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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.

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28 views

Proving $\phi_1(t)=\phi_2(t)$ for if both satisfy $y'=t+y^2,y(0)=0$ on $[0,1]$

Suppose $\phi_1(t)$ and $\phi_2(t)$ are solutions to a particular initial value problem: $y'=t+y^2, y(0)=0$ on the interval $[0,1]$. Show that $\phi_1(t)=\phi_2(t)$ for all $t\in[0,1]$. If I use ...
-1
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0answers
18 views

Find General Solution with initial value [on hold]

How can I determine the general solution for dy/dx = (x^2-y^2)/(2xy-2y) with initial value of y(0)=1? Sorry I am stuck so would appreciate any help.
2
votes
1answer
161 views

Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$. I ...
1
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0answers
64 views

Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
2
votes
1answer
79 views

The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.) For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$ ...
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0answers
71 views
+50

Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+, $$ ...
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0answers
37 views

Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$: \begin{equation}...
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0answers
46 views

Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...
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0answers
54 views

Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$. Which vector fields $F$ are so that ${\rm div} (F)= f$ ?
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0answers
38 views

Solutions of nonlinear equations with multiple parameters

In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form: $$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...
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0answers
70 views

One-point partition

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$. $$ \mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...
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0answers
66 views

Solutions to the Bond Pricing Equation

Consider a spot rate of the form: $dr = (\eta - \gamma r) dt + \sqrt{\alpha r + \beta} dW$ where all parameters are constants. Lets look for a solution of the form $Z(r; t) = e^{A(t;T) - r B(r; T)...
3
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0answers
119 views

Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
1
vote
1answer
55 views

Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations: $$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$ has a unique Caratheodory solution ...
0
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1answer
99 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 ...
3
votes
1answer
351 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} ...
0
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0answers
43 views

The limit of infinite ODE solver iteration with zero time step

Suppose I am trying to find a solution of an ordinary differential equation: \begin{equation} \begin{aligned} y'(x) &= f(y(x))\\ y(0) &= y_0 \end{aligned} \end{equation} on ...
2
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0answers
37 views

Floquet stochastic process

Let $X_t$ be defined by the SDE $$ dX_t = A(t, X_t)dt + dW_t $$ where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
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0answers
17 views

Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
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0answers
65 views

Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
5
votes
1answer
207 views

Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...
1
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0answers
58 views

Solve simple stochastic variational inequality

Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
-2
votes
3answers
169 views

A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Is there an analytical solution satisfying these 3 equations with non-constant z? $$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...
2
votes
1answer
152 views

Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
1
vote
1answer
95 views

Slow and fast forming singularities of the mean curvature flow

Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form. We have a type I singularity if $$ \max_{p \in M} |A(p,...
4
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0answers
261 views

Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...
1
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0answers
52 views

Matrix exponential with particular structure

Context I'm trying to numerically solve the following differential equation: $\frac{\mathrm{d} u}{\mathrm{d} t} = -Au + f$, where $u$ and $f$ are vectors, and $A$ is an $N \times N$ matrix, with $N &...
2
votes
3answers
274 views

Non-linear Basis for PDE's

Asked this on stack exchange and got no response, so I'll try here. An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...
0
votes
1answer
59 views

Does differentiating an integro-differential equation results in equivalent stability of the solution?

I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation: ...
4
votes
0answers
102 views

An embedding question: Morrey spaces

Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
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0answers
76 views

Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...
3
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1answer
349 views

Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
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0answers
21 views

Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
3
votes
1answer
250 views

Asymptotic solution for a first order ODE

Simplified question*: Given $f(t)$ that satisfies $f'(t)>0$, $f'(t)=\omega\left(t^{-1}\right)$, $\log\left(f'(t)\right)=o\left(f(t)\right)$ we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(...
3
votes
1answer
138 views

Exponential map/ Lie derivative in variation for constant formula for ODE

In short: The question is how to go from the first equation on page 8, of this paper to the second equation. Some background I'm working in optimization and I am currently reading a paper see page ...
2
votes
2answers
128 views

Criteria for Schrödinger operator on real line to have simple spectrum

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
1
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0answers
37 views

First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
3
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0answers
59 views

Limits of a simple damped system

Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$. Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$. Ideas: Let $G_n(s)=\frac{1}{s^{n+...
12
votes
4answers
707 views

History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
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0answers
109 views

Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
2
votes
4answers
211 views

Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
1
vote
1answer
73 views

Solve nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
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0answers
52 views

Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
1
vote
1answer
28 views

Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations: \begin{equation} \begin{aligned} \dot{x}_1 &= f_1(x_1,\ldots,x_n) \\ \vdots \\ \dot{x}_n &= f_n(x_1,\ldots,x_n) \end{aligned} \end{equation} In ...
0
votes
1answer
55 views

Galerkin Finite element for solving third order time dependent partial differential equation inti weak form

How to solve the third order time dependent partial differential equation (i.e. u_t + 6u_x + u_xxx = 0) into weak form using galerkin finite different method?
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2answers
89 views

One inequality connected with the linear second order ODE

Is the following statement true? Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation $ \ddot{x}+a \dot{x}+bx=h$ with initial conditions $x(0)=u<0 , \dot{x}(0)...
5
votes
0answers
102 views

Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
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0answers
67 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
6
votes
1answer
123 views

Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
1
vote
0answers
26 views

Mean of a periodic velocity field and trajectory displacement bound

Suppose $u(t,x)$ is a smooth velocity field on $[0,\infty)\times \mathbb{R}$ and periodic in space, i.e., $u(t,0)=u(t,1)$ $\forall t$. Assume that $\int_0^1 u(t,x) \,dx = c$, independent of time. Let $...