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

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-4
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0answers
28 views

First-order nonlinear ordinary differential eqauation [on hold]

can someone help me to solve this equation? I have been trying a few methods. Thanks. y'=(y/x)*((xy + 1)/(xy - 1))
4
votes
1answer
125 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 ...
1
vote
1answer
91 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
0answers
35 views

Classification of PDEs range of influence domain of dependence [closed]

Text books on numerical methods usually give the classic example of classification of a generic 2nd order scalar PDE (Afxx+Bfxy+Cfyy+Dfx+Efy+Ff=G) depending on the value of the discriminant B^2-4AC. ...
-3
votes
0answers
38 views

Topological conjugacy [closed]

Prove that any two linear systems with the same eigenvalues +/-ibeta, beta not equal to 0 are conjugate. What happens if the systems have eigenvalues +/- ibeta and +/- i*gamma with beta not equal to ...
0
votes
1answer
40 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
0answers
63 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
2answers
254 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
1answer
75 views

Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level. My initial situation is the following. Consider two SDEs: \begin{align} ...
0
votes
0answers
43 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation? [migrated]

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
1
vote
1answer
30 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
1answer
149 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 ...
2
votes
0answers
111 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
0answers
27 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
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0answers
80 views

Lifting a quadratic system to a non vanishing vector field on $S^{3}$

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 ...
17
votes
8answers
5k 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 ...
0
votes
0answers
33 views

A hyperbolic partial differential equation

How solve this equation (numeral or analytical)? $u(t,x)=\int_{t-x}^{t}{a \cdot e^{b \cdot s} \cdot \int_{0}^{s-(t-x)}{u(s,z)dz}ds}+\int_{t-x}^{t}{c \cdot e^{d \cdot s} \cdot ...
0
votes
1answer
90 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 ...
11
votes
2answers
782 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"?
0
votes
1answer
120 views

Question on the partial differential equations in complex space

As is known most of the theory now developed for partial differential equations is in the real space, especially function space like Sobolev space, BMO space, $L^p$ space, etc. However is there some ...
6
votes
2answers
176 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
0answers
45 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 ...
1
vote
0answers
42 views

How to find a Lax Pair for the modified KdV equation

Question I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as: $ X = ...
0
votes
1answer
139 views

Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...
3
votes
0answers
92 views

Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$. Then we know that the eigenvalues of $-\Delta$ form an ...
11
votes
2answers
1k views

How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
40
votes
2answers
1k views

What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE. In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...
0
votes
0answers
21 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 ...
-2
votes
1answer
61 views

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

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 ...
8
votes
1answer
372 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
votes
2answers
90 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 ...
4
votes
1answer
50 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, ...
3
votes
1answer
363 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 ...
0
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0answers
30 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
votes
2answers
103 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
votes
0answers
69 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 ...
-1
votes
1answer
60 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
votes
2answers
210 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} & ...
0
votes
1answer
69 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 $SU(n)$. ...
6
votes
0answers
135 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
vote
1answer
43 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
votes
2answers
61 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
30 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, ...
1
vote
0answers
112 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
votes
2answers
153 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) & ...
0
votes
0answers
80 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 ...
0
votes
0answers
97 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 ...
1
vote
0answers
57 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 ...
0
votes
0answers
104 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, ...