Let $K(r)$ be the piecewise function                        &nbs …

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)=\f …

What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books …

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 bot …

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 o …

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) pap …

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what is a non-integrable system.) In particular, is there a dichotomy bet …

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 m …

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 gra …

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 …

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 …

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 bra …

Hello, I've tried to solve the differential equation xy'' + y' + xy = 0 using series. First, I assumed the solution: $y=a_0+a_1x+a_2x^2+a_3x^3+...$ And from that I tried to solv …

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, m …

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 alo …