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

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50 views

How do I solve this differential equation by differentiation? [on hold]

How do I solve this equation $$ f(x) = \int_{y=0}^{1-x}f(x+y)\, dy + \int_{y=1-x}^1 y\, dy $$ by differentiation?
-2
votes
1answer
73 views

BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better? The BDF2 method requires the ...
0
votes
1answer
55 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 ...
7
votes
2answers
164 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
1answer
106 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
2answers
102 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
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1answer
110 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 ...
2
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0answers
39 views

Non-autonomous O.D.E with discontinuous and not integrable R.H.S

Consider the non-autonomous O.D.E $\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$ where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$ and $h:\Bbb R^d \times S \to \Bbb R^d$ where ...
0
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0answers
41 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
1answer
97 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 ...
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34 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
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1answer
163 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
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0answers
63 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
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1answer
173 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 ...
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1answer
50 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
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0answers
31 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 ...
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0answers
49 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
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2answers
71 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
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1answer
41 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 + ...
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0answers
67 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, ...
9
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1answer
216 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
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0answers
59 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 ...
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0answers
26 views

SIRS Stability Analysis

I have set up the following ODE's for a SIRS model: $$\frac{dS}{dt} =-\alpha SI + \zeta R$$ $$\frac{dI}{dt} = \alpha SI - \beta I - \rho I$$ $$\frac{dR}{dt} = \beta I - \zeta R$$ ...
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46 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
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0answers
91 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
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0answers
131 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 ...
0
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0answers
29 views

How to switch from the spectral density of the differential equation

I am modeling random process. It is described with the function of the spectral density, where $\alpha_x$ and $\beta_x$ are damping coefficient and the average frequency of the correlation function of ...
7
votes
0answers
272 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
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1answer
254 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 _+ ...
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0answers
99 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 ...
5
votes
1answer
243 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 ...
7
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0answers
261 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
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1answer
73 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
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3answers
116 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: ...
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0answers
72 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 $\phi_+(0)=0(\phi_-(0)=0)$. ...
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0answers
85 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
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0answers
86 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 ...
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vote
2answers
179 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
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0answers
106 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
1answer
170 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 ...
2
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1answer
103 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
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1answer
59 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
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0answers
65 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
286 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 = ...
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1answer
34 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 ...
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votes
1answer
156 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 ...
3
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0answers
135 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 ...
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0answers
36 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 ...
0
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0answers
188 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 ...
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8answers
6k 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 ...