# Tagged Questions

**0**

votes

**0**answers

15 views

### Existence and Uniqueness of solution of volterra integral equation of the first kind

$$
\int_0^t k(s,t)f(s)ds=g(t)
$$
To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind.
...

**-2**

votes

**2**answers

88 views

### Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [on hold]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?

**4**

votes

**1**answer

738 views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**1**

vote

**0**answers

90 views

### Calogero-Moser eigenfunction

The folllowing function
\begin{equation}
J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...

**0**

votes

**0**answers

30 views

### existence of solution of volterra integral equation for the first kind

$$
\int_0^t k(s,t)f(s)ds=g(t)
$$
To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind.
...

**-2**

votes

**0**answers

46 views

### Differential equations [closed]

For k > 0,x = x(t),y = y(t) Solve this system:
\begin{array}{l}
x\frac{{{d^2}x}}{{d{t^2}}} = k\frac{{dy}}{{dt}}\\
x\frac{{{d^2}y}}{{d{t^2}}} = - k\frac{{dx}}{{dt}}
\end{array}

**0**

votes

**0**answers

98 views

+50

### A derivational approach to the Poincare Bendixson Theorem

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a smooth vector field on the plane. Assume that $K\subset \mathbb{R}^{2}$ is a compact subset (not necessarily invariant under ...

**0**

votes

**0**answers

29 views

### numerical solution of volterra integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation
$$
\int_{a}^{x}f(s)g(s,x)ds=h(x)
$$
...

**0**

votes

**0**answers

26 views

### How to solve integral equation? [duplicate]

I have an integral equation such that
$$
\int_t^T f(s)g(s,t)ds= h(t)
$$
where g and h is given. we want to know function f explicitly. As i know, this type of question is about the integral equation. ...

**3**

votes

**1**answer

89 views

### Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude).
Given $g \in (0,\infty), k \in C^1( [0, ...

**-6**

votes

**1**answer

71 views

### Non Linear PDE's [closed]

I want to solve two systems of questions being
$$\frac{C(r,y)''}{C(r,y)} + \frac{C(r,y)'^2}{C(r,y)^2}=0$$
and
$$A(r,y)'' + 2A(r,y)' \frac{C(r,y)'}{C(r,y)}=0$$
where ' is differentiating with respect ...

**1**

vote

**1**answer

102 views

### Global version of the Picard-Lindelöf theorem [closed]

Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) intervall and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ ...

**1**

vote

**1**answer

160 views

### Quadratic PDE Systems

(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)
I have a problem that leads me to the following quadratic system of PDEs:-
$
c_1 ...

**1**

vote

**0**answers

27 views

### Backlund transformation related to two NL differential equations

I'm looking for a Backlund transformation linking the following two nonlinear differential equations for real $t$:
$$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$
...

**5**

votes

**0**answers

103 views

### The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...

**3**

votes

**0**answers

70 views

### Is there any weighted sobolev embedding with non-decaying weight

Is there any weighted sobolev embedding like
$$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$?
Here $s>0$, and for some appropriate $p, q$.

**-1**

votes

**0**answers

64 views

### if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity?

if $X$ is a vector field in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? And in dimension $n$?
I know there is singularity when $n=2$.

**2**

votes

**0**answers

70 views

### Variant form of the gronwall inequality

I know the following statement for gronwall inequality:
Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have,
$f' \leq \phi f$ and $f(0)=0$ then $f=0$
Now is ...

**1**

vote

**0**answers

56 views

### Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...

**2**

votes

**0**answers

74 views

### What transformations preserve the von Mises distribution?

The von Mises distribution is entirely defined on the circle with a density given by
$$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$
where $x$ is in an arbitrary real interval of ...

**1**

vote

**0**answers

187 views

### The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**2**

votes

**1**answer

61 views

### How to find the general solution of Mathieu differential equation?

Two solutions of the equation are the Mathieu sine and cosine and they form an orthogonal basis. The equation has arisen from the wave equation for an infinite string with a sinusoidal cross-section. ...

**0**

votes

**0**answers

91 views

### How to choose appropriate norms in the analysis of PDEs?

In many papers on the analysis of nonlinear PDEs, the authors could always choose appropriate norms to do their estimates. Sometimes these norms look very odd, which I don't know how these authors ...

**1**

vote

**3**answers

119 views

### Harmonic Function with special property

I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...

**1**

vote

**1**answer

71 views

### A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...

**4**

votes

**1**answer

70 views

### Explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to add spin into account in this scheme, it ...

**13**

votes

**2**answers

1k 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 ...

**0**

votes

**0**answers

94 views

### projecting Laplacian onto tangent and normal bundles of submanifold

If I have a simple linear differential equation involving covariant derivatives such as $\nabla^2 g_{\mu\nu}+ 2g_{\mu\nu}=0$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) ...

**2**

votes

**1**answer

61 views

### First order ODE and possible periodic solutions

I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions:
$\partial_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$.
This equation roughly ...

**0**

votes

**1**answer

109 views

### Legendre differential equation with additional term

In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( ...

**7**

votes

**1**answer

137 views

### Differential analogue of the newton polygon

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows:
Let us recall the ...

**3**

votes

**1**answer

76 views

### Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...

**3**

votes

**2**answers

202 views

### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

**44**

votes

**10**answers

11k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**0**

votes

**1**answer

95 views

### first order differential equation

Is there any solution formula for a differential equation like this
$$
y'(x)=f(x)y(x)+g(x)y(ax)\qquad \text{where}\qquad a\gt1.
$$
I have a differential equation a little more complicated than this. ...

**2**

votes

**1**answer

78 views

### General four-term recurrence relations

I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form
...

**1**

vote

**0**answers

63 views

### Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of ...

**4**

votes

**4**answers

946 views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**1**

vote

**0**answers

78 views

### G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...

**0**

votes

**0**answers

46 views

### how can i solve this nonlinear problem?

let be the differential equation in dimension $ n > 3 $ or $ n =3 $
$$ -\Delta u =|grau|^{2}u $$ (1)
with the constraint that the vector 'u' $ |u|=1 $
then how could i prove that the unitary ...

**35**

votes

**9**answers

6k 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 $f(t)$, ...

**2**

votes

**2**answers

177 views

### Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...

**4**

votes

**1**answer

395 views

### Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange.
My question is where the following differential equation arises naturally and where it ...

**1**

vote

**3**answers

318 views

### Linearization of vector fields

Simply, we discuss things on $C^{2n}$ ($or R^{2n}$) but not on manifolds. Given an anayltic (or real analytic) vector field $V$ on $C^{2n}$ ($or R^{2n}$), with a zero at the origin . And $V_{0}$ is ...

**1**

vote

**1**answer

2k views

### Criterion for finite time blowup of an ODE

Hi I'm trying to understand the most general conditions under which I can conclude finite time blow up of an ODE of the form $\dot{x} = f(x)$ with initial condition $x_0 > 0$ and $f(x) \geq 0$ for ...

**0**

votes

**1**answer

191 views

### Evolution equation on Banach space

Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...

**1**

vote

**0**answers

49 views

### Single parameter bifurcations caused by a simple additive term

Note: I asked this question on Math.SE over two months ago, and it has not received any answers.
Motivation: A practical dynamical system is often described by an ODE that has a parameter that ...

**1**

vote

**0**answers

46 views

### Allen Cahn Equation with Dirichlet Data

consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions ...

**0**

votes

**1**answer

148 views

### Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ ...

**2**

votes

**0**answers

78 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...