# Questions tagged [differential-equations]

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

1,224
questions

**0**

votes

**0**answers

24 views

### Peano Baker solution to a LTV differential equation

The solution to the following transition matrix differential equation
$$
\dot{\Phi} (t,t_{0}) = A(t)\Phi (t,t_{0}), \Phi (t_{0},t_{0}) = I
$$
is given by:
$$
\dot{\Phi} (t,t_{0}) = I + \int_{t_{0}}^{t ...

**1**

vote

**0**answers

78 views

### Show that the manifold interior is invariant under this flow

Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...

**1**

vote

**0**answers

30 views

### Non constant delay differential equations

Given $\varphi:[0,1] \to [0,1]$ a continuous function, let $(E)$ be the delay differential equation (I am not sure about the terminology, as the delay is non constant): $y'(t) = y(\varphi(t))$. It is ...

**3**

votes

**1**answer

320 views

### What is Bouziani space and what are its applications in mathematics?

I have accrossed a new topological space seems were derived from Hilbert Space and it used to solve some boundary value problem for PDE and ODE , Inspired by this paper (page 4, Definition 3.1) , The ...

**-2**

votes

**0**answers

42 views

### Discretization formula for system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation
\begin{equation}
dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1}
\end{equation}
where $A$, $B$ and $C$ are parameters ...

**13**

votes

**2**answers

780 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

518 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 ...

**1**

vote

**0**answers

72 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 ...

**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

218 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 $1 \le p <2$, ...

**1**

vote

**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, \; \...

**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, ...

**1**

vote

**1**answer

53 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}{\...

**7**

votes

**1**answer

114 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}...

**1**

vote

**0**answers

74 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)...

**2**

votes

**1**answer

85 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

210 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],\...

**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**

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}{...

**1**

vote

**0**answers

88 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{...

**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 ...

**5**

votes

**0**answers

72 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)-...

**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 ...

**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 -...

**4**

votes

**2**answers

171 views

### Approximated solutions of SEIR models

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

**7**

votes

**2**answers

409 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 ...

**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

113 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 ...

**5**

votes

**0**answers

106 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 $\...

**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, ...

**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

**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

**0**answers

37 views

### A Mathieu-like equation with a quadratic term

I have the following equation:
$$y''(x)+\left[a-bx^2-c\cos(2x)\right]y(x)=0.$$
This equation is the Schrodinger equation for a quantum pendulum with a quadratic term. If we set $b=0$, the solutions ...

**1**

vote

**0**answers

48 views

### propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$
$$ \partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$
with given initial condition $u_0(x)$.
Assume that:
$u_0$ is rotation invariant, ...

**3**

votes

**0**answers

73 views

### Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...

**0**

votes

**0**answers

28 views

### Polynomial approximations of the vector field and distance between generated flows

Let $\textbf{h} = (h_1,...,h_n)$ be a $C^1$ system of ODEs defined everywhere on on some compact subspace $\mathbb{X} \subset \mathbb{R}^n$.
Suppose we have a polynomial approximation $\textbf{p} = (...

**2**

votes

**1**answer

97 views

### Dynamical system described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...

**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; \...

**30**

votes

**3**answers

2k views

### What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...

**1**

vote

**1**answer

82 views

### Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...

**2**

votes

**0**answers

93 views

### Solutions of the differential equation $f'=(f^{-1})^{[n]}$

For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here)...

**0**

votes

**0**answers

26 views

### What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...

**7**

votes

**1**answer

162 views

### Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...