Questions tagged [differential-equations]

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
120 views

Trace entropies

I'm studying relationships between trace entropy functionals and combinatorics and I'm faced with the following problem. Lets $\mathcal {D}$ be the following differential operator $1 -x\cdot \cfrac{d}{...
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2answers
744 views

Reference request: the theory of currents

I am a graduate student and want to study the theory of currents. What is a good reference for a beginner? I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
7
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1answer
475 views

Resonance arising when harmonic oscillator is excited using sawtooth

Solutions to the differential equation $my'' + ky = F \sin \omega t$ show resonance when the driving frequency $\omega$ equals the natural frequency $\sqrt{k/m}$. That is, solutions are unbounded when ...
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1answer
583 views

Conserved positive charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
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0answers
65 views

Is this definition of a Fuchsian operator correct?

In Bjork, Analytic D-modules and applications, the following definition of a Fuchsian operator is given: Here, I believe, $D(0)=\mathcal{O}$, the zeroth filtered piece of the ring of germs of ...
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1answer
161 views

Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
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0answers
38 views

What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?

When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
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3answers
216 views

How to show continuity and monotonicity of solutions to this parametrized equation?

Let $1 \le p <2$ be a parameter. Consider the equation $$ \frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1} $$ I am rather certain that for each $1 \le p <2$, ...
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9answers
21k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...
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0answers
22 views

Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem \begin{equation}\label{fainait ve} \dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...
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0answers
1k views

Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...
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0answers
58 views

Control and observability of Clifford algebra and quaternion valued systems?

Good evening everybody, I'm asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, ...
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1answer
169 views

Time of peak of an SIR epidemic

I've learned some classical results on the peak and the attack rate of an idealized epidemic which evolves according to a SIR model $\dot{s} = -\beta\cdot i \cdot s$ $\dot{i} = +\beta\cdot i \cdot s -...
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1answer
51 views

Relation between separation of variables and Hessian properties

I have a function $f(x,y)$, where both $x$ and $y$ are $n$-dimensional vectors, $n\ge 2$. I know that this function has the following property: $$ \frac{\partial}{\partial x_j} \frac{\partial}{\...
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0answers
148 views

An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
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0answers
72 views

Shape derivative at manifold $M$ in direction $v$ is equal to the shape derivative at $\partial M$ in drection $\langle v,n\rangle n$

Let $\tau>0$ and $d\in\mathbb N$. Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
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1answer
186 views

Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold

Let $\tau>0$; $d\in\mathbb N$; $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
7
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1answer
112 views

On Integrals of the Airy function

Let $Ai$ be the classical Airy function and let $(a_j)_{j\ge 1}$ be the strictly decreasing sequence of its zeroes: we have $a_{j+1}<a_j<\dots <a_2<a_1<0$, $\lim_{j\rightarrow +\infty}...
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0answers
121 views

Non linear second order PDE involving max operator (Dynamic Programming)

I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$) $$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
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0answers
129 views

Solution of $s\zeta'(s)-\zeta(s)=0$ for some infinite strip $\subset\{0<\Re s<1\}$, with $\Im s>0$

I did the following calculation, first I take the $k$-th derivative with respect $s$ of the Mellin transform of the fractional part $\{\frac{1}{t}\}$ defined for $0<\Re s<1$ as it is showed in ...
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1answer
83 views

Analytical solution to a specific differential equation

I was wondering whether there is an analytical solution to the ODE \begin{equation} -n\int xy(x)dx + ihy'(x) + (x^2+k)y(x) = 0, \end{equation} where $n=0,1,2,...$, $h \in \mathbb{R}$, and $k=+1,0$ or $...
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1answer
307 views

An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...
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1answer
96 views

A solution of a system of equations that involve directional derivatives

[Edited on 29-March-2020 to make the question clearer] Let $f, g : [0,1]^2 \to \mathbb{R}$ be two smooth functions, which are strictly increasing and concave in each coordinate. That is, for every $0 ...
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0answers
34 views

Conditions for liftings of smooth functions to converge to a smooth function

Assume we are given a uniformly convergent series of analytic functions $(f_n)_n$ in $C^\infty(\mathbb{R}^3, \mathbb{R})$ and these converge to a smooth function $f$. Further assume that we have ...
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0answers
87 views

Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?

