Questions tagged [differential-equations]

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

Filter by
Sorted by
Tagged with
18 votes
2 answers
2k views

Big Picture: What is the connection of Malliavin calculus with differential geometry?

I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
vitp's user avatar
  • 293
17 votes
3 answers
4k views

Is there a Poincare-Hopf Index theorem for non compact manifolds?

Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If ...
Michał Oszmaniec's user avatar
17 votes
5 answers
2k views

2- and 3-body problems when gravity is not inverse-square

Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing as $1/d^p$ for distance separation $d$ and some power $p$. Two questions: Presumably the 2-body ...
Joseph O'Rourke's user avatar
17 votes
6 answers
3k views

Catenary curve under non-uniform gravitational field

The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ...
Joseph O'Rourke's user avatar
17 votes
1 answer
1k views

D-modules over algebraic curves VS differential Galois theory

Disclaimer: I know very little about both of the fields in question. My question is pretty simple: What's the relation between differential Galois theory and D-modules over algebraic curves? ...
Saal Hardali's user avatar
  • 7,549
17 votes
2 answers
1k views

The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
jdaw1's user avatar
  • 199
17 votes
1 answer
2k views

Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
Ricardo Buring's user avatar
17 votes
1 answer
1k views

Vector field built from connection and metric

Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by $\...
Martin Hairer's user avatar
16 votes
5 answers
2k views

does the j-invariant satisfy a rational differential equation?

Let $j$ be the Klein $j$-invariant (from the theory of modular functions). Does $j$ satisfy a differential equation of the form $j^\prime (z) = f(j(z),z)$ for any rational function $f$?
Michael Beeson's user avatar
16 votes
3 answers
3k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
zeraoulia rafik's user avatar
16 votes
5 answers
2k views

Are there alternative proofs for existence/uniqueness of ODE solutions?

Consider the differential equation $\dot x = f(x)$. The standard proofs are The Picard iteration based proof of existence/uniqueness for Lipschitz $f$. The Peano existence theorem for continuous $f$...
Sujit_Nair's user avatar
16 votes
2 answers
2k views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
Armin Straub's user avatar
  • 1,372
16 votes
1 answer
1k views

What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
David Spivak's user avatar
  • 8,549
16 votes
0 answers
2k views

The radius of convergence of the p-adic exponential function.

As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is $$\rho = p^{-1/(p-1)}.$$ This is typically proven by computing ...
Marty's user avatar
  • 13.1k
15 votes
5 answers
3k views

Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ) )$?

I am trying to solve the following differential equation $$f ' (x) = f( f( x ) ),$$ but I have no idea how. I don't think the chain rule is useful for this. Although I don't think this differential ...
frigen's user avatar
  • 253
15 votes
2 answers
1k views

What justification can you give for the fact that "most ODEs do not have an explicit solution"?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?
user3078439's user avatar
15 votes
2 answers
2k 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^...
Pixie's user avatar
  • 151
15 votes
9 answers
4k views

Newton equations, second order equation and (im)possible motions

I am am currently studying Newtonian mechanics from a conceptional and axiomatic point of view. Now, if I am not mistaken, one (but surely not all) statement of Newtons second law about nature is, ...
student's user avatar
  • 1,212
15 votes
6 answers
1k views

An example of a series that is not differentially algebraic?

Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
Vladimir Dotsenko's user avatar
15 votes
2 answers
2k views

Solvability in differential Galois theory

It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative. The proof I know goes as follows: Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
the L's user avatar
  • 1,204
15 votes
1 answer
794 views

Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
Tom Price's user avatar
  • 804
14 votes
2 answers
878 views

Solution to differential equation $f^2(x) f''(x) = -x$ on [0,1]

I'd like to solve a differential equation $$ f^2(x) f''(x)=-x $$ where $f(x)$ is defined on $[0,1]$ and has a boundary condition $f(0)=f(1)=0$. I somehow found out that the solution is fairly close ...
Seungki Min's user avatar
14 votes
1 answer
2k views

On the non-rigorous calculations of the trajectories in the moon landings

In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible ...
teil's user avatar
  • 4,261
14 votes
1 answer
839 views

When can Witten-esque moduli spaces be used to define invariants of geometric structures?

