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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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 ...
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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 $L^{1, \delta}$?
T. Amdeberhan's user avatar
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Distributional PDE solutions as topological linear duals of PDE solutions

Let $$ P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast) $$ be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
Urs Schreiber's user avatar
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On modified Bessel solutions to complex ODE's using Kummer's series

I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...
Spoilt Milk's user avatar
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Basin of attraction of gradient flow

Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
Rikimaru's user avatar
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When the integral of the product of two matrix exponentials is singular?

Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $...
nadia's user avatar
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Symbol of differential operator and change of variables [closed]

Recently I posted the following question on stack exchange, but it remained with no answer https://math.stackexchange.com/questions/1863658/symbol-of-differential-operator-and-change-of-coordinates I ...
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Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...
Mehmet Ozan Kabak's user avatar
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Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul Sutcliffe. ...
Joseph O'Rourke's user avatar
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Models used for the Zika virus?

I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ...
Kimball's user avatar
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Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
Idempotent's user avatar
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Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
Basil's user avatar
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System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$: $$ \begin{align}x G''(x)=&\mathscr{...
Eckhard's user avatar
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Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
Felice Iandoli's user avatar
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May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa)

By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ...
Vesselin Dimitrov's user avatar
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4 answers
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Nonlinear second order ODE $y''+f(x)y=g(x)y^3$

I encountered the following ODE in order to find a solution for Einstein equation $$y''+f(x)y=g(x)y^3.$$ It seems to me that it is not among the solvable nonlinear second order differential equations....
Masoud's user avatar
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4 answers
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Solution of second order differential equation with singularities at 0,1, and ∞

I am trying to solve the following equation; $$ U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0 $$ where U is a function of t and C is constant. The above ...
Ravinder Singh's user avatar
3 votes
3 answers
305 views

Generalized Fuchsian-type PDE?

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
Math2024's user avatar
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2 answers
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Application for Differential Equation of higher order [closed]

We found some interesting insights in differential equations of the form $y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$, i.e. for ordinary differential equations of $n$-th order with $n\geq2$....
Ben's user avatar
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existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference. Let $k$ be some (algebraically closed, if needed) field. There is a ...
Dima Sustretov's user avatar
3 votes
2 answers
274 views

Existence and uniqueness of solutions to a distributional ordinary differential equation

Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation $$ \dot{x}(t) = v(x(t)) $$ which I might interpret as an integral equation $$ -\...
cheshircat's user avatar
3 votes
2 answers
463 views

Second order differential equation with oscillating behavior

I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
Sasha's user avatar
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2 answers
551 views

Inverted harmonic oscillator

I am looking for the spectrum of th inverted oscillator $H=-\frac{d^2}{dx^2}-x^2$. Thanks in advance.
Fadil Kikawi's user avatar
3 votes
1 answer
333 views

Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response. For $i=1,\ldots, r$, ...
A. J. Pan-Collantes's user avatar
3 votes
2 answers
382 views

Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$

The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function). $$(x^2y')'-x^2y=\lambda \;y$$ Now for a higher-degree ...
Bertrand's user avatar
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3 votes
2 answers
328 views

Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations

In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...
UserA's user avatar
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Reference request - existence of movable essential singularities

On the Wikipedia page regarding the Painlevé transcendents it says: Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of ...
user1337's user avatar
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An explicit formula for a flat metric compatible to certain polynomial vector field with center

Let $X$ be the following vector field on the plane: $$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$ The vector field $ (X)$ has a non isochronous center at the origin.The ...
Ali Taghavi's user avatar
3 votes
1 answer
2k views

closed form solution of an ODEs

I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help ...
Steven's user avatar
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2 answers
805 views

How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$. Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$. Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
Yuchen Liu's user avatar
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3 votes
2 answers
436 views

Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation: \begin{eqnarray} (1-x^2)y''-2xy'+l(l+1)y=0. \end{eqnarray} Doing some calculations, we ...
Leonardo S. Vieira's user avatar
3 votes
1 answer
942 views

Continuation (extension) of harmonic functions

Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
Ali's user avatar
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3 votes
1 answer
231 views

Are there fundamental solutions of the laplacian that decay rapidly?

The question I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function $$ \...
ClemensB's user avatar
3 votes
2 answers
381 views

Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
7iat's user avatar
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3 votes
2 answers
906 views

Herpolhode equation

Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of ...
quantropy's user avatar
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3 votes
2 answers
320 views

parallel transport along $W^{1,2}$-curves

Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t_0\in I$ and $\xi_0\in T_{c(t_0)}Q$ be arbitrary. I am looking for a citeable reference ...
Orbicular's user avatar
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3 votes
3 answers
829 views

Analytic ODE with complex time

Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r. I would like to understand: 1) if there exists an analytic flow $\phi_t(x)...
Marco Disce's user avatar
3 votes
3 answers
2k views

Error analysis of implicit functions

I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms): $$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
tralston's user avatar
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3 votes
1 answer
273 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],\...
0xbadf00d's user avatar
  • 161
3 votes
2 answers
3k views

General formula for integrating factor of an homogeneous differential 1 form

This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
Hector Blandin's user avatar
3 votes
2 answers
393 views

Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator $$ -\frac{d^2}{dx^2} + V(x) $$ Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $. Now of course,the ...
Surajit's user avatar
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3 votes
2 answers
124 views

How to find an annihilating linear ode for a class of integrals depending on a parameter?

Given two polynomials $a(t)$ and $b(t)$, consider the integral $$C(z)=\int_0^1 \frac{dt}{\sqrt{(z-b(t))^2+a(t)}}.$$ For $z$ near $\infty$, $C(z)$ is a well-defined analytic function which can be ...
user53092's user avatar
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1 answer
361 views

Differential equations → predicate logic mapping

I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark): I was gonna say, there was a book I ...
Matěj G.'s user avatar
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1 answer
278 views

On Wazewski's theorem on system of differential inequalities

According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem: Consider the system of differential inequalities given by $$ \...
Tadashi's user avatar
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3 votes
1 answer
9k views

how to solve this system of nonlinear differential equations

I want to solve this system \begin{align*}\tag{*} x'(s)=x^2(s)+y(s), y'(s)=x(s)y(s) \end{align*} with initial conditions $$x(0)=t, y(0)=t,$$ where $t\not=0.$ With the help of Maple, the solution is ...
azhi's user avatar
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3 votes
1 answer
421 views

Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation: $$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$ for $x>1$. If that is not possible, I would at ...
Ivan's user avatar
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3 votes
2 answers
705 views

Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
Tom LaGatta's user avatar
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3 votes
1 answer
123 views

A type of singular limit for systems of differential equations

Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles), $$\begin{cases}\dot{x}_{i,\...
Matt Rosenzweig's user avatar
3 votes
1 answer
199 views

Is there a theory of "elementary closed form solution" at the operator level for differential equations?

We begin by considering the usual general first order linear equation of the form $$ a_0 y' + a_1 y + a_2 = 0 $$ Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
Sidharth Ghoshal's user avatar
3 votes
2 answers
426 views

A proposition for summing divergent series, but how should partial summation be defined at non-natural values?

Introduction I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
Caleb Briggs's user avatar
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