Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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7 views

Appropriately choosing the parameter so that one function maximizes while other minimizes

Suppose we have two functions $f$ and $g$ of some variables $x,y,z\in R$. Our goal is to appropriately choose the values of $x,y,z$ such that $f$ is maximized while $g$ is minimized. Here, I am not ...
0
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0answers
23 views

reducing an n-order differential equation to a first order system of equations using either sagemath or sympy

I want to reduce a n-order ordinary differential equation into a first order system of equations. This is in preparation for numerical analysis. I use both Sympy and Sagemath for Computer Algebra, but ...
7
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1answer
251 views

Examples of continuous differential equations with no solution

Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem. ...
2
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1answer
24 views

Making a system of second-order ODEs chaotic

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have X'' ...
4
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1answer
32 views

What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
2
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0answers
59 views

Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is \begin{align} \nabla \times B + i\omega E &= 0\\ \nabla \times E - i\omega B &= 0 \\ \nabla \cdot B &= 0 \\ \nabla ...
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0answers
41 views

Good Intermediate text on ODEs for self-study (meaning solved problems) [on hold]

So I am just about to finish Tenenbaum and Pollard's introductory text on ODEs and I wanted to go to the next level. I am doing this as a self-study, so I really appreciate the fact that T&P have ...
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0answers
27 views

Simple monotone differential operators

Where one may find any reference to lemmas the following kind: If x(t) is C1 in [0,T], x(0)>0, dx/dt + c(t)x(t) >0 in [0,T] then x(t) > 0 in [0,T]. There is a version with weak inequalities. This ...
-1
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2answers
91 views

Blow up solution for a Riccati's equation

I apologize for the problem too simple, but I'm not able at solving it. Consider the Cauchy problem $$ \left\{ \begin{array}{l} \dot x=x(t)^2+t\\ x(0)=0 \end{array} \right. $$ Show that its solution ...
2
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0answers
62 views

Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$. It seems to me that something like the ...
2
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0answers
27 views

When is the solution to a n initial value problem matrix differential equation invertible? [migrated]

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
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1answer
54 views

Solution to simple first-order partial differential equations [closed]

Is there a general solution for first-order partial differential equations of the form $$m(x) \partial_x f(x,y) = n(y) \partial_y f(x,y)$$ for given $m(x),n(y)$ and reasonable boundary conditions ...
5
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2answers
159 views

Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II" They have the following estimates for derivatives of Bessel functions: For $k \geq 2$ \begin{align} & ...
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1answer
57 views

equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
3
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0answers
73 views

Toda Flow Embeddings

What are strategies for generating the following types of pictures: Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are: ...
1
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1answer
39 views

Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
3
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2answers
58 views

Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate ...
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0answers
23 views

Uniqueness of Newton (modulo a constant) series on a compact set

Good morning everybody. My question is as follows: Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$. Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It ...
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0answers
27 views

Solution of ODE related to normal density

This question below is from mathse. I reqeustioned here because very limited respond there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...
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0answers
107 views

A differential equation on toric Kahler manifolds

Let $M$ be a toric Kahler manifold with $\text{dim}_{\mathbb{R}} = 4$. Let $V$ be a Killing vector field associated to the action of the two-torus $\mathbb{T}^2$ on $M$. We also assume the existence ...
2
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2answers
144 views

How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations: $$ \left[ \begin{array}{ccccccc} \text{No.}& t & y_1(t)&y_2(t) & ...
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0answers
78 views

What are the most important papers to read about Ricci flow? [duplicate]

I would like to know, which papers are important to read, if one wants to work about Ricci flow. What papers have to be known by every person conducting research in this field? Can you provide me with ...
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0answers
90 views

The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$ It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
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0answers
46 views

Smooth normal forms of vector fields (the path method)

I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of ...
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0answers
20 views

Condition of existence and uniqueness of solution for abel integral equation [migrated]

It is well known that Abel integral equation has a unique continuous solution. For example, $$ f(t)=\int_0^t\frac{g(s)}{(s-t)^{\alpha}}ds , 0<\alpha<1 $$ where f(t) is known. Specifically, ...
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93 views

Existence of the Dirichlet heat kernel for arbitrary open subsets?

consider first of all an open and bounded subset $\Omega\subset\mathbb{R}^n$, s.t. the boundary $\partial \Omega$ is a manifold of class $C^2$. Then I know that there exists a Dirichlet heat kernel, ...
7
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3answers
356 views

Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...
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0answers
47 views

Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution $\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$? ...
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1answer
141 views

Interior gradient estimate for uniformly elliptic equations

I am struggling with a problem like this: In dimension $n\geq 3$, For the following uniformly elliptic equation, do we have interior gradient estimates? $$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where ...
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2answers
108 views

Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
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0answers
104 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
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1answer
139 views

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 ...
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1answer
74 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
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1answer
108 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 $I\times \mathbb{R}^{n}$ ...
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0answers
28 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 $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ ...
6
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0answers
127 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 ...
1
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1answer
177 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 ...
3
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0answers
71 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 $p, q$.
2
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0answers
77 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
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1answer
86 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 ...
3
votes
1answer
104 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, ...
2
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0answers
79 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 ...
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0answers
194 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
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1answer
90 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. ...
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96 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
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1answer
74 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 ...
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3answers
129 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 ...
4
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1answer
72 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 ...
2
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1answer
73 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
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0answers
96 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) ...