Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,656
questions
3
votes
0
answers
191
views
Nonexistence of Limit Cycle
Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where $...
3
votes
0
answers
130
views
Approximating solutions of non-linear differential equations
I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
$$\frac{\mathbb{d}z_k}{\mathbb{d}t}=(1-\alpha)y_k\sum_s{s^az_s}...
- 171
3
votes
0
answers
166
views
How to find solutions of non-linear ODE with particular BCs
What are some methods, numerical or otherwise, of finding solutions to nonlinear ODEs that satisfy particular boundary conditions? In particular, I'm looking for curves y(s) constrained to a ...
- 151
3
votes
0
answers
586
views
Differential Equations vs Difference Equations
My question is:
Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain?
I mean if I know a solution to the difference ...
- 1,524
3
votes
0
answers
309
views
Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$
Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\...
user21574
3
votes
1
answer
656
views
Short time existence on Hyperbolic Ricci flow in non-compact case
We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...
user21574
3
votes
0
answers
228
views
A priori estimate for Yamabe solution
We know Schoen's compactness on Yamabe problem:
Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension
$3\leq n\leq 24$ without boundary. Denote $\Phi$ to be the full set
of arbitrary ...
- 303
3
votes
0
answers
323
views
Periodic orbits of Hamiltonian systems
Consider a second order Hamiltonian system of the type
$$
\ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N.
$$
Under very `natural assumptions' it is possible to prove the existence of a non constant $T$-...
- 397
3
votes
0
answers
2k
views
Bessel functions in wave propagation and scattering
Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values of n) and small arguments....
- 41
3
votes
0
answers
484
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
3
votes
0
answers
204
views
closure properties of q-differential equations
I am interested in q-differential equations of the form
$p(f(z), f(qz),\dots,f(q^kz))=0$
where $p$ is a polynomial and $k$ an nonnegative integer. I wonder about the closure properties of the class ...
- 5,563
3
votes
0
answers
1k
views
(Approximate) analytic solutions to the Mathieu equation
I'm trying to solve the driven Mathieu equation
$x''+\beta x'+(a-2q\cos{\Omega t})\frac{\Omega^2}{4}x=f(t)$
for both zero and non-zero $\beta$.
I can write down an analytic solution using the ...
- 31
3
votes
0
answers
1k
views
Transforming boundary conditions into initial conditions...
Hello
Is there any methods available for transforming a 2nd order Boundary value problem such as
$F\left(x,y,\frac{\text{dy}}{\text{dx}},\frac{d^2y}{\text{dx}^2}\right)=0$
$y(a)=y_0$ and $y(b)=...
- 569
3
votes
0
answers
359
views
Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs
$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...
- 2,044
2
votes
2
answers
284
views
Analytic function on $\mathbb{C}$
Is there is a holomorphic function $g:\mathbb{C}\to \mathbb{C}$ so that $$\frac{|g'(z)|}{1+|g(z)|^2}=\frac{c}{1+|z|^2}$$ for some $c>1$ and all $z\in \mathbb{C}$?
- 719
2
votes
4
answers
6k
views
Undergraduate Derivation of Fundamental Solution to Heat Equation
It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
- 5,875
2
votes
4
answers
833
views
Complex differential equations
I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...
- 3,554
2
votes
2
answers
552
views
Will the eigenvalue of the dirac operater tend to negative infinity?
Question: If M is a spin manifold. Condider the dirac operator on a spinor bundle. Can the eigenvalue of this operater tend to negative infinity? If it can, can we choose a riemann matrix such that ...
- 283
2
votes
2
answers
587
views
calabi conjecture on compact manifolds
hi,
is the calabi conjencture formulated for compact manifolds with boundary ? or only for those without boundary ? excuse me if the question is too trivial but in my literature it isn't mentioned ...
- 213
2
votes
3
answers
5k
views
Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?
Consider Schrödinger's time-independent equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called potential jumps.
Outside ...
- 3,144
2
votes
2
answers
286
views
Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$
Consider the ode
$$
f''(t)-e^{-2t} f(t)=0.
$$
What is the general behaviour of $|f|$ for large $t$s?
Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(...
- 115
2
votes
2
answers
317
views
Criteria for Schrödinger operator on real line to have simple spectrum
Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum
$-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
- 891
2
votes
3
answers
553
views
CAS for finding closed form solutions to PDEs and SDEs?
Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to
a) partial differential equations,
b) stochastic differential equations?
If yes, what ...
- 5,875
2
votes
3
answers
252
views
How to show continuity and monotonicity of solutions to this parametrized equation?
Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, ...
- 6,621
2
votes
3
answers
2k
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 ...
- 21
2
votes
3
answers
638
views
Criteria for Involutive Subbundles
Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle
$VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under
the Lie bracket ...
- 2,014
2
votes
2
answers
288
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 6,730
2
votes
2
answers
255
views
One question about a specific first-order differential equation
Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below:
$$f'(x)=f(x+c),$$
where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.
I found two families of ...
- 284
2
votes
2
answers
273
views
Euler method (and others) for unbounded intervals
In the course of my research I'm confronted with performing a numerical approximation of the solution of an initial value problem
$$\begin{cases}
y'=f(y,t),\\
y(t_{0})=y_{0}
\end{cases}\quad\quad(1)$$
...
- 419
2
votes
3
answers
247
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: ...
- 7,521
2
votes
1
answer
268
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 ...
user21574
2
votes
3
answers
3k
views
Jacobi method on first order partial differential equations
Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance
- 23
2
votes
2
answers
341
views
Closed forms for Monotonic polynomial recurrences?
I have a monotonic polynomial recurrence of the following form:
c_n = 1-p + p*(c_n-1)^2
This comes from the probability that a specific branching process (Galton-Watson) will be extinct before the ...
- 2,353
2
votes
1
answer
494
views
Underdetermined system of linear PDEs
Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I ...
- 935
2
votes
1
answer
3k
views
Precise versions of "differential operators are unbounded but closed linear operators"
I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...
- 1,162
2
votes
1
answer
184
views
Frobenius theorem and the size of integral manifold
Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and
$[X,Y]:=XY-YX=0$.
Then by ...
- 83
2
votes
1
answer
288
views
An ODE is linear if and only if the maximal solutions are a linear space?
Let $I$ be a non trivial interval of $\mathbb R$, let $f : I \times \mathbb R^n \to \mathbb R^n$ and consider the following ordinary differential equation (ODE):
\begin{equation}\tag{$\mathscr E$}\...
- 605
2
votes
1
answer
158
views
A vector field $X$ on $\mathrm{GL}(n,\mathbb{R})$ with $\begin{cases} X.\mathrm{trace}=\mathrm{Det} \\X.\mathrm{Det}=-\mathrm{trace} \end{cases}$
Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition:
$$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\...
- 312
2
votes
1
answer
181
views
Functional equation $\int_z^{2z} [f(x)-f(z)] dx = 0$
Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$,
$$\int_z^{2z} [f(x)-f(z)] dx = 0.$$
It is clear that a constant ...
- 225
2
votes
1
answer
179
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 864
2
votes
1
answer
817
views
The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\
&...
- 271
2
votes
1
answer
459
views
General four-term recurrence relations
I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form
$$A(n)u_{n+3}+B(n)u_{n+2}+C(n)u_{...
2
votes
2
answers
1k
views
What is the right initial domain for the Dirichlet-Laplacian on a bounded domain?
Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian.
Some books ...
- 665
2
votes
3
answers
701
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 ...
- 885
2
votes
1
answer
190
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 ...
- 280
2
votes
1
answer
416
views
Differential structures on unit-root Frobenius modules
Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n$, $a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...
- 491
2
votes
2
answers
308
views
Complex harmonic oscilator
I was wondering if anybody could give me some references to already
existing literature for the following open ended problem.
Namely, I am interested in studying the equation of
"complex harmonic ...
2
votes
1
answer
238
views
Existence of non-trivial solutions to Dirichlet problem with a potential lying between eigenvalues.
I'm consideirng the example of
$-\Delta u + V(x) u = 0$ in $\Omega$ with $u = 0$ on $\partial \Omega$. I'm trying to see if it's true that if $-\lambda_1 < V(x) < -\lambda_2 < 0$ on $\...
- 2,601
2
votes
1
answer
130
views
Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane
Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...
- 7,161
2
votes
2
answers
230
views
Is there any work on distributional vector fields?
I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we start with an integral equation, ...
- 182