**0**

votes

**0**answers

92 views

### Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function.
When I consider price of American ...

**9**

votes

**2**answers

282 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 ...

**0**

votes

**1**answer

174 views

### Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...

**4**

votes

**2**answers

175 views

### Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote:
...

**0**

votes

**2**answers

401 views

### General description of surface with zero gaussian curvature

Let suppose function $g(s,t)$ satisfies partial differential equations:
$g_{ss} g_{tt} - g_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution ...

**0**

votes

**0**answers

51 views

### Shooting method for non-oscillatory solutions

I am having some trouble with using the shooting method :
I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I ...

**2**

votes

**1**answer

172 views

### Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation
$$\frac{\partial f ...

**1**

vote

**0**answers

64 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**0**

votes

**1**answer

182 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**1**

vote

**0**answers

132 views

### Floquet-Bloch solutions of the quasiperiodic Schrödinger equation

I am concerned about the Schrödinger equation
$-x''(t)+q(t)x(t)=Ex(t).$
Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. That is, we let $T^d$ be the $d$-dimensional ...

**1**

vote

**1**answer

191 views

### Vector fields whose divergence are proper maps

Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of ...

**-1**

votes

**1**answer

69 views

### Integrating factors and integrability of an ODE system

The following argument is from a paper about the Bendixson-Dulac Theorem.
Consider a smooth differential equation on the plane
$$
x'=g(x,y),\quad y'=h(x,y).
$$
Suppose there exists a function ...

**0**

votes

**0**answers

55 views

### Perturbation method of a boundary value problem

Let $u(x, \epsilon, \theta)$ be the solution of $$u''+(\epsilon \cos(x)+\theta-u)u=0$$ with boundary conditions $u'(0)=0$ and $u'(\pi)=0$. Here $\theta\in [0, 1]$.
I tried to put the solution in ...

**1**

vote

**0**answers

64 views

### Regularity of Schrödinger Resolvent

The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...

**1**

vote

**2**answers

91 views

### Finding conditions to guarantee existence of solutions to IVP [closed]

Consider the following IVP:
$x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.
Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.
In order for the ...

**0**

votes

**1**answer

51 views

### Solution of a second order nonlinear ode [closed]

I encountered the following ode in the attempt to solve the cauchy problem of Liouville equation. I have tried for a long time to give it a solution but failed.
$(K e^f h + ...

**0**

votes

**0**answers

72 views

### Dipole Transition Integrals - Acceleration Form, What's Wrong?

I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses.
The following question may sound stupid, since I'm sure I was wrong somewhere, ...

**11**

votes

**1**answer

309 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 ...

**3**

votes

**0**answers

80 views

### Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...

**1**

vote

**1**answer

61 views

### SIRS Stability Analysis

I have set up the following ODE's for a SIRS model:
$$\frac{dS}{dt} =-\alpha SI + \zeta R$$
$$\frac{dI}{dt} = \alpha SI - \beta I - \rho I$$
$$\frac{dR}{dt} = \beta I - \zeta R$$
...

**0**

votes

**0**answers

49 views

### Mathematical simulation of viscous material behaviour

I have a non linear first order differential equation of the type:
$[y(t)]^n + a \frac{dy(t)}{dt} = b(t)$
where $y(t)$ is a real function, the exponent n is a real number greater than $2$, but not ...

**2**

votes

**0**answers

102 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

**3**

votes

**0**answers

151 views

### System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x ...

**7**

votes

**1**answer

367 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**0**

votes

**1**answer

265 views

### About an integral equation

I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...

**1**

vote

**0**answers

111 views

### Is there a unique solution? [closed]

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...

**7**

votes

**1**answer

424 views

### Killing vector fields on sphere

Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.
$$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$
Is $u$ necessarily ...

**12**

votes

**1**answer

436 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 ...

**1**

vote

**1**answer

93 views

### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...

**2**

votes

**3**answers

158 views

### closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2

Let
$$
(\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0,
$$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$.
Q: ...

**0**

votes

**0**answers

93 views

### How to solve a couple of ODEs

Let $\phi_+ (\phi_-)$ be a strictly increasing (decreasing) function defined on $R_+$ such that $\phi_+(\phi_-)\in\mathcal{C}^0(R_+)\cap\mathcal{C}^1(R_+^{\ast})$ and $\phi_+(0)=0(\phi_-(0)=0)$. ...

**1**

vote

**0**answers

92 views

### Difference Quotients Evans

There is a theorem in Evans partial differential equation book as follows:
if $u \in W^{1,p}(U)$ then for each compact $V$ in $U$ we have that:
$ |D^hu|_{L^p(V)} \leq C |Du|_{L^p(U)} $
for all $ ...

**2**

votes

**0**answers

127 views

### Differential equation related to the Schwarzschild metric

How can one find solutions of the following second-order diﬀerential equation
$$\frac{d^2W}{dr^2}-\frac{1}{r}\frac{dW}{dr}=\frac{C}{W^2}\frac{dW}{dr}$$
with the boundary condition $W(r)\to r^2$ at ...

**1**

vote

**2**answers

204 views

### Two questions related to $TS^{2}$ as a holomorphic manifold

We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...

**2**

votes

**0**answers

124 views

### A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...

**4**

votes

**1**answer

197 views

### Reference request: Invariant sets of dynamical systems

(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...

**3**

votes

**1**answer

114 views

### hypergeometric at nearest singularity

Reference request. A prototype case:
In
$$
{}_2F_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the ...

**0**

votes

**1**answer

88 views

### A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable

First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
...

**2**

votes

**0**answers

68 views

### Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...

**2**

votes

**2**answers

323 views

### Monge-Ampere type PDE

NB: I have edited this question to clarify what the OP is asking – Robert Bryant
Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = ...

**1**

vote

**1**answer

39 views

### Can we implicitly fit a system of linear ODEs by reduced information?

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...

**-3**

votes

**1**answer

173 views

### The logarith map as a contraction [closed]

Two Questions:
(1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping?
Or more ...

**4**

votes

**1**answer

216 views

### A special non vanishing vector field on $S^{3}$

Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for ...

**1**

vote

**0**answers

45 views

### singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation
$$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$
where $\gamma\in (1, 2]$ is some ...

**1**

vote

**0**answers

233 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**21**

votes

**8**answers

7k views

### Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such.
Yet, they could have been just ...

**7**

votes

**5**answers

455 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 ...

**13**

votes

**2**answers

904 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"?

**10**

votes

**2**answers

245 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, ...

**1**

vote

**0**answers

69 views

### Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...