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

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3
votes
1answer
189 views

Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation: $$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$ for $x>1$. If that is not possible, I would at ...
1
vote
0answers
53 views

On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by $$\left\{ \begin{array}{l} \dot{\theta}_1 = \omega_1 - ...
1
vote
1answer
269 views

solving elliptic system of first-order linear PDE's

I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which ...
2
votes
3answers
273 views

Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...
2
votes
1answer
212 views

Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
3
votes
0answers
107 views

Approximating solutions of non-linear differential equations

I have met a system of non-linear equations as follows, $$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$ ...
4
votes
2answers
200 views

Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
2
votes
2answers
164 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
4
votes
1answer
244 views

Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote: The derivation of the conditions of exact integrability of an ...
1
vote
1answer
171 views

Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$. Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...
4
votes
1answer
343 views

A Differential Equation with Nested Functions

This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems. I would like to know whether there exists a differentiable function from the ...
0
votes
1answer
600 views

Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ? assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...
4
votes
1answer
431 views

Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.

Suppose that a continuous function $f$ on the line and satisfies $$ |f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1] $$ ...
1
vote
1answer
125 views

A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) ...
1
vote
0answers
92 views

growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation $$ A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2 $$ where $$ A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2 + ...
1
vote
1answer
99 views

whether there are some books and original papers ergodic theory approach to ODE

Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications. People always said that most of the ideas in ...
3
votes
1answer
327 views

Path integrals for stochastic equations

Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example http://arxiv.org/abs/hep-ph/9912209v1 For imaginary time rigorous ...
1
vote
1answer
106 views

Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$. Minimal ...
2
votes
1answer
161 views

Total variation distance between diffusion processes with different volatility coefficient

Preamble: This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
6
votes
1answer
343 views

Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$. It seems that the solution of linear ODE $$\dot x= Ax$$ can not be written in a closed form using $\int$, $A$, $x(0)$ ...
2
votes
1answer
121 views

Is autonomous dynamical system equivalent to one single higher-order ode?

We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a ...
13
votes
5answers
1k views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
2
votes
0answers
126 views

Solving ODE with negative expansion power series

I'm moving this here, as suggested from physics.stackexchange. The original is here. So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...
5
votes
1answer
279 views

Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?

First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...
3
votes
0answers
217 views

May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa)

By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ...
1
vote
0answers
139 views

weak form in Hamilton Jacobi PDE and to include an Dirichlet bound conditions

I tried to solve the next nonlinear PDE with finite elements, therefore I need to make a weak form for: $$\frac{\partial u}{\partial t}+\nu(\vec{x},t) \|\nabla u\| = 0$$ and to integrate the ...
-1
votes
1answer
104 views

Proving convergence of an integral-differential equation [closed]

I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating. My question is where can I ...
2
votes
0answers
133 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
228 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
38 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
102 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
176 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
74 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
251 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
251 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
81 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
262 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
910 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
119 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
600 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
176 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
78 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
656 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
328 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 ...
19
votes
0answers
281 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
187 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
256 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
273 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
292 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 $g(x)$, $f' + af + bg = S_1(f,g) + S_2(f,g)$ ...