It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$ \begin{equation} \tag{1} \left(p y' \right)' - qy \; = ...
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0answers
33 views

Asymptotic ODE behaviour

I'm interested in the behavior of a solution to the ode system \begin{equation} \varepsilon \dot x(t) = f(x(t),t) \text{ for } t \in [0,1] \text{ with } x(0) = x_0 \hspace{1cm} \text{(*)} \end{...
7
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1answer
708 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations $$ \dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
5
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0answers
71 views

Analysis of solutions to a system of nonlinear ODEs arising from differential geometry

Consider the system of ODEs: \begin{equation} \varphi''\varphi'^{q-1}\psi'^{p-2}=\varphi^{p-1}\psi^{q-1}, \end{equation} \begin{equation} \varphi'^2+\psi'^2=1, \end{equation} where $\varphi>0$, $\...
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0answers
66 views

Proving positive invariance

I need to prove that set $D$(A picture for Set $D$) given by $$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system: $$\dot{x}=k_1(R_0-x-y)(L_0-x)-...
1
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1answer
248 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)}\...
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0answers
119 views

Solutions to the differential equation $f'(x) = f^{-1}(x)$ [duplicate]

I recently got interested in this innocent looking equation : $f'(x) = f^{-1}(x)$, or equivalently $f \circ f' = x$ I was only able to find the following two solutions, thanks to the equalities $\...
1
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1answer
274 views

Is this curve well known?

I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that $\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$ $\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$ $a,b\in\mathbb{R}$ Is the ...
4
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2answers
170 views

Approximated solutions of SEIR models

Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof – $\dot{S} = - N$ $\dot{E} = + N - E/\lambda$ $\dot{I} = + E/\lambda - I/\delta$ ...
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1answer
183 views

Delay equations

In an effort to solve a delay partial differential equation $$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$ with $$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$ Where $\alpha$ is the delay ( a real ...
0
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1answer
247 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-...
7
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2answers
407 views

Finding solutions of the differential equation $x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$

In my research I have come across the following non-linear differential equation: $$x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$$ I want to find the general solution of this equation ...
5
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0answers
105 views

Uniqueness of solutions of Young differential equations

Consider the following one dimensional Young differential equation: $$ Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1]. $$ Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\...
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0answers
54 views

Differential equation

Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation $$\partial_x u - u\partial_x (\phi-\psi)=0$$ has a trivial solution $u(\phi,\...
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1answer
112 views

Is there a standard definition of weak form of a nonlinear PDE?

Comments on the question Are those distributional solutions that are functions, the same as weak solutions? suggest there might not be a standard definition of the weak form of a non-linear PDE. Is ...
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0answers
112 views

Are those distributional solutions that are functions, the same as weak solutions?

There are two closely related concepts and I am not sure exactly how close. Consider a partial differential equation. (The coefficients need not be constant but assume they are functions, and not ...
2
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1answer
66 views

How to solve a differential equation in the form $\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$?

How to find the general solution of a differential equation with a shift, in the following form? $$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$$ where $\...
2
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1answer
89 views

$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ have periodic solution $\iff\; \frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $

I have a research work concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$ f and g are defined and continuous in $\mathbb R$ and with values ​​in $\mathbb R$. ...
6
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3answers
494 views

What is an “exact solution” to a PDE?

Wolfram MathWorld says As used in physics, the term “exact” generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate, ...
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2answers
6k views

How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations: $$ \left[ \begin{array}{ccccccc} \text{No.}& t & y_1(t)&y_2(t) & ...
1
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1answer
66 views

BSDE without volatility

Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE $$ dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t $$ with terminal condition ...
0
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0answers
62 views

Manifold flows and higher-order tangent bundles

Consider the flow on a manifold $\mathcal{M}$ defined by $\dot{x} = f(x)$ with $x\in\mathcal{M}$ and $f : M\rightarrow TM$. Associated to this flow I can define the variational dynamics $\delta \dot{x}...
1
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2answers
159 views

Backward stochastic differential equation

Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
1
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1answer
48 views

Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation \begin{equation} \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
1
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0answers
69 views

Fredholmness of elliptic operator on Hölder spaces

Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
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1answer
147 views

How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...

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