Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,645
questions
4
votes
0
answers
397
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
4
votes
0
answers
142
views
An embedding question: Morrey spaces
Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
4
votes
0
answers
173
views
Distributional PDE solutions as topological linear duals of PDE solutions
Let
$$
P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)
$$
be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
4
votes
0
answers
377
views
On modified Bessel solutions to complex ODE's using Kummer's series
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...
4
votes
0
answers
255
views
Basin of attraction of gradient flow
Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
4
votes
0
answers
708
views
When the integral of the product of two matrix exponentials is singular?
Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $...
4
votes
0
answers
651
views
Symbol of differential operator and change of variables [closed]
Recently I posted the following question on stack exchange, but it remained with no answer https://math.stackexchange.com/questions/1863658/symbol-of-differential-operator-and-change-of-coordinates
I ...
4
votes
0
answers
106
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Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
4
votes
0
answers
146
views
Connection between cardiac equations and untangling knots?
I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe. ...
4
votes
0
answers
579
views
Models used for the Zika virus?
I am currently teaching an ordinary differential equations course, and am thinking about doing a module on infectious disease models, e.g. SIR/SIRS. I thought, if possible, it would be nice to ...
4
votes
0
answers
164
views
Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?
My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
4
votes
0
answers
429
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 ...
4
votes
0
answers
202
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 G''(x)=&\mathscr{...
4
votes
0
answers
429
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
4
votes
0
answers
349
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 ...
3
votes
4
answers
532
views
Nonlinear second order ODE $y''+f(x)y=g(x)y^3$
I encountered the following ODE in order to find a solution for Einstein equation
$$y''+f(x)y=g(x)y^3.$$
It seems to me that it is not among the solvable nonlinear second order differential equations....
3
votes
4
answers
816
views
Solution of second order differential equation with singularities at 0,1, and ∞
I am trying to solve the following equation;
$$
U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0
$$
where U is a function of t and C is constant.
The above ...
3
votes
3
answers
305
views
Generalized Fuchsian-type PDE?
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
3
votes
2
answers
2k
views
Application for Differential Equation of higher order [closed]
We found some interesting insights in differential equations of the form
$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,
i.e. for ordinary differential equations of $n$-th order with $n\geq2$....
3
votes
2
answers
799
views
existence and uniqueness of solutions for ODEs in formal power series?
I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a ...
3
votes
2
answers
274
views
Existence and uniqueness of solutions to a distributional ordinary differential equation
Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation
$$
\dot{x}(t) = v(x(t))
$$
which I might interpret as an integral equation
$$
-\...
3
votes
2
answers
463
views
Second order differential equation with oscillating behavior
I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
3
votes
2
answers
551
views
Inverted harmonic oscillator
I am looking for the spectrum of th inverted oscillator $H=-\frac{d^2}{dx^2}-x^2$.
Thanks in advance.
3
votes
1
answer
333
views
Existence of solution to linear inhomogeneous first order PDEs systems
Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...
3
votes
2
answers
382
views
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$
The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree ...
3
votes
2
answers
328
views
Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations
In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...
3
votes
1
answer
126
views
Reference request - existence of movable essential singularities
On the Wikipedia page regarding the Painlevé transcendents it says:
Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of ...
3
votes
1
answer
193
views
An explicit formula for a flat metric compatible to certain polynomial vector field with center
Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The ...
3
votes
1
answer
2k
views
closed form solution of an ODEs
I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help ...
3
votes
2
answers
805
views
How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?
To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
3
votes
2
answers
436
views
Legendre equation: An interpretation [closed]
I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...
3
votes
1
answer
942
views
Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
3
votes
1
answer
231
views
Are there fundamental solutions of the laplacian that decay rapidly?
The question
I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function
$$ \...
3
votes
2
answers
381
views
Nonlinear ODE system: stability
I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
3
votes
2
answers
906
views
Herpolhode equation
Poinsot’s construction describes the motion of a freely rotating rigid body in terms of an ellipsoid rolling on a plane. (http://www.phys.ttu.edu/~huang24/Teaching/Phys5306/CH5C.pdf), and the path of ...
3
votes
2
answers
320
views
parallel transport along $W^{1,2}$-curves
Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t_0\in I$ and $\xi_0\in T_{c(t_0)}Q$ be arbitrary. I am looking for a citeable reference ...
3
votes
3
answers
829
views
Analytic ODE with complex time
Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
3
votes
1
answer
273
views
Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
3
votes
2
answers
3k
views
General formula for integrating factor of an homogeneous differential 1 form
This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
3
votes
2
answers
393
views
Minimum eigenvalue of One-dimensional Schrodinger Operator
Consider the One dimensional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + V(x)
$$
Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $.
Now of course,the ...
3
votes
2
answers
124
views
How to find an annihilating linear ode for a class of integrals depending on a parameter?
Given two polynomials $a(t)$ and $b(t)$, consider the integral
$$C(z)=\int_0^1 \frac{dt}{\sqrt{(z-b(t))^2+a(t)}}.$$
For $z$ near $\infty$, $C(z)$ is a well-defined analytic function which can be ...
3
votes
1
answer
361
views
Differential equations → predicate logic mapping
I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark):
I was gonna say, there was a book I ...
3
votes
1
answer
278
views
On Wazewski's theorem on system of differential inequalities
According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem:
Consider the system of differential inequalities given by
$$ \...
3
votes
1
answer
9k
views
how to solve this system of nonlinear differential equations
I want to solve this system
\begin{align*}\tag{*}
x'(s)=x^2(s)+y(s),
y'(s)=x(s)y(s)
\end{align*}
with initial conditions $$x(0)=t, y(0)=t,$$
where $t\not=0.$
With the help of Maple, the solution is
...
3
votes
1
answer
421
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 ...
3
votes
2
answers
705
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
3
votes
1
answer
123
views
A type of singular limit for systems of differential equations
Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles),
$$\begin{cases}\dot{x}_{i,\...
3
votes
1
answer
199
views
Is there a theory of "elementary closed form solution" at the operator level for differential equations?
We begin by considering the usual general first order linear equation of the form
$$ a_0 y' + a_1 y + a_2 = 0 $$
Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
3
votes
2
answers
426
views
A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Introduction
I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...