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

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2
votes
1answer
239 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
74 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
213 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 ...
0
votes
0answers
71 views

Existence and uniqueness of a matrix differential equation with L^1 coefficients

I came across the following differential equation when considering some direct scattering problems: $$ N'_x(x,z)=G(x,z)N(x,z) $$ where $N(x,z)$ is a $2\times2$ complex matrix with variables $x$ and ...
8
votes
4answers
720 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
95 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
468 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
169 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
545 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
0
votes
0answers
46 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let ...
16
votes
0answers
259 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
179 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
241 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
245 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
264 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
238 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
239 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
337 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
397 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
164 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
70 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
131 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 & ...
18
votes
0answers
825 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 ...
0
votes
0answers
222 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
476 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
236 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
741 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
278 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
346 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
136 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
153 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
276 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
644 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
350 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
627 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: $r'^2+r^2=(kt)^2$, ...
0
votes
3answers
340 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
212 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
334 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
233 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
85 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
273 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
331 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) ...
3
votes
1answer
205 views

A partial differential equation on $\mathbb{CP}^1$

Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation $\frac{\partial}{\partial z} f = ...
1
vote
0answers
359 views

Does the following differential equation have a non-trivial solution:(f'')^4+(f)^4=2(f')^4 ? [closed]

Hello! Currently,I meet the following differential equation: ${f''}^4+f^4=2f'^4$ we can easily find the trivial solutions:$a e^{bx}$,where $a$ is an arbitrary constant,$b=\pm i$ or $b=\pm 1$,does ...
5
votes
1answer
446 views

Numerical calculation of Arnold tongue

Hello. I am working on investigation of family of dynamical systems on the torus $$\dot{x}=\cos(x)+b\cos(t)+a$$ $$\dot{t}=1$$ and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$ I need to find ...
2
votes
0answers
137 views

Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation $$ \text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t) $$ with ...
3
votes
1answer
407 views

Are (Frobenius) integrability conditions covariant?

Starting with a system of partial differential equations for functions on $\mathbb{R}^n$, Frobenius' theorem gives a bunch of integrability conditions (on e.g. functions which are 'data' for the ...
13
votes
4answers
864 views

ODE's without a Lipschitz condition

When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is ...
0
votes
1answer
66 views

Self-inhibitions are diagonal matrix

In all the discontinuous neural networks models, the self-inhibitions are diagonal matrix, what is the reason for this assumption?