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

learn more… | top users | synonyms

0
votes
1answer
73 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 ...
5
votes
1answer
892 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
0answers
17 views

Floquet-Bloch solutions of the quasiperiodic Schrödinger equation

I am concerned about the Schrödinger equation $-x''(t)+q(\omega t)x(t)=Ex(t).$ Here, the potential $q$ is real and quasiperiodic with frequency vector $\omega$. Equivalently, we may formulate this ...
2
votes
0answers
403 views
+50

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
0answers
28 views

Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
1
vote
1answer
157 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
1answer
43 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
0answers
30 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 ...
0
votes
0answers
34 views

How to solve this partial equation using weighted Laplacian Matrix?

I have to solve the following system of partial differential equations: \begin{equation*} \begin{array}{rll} Z(t,x) & = & \nabla_w T(x,t) \\ Z_t(t,x) & = & \nabla_w L_w . ...
-1
votes
0answers
53 views

Differential equation with many unknowns [closed]

Let $f_1(x), \dots, f_n(x)$ be unknown functions, and $a_0, \dots, a_n$ and $b_1, \dots, b_n$ be some real numbers. Also assume $b_i \neq 0$ for all $i$. The following differential equation holds. $$ ...
6
votes
0answers
234 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
votes
0answers
29 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): $$\left\{\begin{matrix} \frac{dx}{dt}=(\alpha-\beta y)x \\ \frac{dy}{dt}=(-\gamma + ...
9
votes
1answer
198 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 ...
0
votes
2answers
2k views

Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution.

I have been working on Riccati Equation. I have tried many different methods to find a closed form for the solution of first order non-linear differential equation ($y'+y^{2}=f(x)$) without knowing a ...
0
votes
1answer
37 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 + ...
1
vote
0answers
47 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
2answers
63 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
0answers
63 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, ...
3
votes
1answer
134 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, ...
3
votes
0answers
52 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 ...
5
votes
2answers
193 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
0
votes
0answers
25 views

SIRS Stability Analysis

I have set up the following ODE's for a SIRS model: $$\frac{dS}{dt} =-\alpha SI + \zeta R$$ $$\frac{dI}{dt} = \alpha SI - \beta I - \rho I$$ $$\frac{dR}{dt} = \beta I - \zeta R$$ ...
6
votes
5answers
245 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 ...
0
votes
0answers
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
votes
0answers
88 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
0answers
126 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 ...
5
votes
0answers
574 views

A question on “The weakened Hilbert 16th problem”

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } ...
2
votes
1answer
143 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. ...
7
votes
0answers
265 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
1answer
174 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 ...
0
votes
1answer
250 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 _+ ...
0
votes
0answers
27 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 ...
12
votes
3answers
772 views

What theorem of Liouville's is Gian-Carlo Rota referring to here?

I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations: "For second order linear differential ...
3
votes
2answers
295 views

delay differential equation

I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE): $ y_x(x) = A y(x-1)^a $ where $ 0 < a < 1 $ and $ A > 0 $ are given ...
0
votes
0answers
184 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 ...
1
vote
0answers
96 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 ...
0
votes
0answers
143 views

Hammerstein integral equation with inverse of the solution

In signal processing theory I found this integral equation that I recognized to be of Hammerstein type: $$u(t)-\int_{0}^{1}d\phi cos(\omega t+\phi)\frac{1}{u(\phi)}=0$$ Unfortunately the solution ...
2
votes
0answers
102 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 ...
5
votes
1answer
206 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 ...
1
vote
1answer
71 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
3answers
111 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
0answers
64 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
0answers
84 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 $ ...
1
vote
0answers
76 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
2answers
177 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. ...
0
votes
0answers
186 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 ...
4
votes
1answer
162 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
votes
1answer
100 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
1answer
53 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
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 ...