# Questions tagged [differential-equations]

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

1,220
questions

**1**

vote

**1**answer

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}{...

**11**

votes

**2**answers

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**

votes

**1**answer

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 ...

**5**

votes

**1**answer

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} ...

**1**

vote

**0**answers

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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**2**

votes

**3**answers

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 $1 \le p <2$, ...

**60**

votes

**9**answers

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 $f(t)$, ...

**1**

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**0**answers

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, \; \...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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, ...

**6**

votes

**1**answer

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 -...

**1**

vote

**1**answer

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}{\...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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)...

**3**

votes

**1**answer

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**

votes

**1**answer

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}...

**0**

votes

**0**answers

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\} - \...

**1**

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**0**answers

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 ...

**2**

votes

**1**answer

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 $...

**3**

votes

**1**answer

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^\...

**0**

votes

**1**answer

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 ...

**-1**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 \; = ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 $\varphi>0$, $\...

**0**

votes

**0**answers

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**

vote

**1**answer

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)}\...

**0**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**2**answers

170 views

### Approximated solutions of SEIR models

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

**0**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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 $\...

**0**

votes

**0**answers

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,\...

**2**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**3**answers

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, ...

**5**

votes

**2**answers

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**

vote

**1**answer

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**

votes

**0**answers

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**

vote

**2**answers

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**

vote

**1**answer

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**

vote

**0**answers

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 ...

**0**

votes

**1**answer

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 ...