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

2
votes
0answers
129 views

A natural tower of bundles over Grassmannian manifolds

There is a well-known exact sequence of vector bundles $$ 0\to U\to T\to N\to 0 $$ over a Grassmannian manifold $Gr(V,n)$: the fibers over $L\in Gr(V,n)$ are, respectively, $L$ itself, the whole $V$ ...
2
votes
1answer
221 views

Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications ...
1
vote
0answers
35 views

requiring reference on the error bounds for the asymptotic expansions of an ODE of order greater than two

Let \begin{equation} \sum_{j = 0}^{n} a_{ j}(z) w^{(j)} = 0 \end{equation} be an ODE of order $n$, with $a_n(z) = 1$, $a_j(z) = a_{j, 0} + a_{j, 1} z^{-1} + ...$ holomorphic at $\infty$. It has a ...
2
votes
0answers
101 views

How to find solutions of non-linear ODE with particular BCs

What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a ...
2
votes
0answers
154 views

Differential Equations vs Difference Equations

My question is: Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain? I mean if I know a solution to the difference ...
0
votes
1answer
109 views

The solvability of a Hölder ODE [closed]

The problem is follow, I want to know weather there is a function $u\in \text{C}^{0}\left((-1,1)\right)$ but not $\text{C}^1((-1,1))$, such that $\lim\limits_{h\rightarrow ...
2
votes
0answers
71 views

Rotation number of perturbated equation

I have a differential equation on torus $(t,x)$ and well studied it's Arnold tongues for Poincare map of the circle $x(t=0) \to x(t=2\pi)$. The question is how changes rotation number when I add small ...
2
votes
1answer
242 views

Geodesics and harmonic map heat flow

I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes: Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] ...
2
votes
1answer
247 views

Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
3
votes
0answers
79 views

Classical parabolic theory (PDEs)

I am reading an articule(http://link.springer.com/content/pdf/10.1007%2Fs00028-010-0085-8.pdf) so I almost done but I dont understand the next argument is in the page 8 "It is known from the ...
7
votes
2answers
246 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
9
votes
4answers
831 views

Is there anyway to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there anyway (or any algorithm) ...
1
vote
1answer
113 views

Solving systems of integral equations using Volterra series

I came across this problem when trying to solve the following integral equations arising in direct scattering: $$ \begin{align} n_{11}(x,z)=1+\int_{-\infty}^xe^{-izy}u(y)n_{21}(y,z)dy, \quad ...
2
votes
1answer
542 views

Derivation of Bessel functions

I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism ...
2
votes
4answers
175 views

Heat integro - differential equation

In the heat equation: $$\partial u(x,t)=D\partial_{xx}u(x,t)$$ the diffusion coefficient $D$ is in general a constant or a given function of $u(x,t)$ in the nonlinear equation. Suppose I have a ...
0
votes
0answers
76 views

elliptic equation

How to prove that (2) is the fundamental solution (1)??? $\frac{\partial}{\partial x}\left(P\frac{\partial\varphi}{\partial x}\right)+\frac{\partial}{\partial y}\left(P\frac{\partial\varphi}{\partial ...
4
votes
5answers
612 views

A simple and good reference about solitons

I want to study gradient Ricci solitons. Can anyone help me to find a simple and good reference about solitons and their applications? Thanks
1
vote
3answers
322 views

Linearization of vector fields

Simply, we discuss things on $C^{2n}$ ($or R^{2n}$) but not on manifolds. Given an anayltic (or real analytic) vector field $V$ on $C^{2n}$ ($or R^{2n}$), with a zero at the origin . And $V_{0}$ is ...
18
votes
0answers
272 views

Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually hoping for answers with something closer to a proof than what I got there.) Suppose we have a unit mass planet at each ...
2
votes
1answer
183 views

Exact or Numerical solutions of a system of differential equatios

I need to solve the following system of differential equations: \begin{align*} x' + 3a \sqrt{ x+ y}x &= -b \sqrt{xy} \\\\ y' + 3c \sqrt{x+y}y &= b \sqrt{xy} \end{align*} where $a$, $b$, ...
0
votes
1answer
251 views

Calculate the inverse of a matrix

Hi I have a equation $$Ax=b,$$ where matrix $A$ is invertible, $b$ is a constant vector, and $x$ is the unknown vector. To obtain $x$, it is obvious $x=A^{-1}b$. Alternatively, if $A$ is Hurwitz, one ...
4
votes
1answer
258 views

Prescribing the Lie derivative of the metric?

