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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.

8
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0answers
567 views

When is the monodromy group of a linear differential equation dense in the Galois group?

Given a system $Y'=A(t)Y$ with only regular singular points, then a theorem of Schlesinger says that the Zariski closure of the monodromy group is equal to the Galois group of the corresponding Picard-...
4
votes
2answers
245 views

Adaptive controllers for stiff ODE and DAE integrators

I'm looking for adaptive controllers (adaptive in both step size and order) for stiff integrators. I have asymptotically correct error estimates for the current method and all candidate methods of ...
11
votes
1answer
819 views

Prescribing Gaussian curvature

Let $K(r)$ be the piecewise function                            &...
2
votes
4answers
4k views

Undergraduate Derivation of Fundamental Solution to Heat Equation

It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
18
votes
2answers
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 ...
1
vote
2answers
385 views

Using Wavelet Transforms to Approximate Matrices

It's a long time since I worked on this kind of problem, so please bear with me. I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...
4
votes
2answers
2k views

Minimizing a function containing an integral

I am trying to optimize a function of the following form: $L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter i.e. I am trying to find the optimum x(t) that minimizes L over all admissible x(...
16
votes
6answers
5k views

PDE on manifolds

I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...
99
votes
1answer
8k views

Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
6
votes
3answers
381 views

Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?

The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does ...
4
votes
3answers
841 views

Jacobi fields on a “bump surface”

Consider a "bump surface" which looks like the following: Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature along the ring (the ...
10
votes
2answers
1k 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 $[X,Y]\subset E$...
2
votes
1answer
975 views

Mathematical modeling - how to calculate the displacement at any point in a membrane

I'm trying to calculate the displacement of a wall at any point due to a point source of vibration. The vibration is considered to be directly perpendicular to the surface of the wall, for calculation ...
12
votes
3answers
890 views

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$? (Asked by bcross at math.iuiui.edu on the Q&A board at JMM.)
41
votes
1answer
4k views

D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
21
votes
5answers
2k views

Sheaves and Differential Equations

How do sheaves arise in studying solutions to ordinary differential equations? EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to ...
2
votes
2answers
314 views

Closed forms for Monotonic polynomial recurrences?

I have a monotonic polynomial recurrence of the following form: c_n = 1-p + p*(c_n-1)^2 This comes from the probability that a specific branching process (Galton-Watson) will be extinct before the ...
25
votes
7answers
2k views

Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself. Say I have a nasty ODE, nonlinear, maybe extremely ...
0
votes
2answers
2k views

Stability analysis of a system of 2 second order nonlinear differential equations

How does one linearize and analyze such a system? Just noticed I could edit this, so from my comment below: I am trying to get a feel for what analysis us used beyond the introduction I have had. ...
10
votes
5answers
6k 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 ...
0
votes
1answer
179 views

Difference Equations & Possible Limits

The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here. If we look at the behaviour of a point in R n under matrix multiplication, we ...
6
votes
1answer
348 views

Is there a theory of differential equations for smooth correspondences?

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth ...
2
votes
2answers
617 views

Is the Hessian of Hamilton's function positive-definite?

Background Consider an electron with mass $1$ moving in $\mathbb R^n$ in under the influence of a static electromagnetic field. Up to identifying vector fields with differential forms, Maxwell's ...
127
votes
14answers
26k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
8
votes
3answers
1k views

What is the term analogous to “Wronskian” for difference equations?

I am currently following a course on differential equations and difference equations (recurrence relations). The teacher tries to make parallels between the two concepts, because the methods for ...
10
votes
2answers
4k views

Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...
0
votes
1answer
145 views

Transformation from domains to half-spaces

In a paper I read, an elliptic boundary value problem on a bounded domain D x (0,T) is solved by first transforming it in a set of equations on half-spaces R^n and then applying partial Fourier ...
2
votes
3answers
302 views

In an n-dimensional linear 2nd-order ODE, why is the transpose-inverse to a system of solutions also a solution?

I'm at a sticky spot in my research. Namely, I have a particular fact, and it ought to have a short proof, but the only way I know how to show it is long and drawn out, and I don't like it and worry ...
10
votes
3answers
574 views

What happens to Newtonian systems as the mass vanishes?

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one. Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $...
9
votes
2answers
3k 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 $L^{\infty}$ ...
6
votes
5answers
451 views

What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

Background Lagrangian mechanics on $\mathbb R^n$ is usually defined by picking a Lagrangian function $L: {\rm T}\mathbb R^n \to \mathbb R$, where ${\rm T}\mathbb R^n = \mathbb R^{2n}$ is the tangent ...
7
votes
1answer
405 views

Is the space of nondegenerate classical paths connected?

I have a fairly specific question. My intuition says the answer is "yes", but there is a natural generalizations in which I take out all the "physics", and then I think the answer is "no". Edit ...
3
votes
3answers
1k 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}{...
11
votes
2answers
696 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 ...
5
votes
1answer
321 views

Are the asymptotics of Fourier coefficients to periodic solutions of ODE known?

The Van der Pol equation, given by $$x'' + x = g x' (1 - x^2),$$ has periodic solutions $x(t)$, with the period $T(g)$ depending on the parameter. Thus, one can expand $x(t)$ as a Fourier series ...
2
votes
3answers
474 views

CAS for finding closed form solutions to PDEs and SDEs?

Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to a) partial differential equations, b) stochastic differential equations? If yes, what ...
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votes
2answers
2k views

Ito's lemma in differential form

Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is ...
3
votes
2answers
666 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...
10
votes
4answers
785 views

easy(?) probability/diff eq. question

I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
15
votes
5answers
4k views

Describing the universal covering map for the twice punctured complex plane

As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map. In a sense, this shows that the logarithm has ...
49
votes
9answers
18k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...