb R_+$;
$\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
0
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0
answers
103
views
One solution is known, how to find another one
This question is driven by pure curiosity as for all practical purposes I can generate numerical or series solutions of the given ODE
\begin{align}
p(a,u) [a'(u)]^2+q(a,u)a'(u)+a(u) &= 0,\\
a(0) &...
7
votes
1
answer
298
views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
2
votes
0
answers
52
views
On improving the regularity of solutions to nonlinear parabolic pde
There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
1
vote
0
answers
132
views
$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$
I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
5
votes
1
answer
360
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Asymptotic solution of a system of ODEs
I have asked this question on math.stackexchange, however, have not got any answer. Therefore, I suspect that this system of ordinary differential equations cannot be solved analytically. But I still ...
0
votes
0
answers
35
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System of equations with one integral equation
Start with a system of three equations such that two of the equations are ordinary or partial differential equations, but one of them is an integral equation as follows:
$C = \int_{0}^{\infty} X \: ...
0
votes
0
answers
45
views
Numerically finding periodic solution to Riccati equation with a scalar unknown parameter
We have a Riccati equation with an unknown scalar parameter $a$. It is proven to have unique real solution pair $\{y(x), a\}$ exists so that $y$ is periodic:
$$
y'=ap_{0}(x)+q_{0}(x)+q_{1}(x)y+q_{2}(x)...
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votes
0
answers
130
views
Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
2
votes
2
answers
132
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Upper bound estimation for second-order variable-coefficient ODE
I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of
$y(t)$ s.t $|y(t)|\le M$. The equation in question ...
5
votes
0
answers
206
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
2
votes
0
answers
137
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
1
vote
0
answers
64
views
Complex integration in a separable EDO for an eigenvalue problem
I'm trying to prove a result from Leon Cohen's book "Time-frequency Analysis", Chapter 18. Namely, I want to verify the solution of the eigenvalue problem $$\mathcal{C}s(t) = c s(t)$$ for ...
0
votes
1
answer
152
views
A self-consistent equation that turns into a differential equation
Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation
\begin{align*}
& f(x,y) = \frac{1}{1-y} + ...
0
votes
1
answer
249
views
How to force my differential equations give a bounded solution?
I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation):
$$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...
2
votes
0
answers
95
views
On the decay of a supersolution of a Sturm Liouville eigenvalue problem
I am not really familiar with Sturm-Liouville theory, so possibly the answer to my question is rather trivial. Consider the SL problem
\begin{equation*}
L \varphi (x) : = \frac{d}{dx}\Big[ (x^2+1)^\...
2
votes
0
answers
108
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
5
votes
1
answer
207
views
Can a solution to this parameterized ODE converge to zero?
Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$?
$$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$
We are also given y(0) = 2/3, $...
1
vote
2
answers
187
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
3
votes
0
answers
95
views
Distance between solutions of differential inclusions
Suppose that we have two differential inclusions
$$\frac{dY^1}{dt}(t)\in b_1(Y^1,t)$$
with $Y^1(0)\in Y_0^1$ and
$$\frac{dY^2}{dt}(t)\in b_2(Y^2,t)$$
with $Y^2(0)\in Y_0^2$.
Can we then control $d(Y^1(...
1
vote
0
answers
63
views
Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
11
votes
4
answers
846
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Interesting examples of systems of linear differential equations with constant coefficients
In this paper, Gian-Carlo Rota wrote:
A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal
processing, even in ...
1
vote
1
answer
169
views
Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?
Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system?
that is, does ...
0
votes
0
answers
20
views
Nonnegative eigenvector form the point of view of variational characterization
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$ $(N\geq 3)$. We denote by $G(x, y)$ the Green function for the boundary value problem
$$
-\Delta_x G(x, y)=c_n \delta(x-y) \quad \text { in } \Omega, ...
0
votes
0
answers
48
views
Sufficient conditions for chain recurrent set equal to set of non wandering points
Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
0
votes
1
answer
102
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Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?
The special form of Riccati equation
$$
\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}
$$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$.
The sufficiency is ...
5
votes
0
answers
211
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
6
votes
1
answer
832
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
22
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
3
votes
0
answers
48
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
0
votes
0
answers
24
views
Coefficients in the series expansion of a central manifold are all zero
I have a system of 4 ODEs, which linearized around the origin gives
$$
\begin{align}
&\dot{q_1}=a\, q_1\\
&\dot{q_2}=b\,q_2\\
&\dot{q_3}=0\\
&\dot{q_4}=c\,q_4
\end{align}
$$
with $a$, $...