I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds. It is striking ...
user11743's user avatar
  • 242
14 votes
1 answer
549 views

Projective-invariant differential operator

This question was originally asked on Math StackExchange. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = ...
user76284's user avatar
  • 1,793
14 votes
1 answer
6k views

Exactness of 2nd-Order Differential Equations via Differential Forms

This (probably very elementary) question came up the last time I taught differential equations, and I've been toying with it for a while with no success: A 1st-order differential equation $M(x,y)dx+N(...
Cam McLeman's user avatar
  • 8,417
14 votes
2 answers
1k views

ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension. The first one I know is the Peano existence theorem. I ...
mathcounterexamples.net's user avatar
14 votes
2 answers
369 views

Is there a singularity theorem in higher-dimensional Newtonian gravity?

In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction ...
Adam B's user avatar
  • 273
14 votes
2 answers
2k views

Picard-Fuchs equations for modular functions

Hello, MathOverflow community! Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil ...
Bruno Joyal's user avatar
  • 3,860
14 votes
1 answer
765 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D Monge-...
14 votes
2 answers
1k views

Modular equations for quasimodular forms

This problem is motivated by this question and by teaching modular polynomials for the classical modular invariant $j(\tau)$. The latter implies that if we consider the fields of modular functions $\...
Wadim Zudilin's user avatar
14 votes
1 answer
989 views

Computing spectra without solving eigenvalue problems

There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
Victor Galitski's user avatar
14 votes
1 answer
1k views

The perturbation of non-Hamiltonian algebraic vector fields

In this question, we are interested in the number of limit cycles which appears in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
Ali Taghavi's user avatar
13 votes
2 answers
2k views

Frobenius Theorem for subbundle of low regularity?

Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$...
Zarathustra's user avatar
  • 1,404
13 votes
2 answers
840 views

Motivation for BMO

At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
MLevi's user avatar
  • 261
13 votes
3 answers
600 views

Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
Joseph O'Rourke's user avatar
13 votes
3 answers
4k views

PDEs, boundary conditions, and unique solvability

I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the ...
Igor Khavkine's user avatar
13 votes
2 answers
1k views

Young-Fibonacci version of Nekrasov-Okounkov

This question addresses a hierarchy of linear recurrences which arise from an attempt to generalize the Nekrasov-Okounkov formula to the Young-Fibonacci setting. A related posting extensions of the ...
Jeanne Scott's user avatar
  • 1,847
13 votes
1 answer
592 views

Poincaré on analytic dependence on parameters of solutions of linear differential equations

There is the following important General Principle: if a parameter enters in a linear differential equation additively, for example $$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$ where the parameter is $\...
Alexandre Eremenko's user avatar
12 votes
5 answers
3k views

Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
sam's user avatar
  • 429
12 votes
3 answers
1k views

Vector field with holomorphic flow

Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms. I know one reference ...
Nick L's user avatar
  • 6,923
12 votes
4 answers
2k 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 ...
Bogdan's user avatar
  • 1,330
12 votes
5 answers
7k views

Applied mathematics Books (graduate level)

What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems? There is a lot of books on ...
12 votes
2 answers
4k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
MLevi's user avatar
  • 261
12 votes
2 answers
2k views

Reference for a nice proof of "undetermined coefficients"

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
Ryan Reich's user avatar
  • 7,173
12 votes
2 answers
358 views

curvature flow for loops in S^2

Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...
Michael Freedman's user avatar
12 votes
1 answer
1k views

applications of C$^*$-algebras in the field of PDEs

I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
12 votes
2 answers
3k views

Derivative of the flow for ODEs on manifolds

Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its flow, i.e. the unique solution to the ...
user111164's user avatar
12 votes
2 answers
746 views

Solving ODE via contact geometry

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential ...
Lukas S's user avatar
  • 221
12 votes
1 answer
2k views

Differential algebraic geometry vs Diffiety theory

Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations. ...
exchange's user avatar
  • 221

1
2
3 4 5
33