# 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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### 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**

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**2**answers

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 ...

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**1**answer

819 views

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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**

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**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 ...

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vote

**2**answers

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

**2**answers

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

**6**answers

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**

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**1**answer

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

**3**answers

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

**3**answers

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**

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**2**answers

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 $[X,Y]\subset E$...

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votes

**1**answer

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 ...

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**3**answers

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

**1**answer

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**

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**5**answers

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

**2**answers

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

**7**answers

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

**2**answers

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**

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**5**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**14**answers

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" ...

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**3**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**3**answers

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

**2**answers

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 $L^{\infty}$ ...

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**5**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**2**answers

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

**4**answers

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 ...

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votes

**5**answers

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

**9**answers

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 $f(t)$, ...