This is a question that arises from my research problem. Suppose $(M,g)$ is a compact Riemannian manifold with boundary and $g$ is smooth up to the boundary (if you like, take $M$ to be diffeomorphic ...
0
votes
0answers
282 views

Green's function of coupled ODEs

For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$, $f' + af + bg = S_1(f,g) + S_2(f,g)$ ...
1
vote
0answers
263 views

How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$, a ...
2
votes
1answer
270 views

Strongly parabolic PDE vs weakly parabolic PDE

In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...
3
votes
1answer
345 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
15
votes
1answer
432 views

The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
5
votes
2answers
179 views

Constructing a linear ODE for a product of two holonomic functions without introducing additional singularities

A function $f$ is called holonomic if it satisfies some linear differential equation with polynomial coefficients $$p_n(x) f^{(n)}(x)+\dots+p_1(x)f'(x)+p_0(x)f(x)=0.$$ Now if $f,g$ are holonomic then ...
1
vote
0answers
74 views

differential equation with delay

Suppose, i have an differential equation with a fixed $\tau$ delay $\frac{dx}{dt}=f(x(t)+\alpha x(t-\tau))$. Here $\alpha$ is a small parameter. When $\alpha=0$ its a simple Self-oscillatory ...
0
votes
0answers
136 views

Time-delay differential equation

Is it possible that the system \begin{equation} \begin{cases} 2\dot{q}(t) + \dot{q}(t-1) + \dot{q}(t+1) = c & \text{if} \hspace{5mm} 0 \le t \le 2 , \dot{q}(t) + \dot{q}(t-1) = k & ...
17
votes
0answers
845 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
1
vote
0answers
224 views

Whether does the following equation have only one finite zero?

Dear MOs, Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ...
0
votes
2answers
492 views

Solution to the fractional differential equation

What is the solution of the fractional differential equation $$ f^{(\alpha-1)}(t) = tf(t) $$ where $(\alpha)$ denotes the fractional derivative of order $\alpha$ EDIT: Background behind this ...
2
votes
1answer
240 views

symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation $v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...
8
votes
2answers
774 views

Reference for a nice proof of “undetermined coefficients”

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
1
vote
0answers
279 views

a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...
1
vote
1answer
394 views

Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...
1
vote
1answer
140 views

finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation $det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$ you can find the definition of effective forms here
1
vote
0answers
156 views

multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\left | \sigma \right |\leqslant ...
2
votes
3answers
309 views

Laurent series expansion for ODE.

OK, then I read Frobenius method in mathworld (I learned when I took ODE 2): http://mathworld.wolfram.com/FrobeniusMethod.html My question is: Are there any ODEs where the solution is given by full ...
2
votes
1answer
890 views

First Order PDE Solution Method Issues

Greetings, I have a rather pedantic problem that's honestly been holding me back for over a month & is something neither stackoverflow nor some of my professors nor some graduate students have ...
2
votes
1answer
381 views

closed form solution of an ODEs

I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help ...
4
votes
1answer
675 views

What is symmetry group of non-linear equation?

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, ...
-1
votes
3answers
355 views

problem related to Airy functions [closed]

I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems: What are the normalization ...
5
votes
2answers
218 views

When is a solution to an ODE determined by its average value?

Fix a smooth vector field $\vec v$ on $\mathbb R^n$. It is well-known that any trajectory $\vec x : [0,1]\to \mathbb R^n$ solving the ODE $\dot x(t) = \vec v(\vec x(t))$ is determined by its ...
3
votes
2answers
351 views

Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi, I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$. Fix $T>0$. ...
1
vote
1answer
245 views

A 'conjecture' on critical elliptic pde

I conjecture the following. Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define $$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$ $E_{\mathbb{R}^3}$ is defined ...
1
vote
0answers
91 views

A critical elliptic PDE

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
1
vote
1answer
281 views

Solution of a PDE and its uniqueness

Hallo, consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE ...
0
votes
2answers
335 views

Solution to differential equation

a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y_0>0$ and $y'(s)=y_1$, $s<0$, $$y''+(2-n)\coth(t) ...