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
107 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
2
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1answer
262 views

Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
2
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0answers
109 views

Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...
2
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0answers
61 views

Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...
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1answer
56 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
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0answers
71 views

Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous function, $ \omega $ be a modulus of continuity and assume $$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$ whenever $ x,h \in \...
-1
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1answer
67 views

Closed formula for a homogeneous second order linear ODE [duplicate]

Let $A, B, C\geq 0$ be constants. Is there an explicit formula to a nontrivial solution to the homogeneous linear ODE $$y''(t) -(A+B\,\sin t)\,y'(t) -C\, y(t)=0$$ for $t\in(0,2\pi)$ with periodic ...
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1answer
84 views

Explicit solution for one-dimensional Gelfand problem

I wonder if the ODE $y''+e^{y}=a$ can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$...
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0answers
55 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[x]...
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0answers
63 views

Scattering in (pseudo-)Riemannian spaces

I will ask my question in a broad way, leaving a lot of freedom for answers. Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
2
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1answer
76 views

Second order linear ODE question with boundary conditions

I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't ...
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0answers
46 views

Explicit solution of a Cauchy-type singular integral equation with regular part

I am doing research on the Riemann boundary value problem for bi-half-planes, and in a certain case I was able to reduce this problem to a linear singular integral equation of the form $$\left(\...
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0answers
61 views

inverse problem to resolution of the identity

Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
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0answers
36 views

Concrete examples of ODEs/PDEs arising in proofs in Complexity Theory and other subfields of CS

Can someone give me specific examples, (if and) where ODEs/PDEs arise in subfields of computer science ? What I'm not looking for are examples from numerical analysis or parts of computer science, ...
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2answers
176 views

Is there a true many-body green's function for interacting systems?

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition: ...
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3answers
250 views

Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form: $(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} {x^2(...
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1answer
54 views

What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for $$ u''+u'-i\lambda V(x)u=0, \, x\in [0,1], $$ What is the ...
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1answer
169 views

Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)

Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying $$ D(a.v)=(t\frac{d}{dt}a)....
1
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1answer
63 views

Frobenius series of Fuchsian PDEs

I'm interested in the analyticity of Frobenius-like series solutions to a PDE in $z=(z_1,\ldots ,z_N)\in\mathbb{C}^N$ with regular singular behavior at $z_\alpha=0$ for all $\alpha=1,\ldots, N$. For ...
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0answers
94 views

2nd order partial differential equation with non-constant coefficients

During my research I came across the following differential equation: $$f(x,y) = \left(y^2 \partial_{x}^{2} + x^2 \partial_{y}^{2} \right)f(x,y)$$ Any ideas how to solve it without using series ...
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1answer
835 views

Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...
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0answers
791 views

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
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1answer
64 views

Trying to solve this non-linear differential equation

I have a second order differential equation given by: $x''(t) = \frac{exp(-\frac{x(t)^2}{4t})}{A \sqrt{t}}$ I would very much like to be able to obtain an analytic solution to this equation, which ...
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0answers
73 views

Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, c=5....
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1answer
108 views

Is there any solution for this PDE system?

‎Let $(\mathbb{R}^2,\langle‎ .‎,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with‎ ‎\begin{align}‎ ‎J_{\...
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0answers
24 views

numerical differentiation of sum of one-dimensional sinusoids with angular frequency close to Nyquist one

Suppose that $f(t) = \sum_i C_i e^{i\omega_i t}$, and $f$ is sampled at certain sampling angular frequency $\omega_s$. All $\omega_i$s are very close to $\omega_s/2$, and thus standard finite ...
3
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0answers
108 views

Weak form of the Laplace-Beltrami operator on closed manifolds

Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have \begin{align} \int_M W \Delta U \mathrm{d} x & = -\int_M \...
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1answer
166 views

An example for affine function [closed]

I'm looking for an example of a non-Euclidean non-compact Riemannian manifold $(M,g)$ such that we could define a non-constant affine function $f:M\rightarrow \mathbb{R}$, namely its gradient vector ...
2
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2answers
212 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta u_{t}+\...
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2answers
170 views

Existance of Integrating Factors, a Constructive Proof

Being a novice with differential equations, I recently learned that if $\mu$ is an integrating factor for $\frac{dy}{dx}f(x,y)+ g(x,y)=0$, then the corresponding 1-form, $\mu fdy+\mu g dx$, is exact. ...
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0answers
100 views

Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form: $\frac{d U_t}{dt} = (A + w(t)B)U_t$ where $A,B \in \mathfrak{su}(n)$ generate the ...
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1answer
135 views

Exact solution to nonlinear differential equation sought

I am looking for an exact solution to equation: $w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants. Of course I can ...
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1answer
144 views

How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely $$x(t)=a_0+b_0e^{-t}+\epsilon(...
0
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0answers
53 views

solution of a mixed difference and differential equation

Is it possible to solve the following difference and differential equation: $a\frac{d f(x)}{d x} = \frac{d f(2x)}{d x}$, where $a<1$ and $\int_0^\infty f(x)=1$.
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70 views

Finite blowup time for a simple ODE

Consider the ODE $$ y'(x) = C y(x)^{2-x},$$ where $C$ is a positive constant. I suspect there is no closed-form solution. I want to understand the constant $C$ such that for a given $y(0) > 0$,...
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0answers
70 views

Is the Wave Function a “Smooth” Function of the Potential?

Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...
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1answer
56 views

Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer: http://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y Consider autonomous ODE $y' = ...
0
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1answer
63 views

Ordinary homogeneous differential equation [closed]

How to solve this one $y''=(2xy - \frac{5}{x})y' + 4y^2 - \frac{4y}{x^2}$ I know it's homogeneous. I've made replacement $x = e^t$ and $y = ze^{-2t}$ but I had no result.
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2answers
90 views

Index Reduction of Differential Algebraic Equations by Hand

I dont really understand how to reduce the index of DAEs ? Does Reducing the index of DAE result in an ODE ? How would I reduce the index of the DAE by Hand ? Say I have : $$ \begin{matrix} ...
4
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1answer
158 views

Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?

The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix $$ \left\|\frac{\partial^2u}{\partial x^i\...
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1answer
99 views

gradient descent in space of functions

Differential equations of the form $$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$ can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...
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2answers
149 views

A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy $-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$, where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
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1answer
159 views

Schauder estimate for the heat equation on compact manifolds

I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here. Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in C^...
2
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1answer
192 views

General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
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2answers
685 views

What does it mean for a differential equation “to be integrable”? [duplicate]

What does it mean for a differential equation "to be integrable"? Are "integrable" and "solvable" synonyms? The first thing that comes to my mind is to say: it's integrable if we can find the ...
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0answers
42 views

Algebraic invariants of linear ODE's with constant coefficients

Let consider linear ODE with constant coefficients: $$y^{(n)}(x) + A_{n-1} y^{(n-1)}(x) + ... + A_1 y'(x) + A_0 y(x)= 0.$$ It admits some equivalence point transformations, that preserves its ...
1
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1answer
332 views

On Harmonic Unit Vector Fields

When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g X=...
5
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1answer
113 views

A special function solution of a fourth-order ODE

I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier ...
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1answer
494 views

A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry. It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
1
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1answer
66 views

Holomorphic vector field with infinite separatrix

Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...