Questions tagged [differential-equations]
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
1,667
questions
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Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 3,154
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votes
0
answers
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views
Bounded solutions of nonlinear third-order ODEs
I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
- 633
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votes
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Bounding norms of symplectic matrix factorisations and non-separable Hamiltonian flows
Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc}
A & C\\
C^T & B\\
\end{array}\right)$ ...
- 21
0
votes
0
answers
54
views
Second order matrix PDE
In my research I came across a second-order PDE of the following form
$$
\biggl(
v
\odot
\Big(
\nabla \rho(x)\,\mathbf{1}^{\top}
+
...
- 362
2
votes
1
answer
158
views
Differential equation involving square root
I am absolutely not familiar with differential equations. However, I am facing the following differential equation:
\begin{equation}
a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)}
\end{equation}
...
- 63
-1
votes
0
answers
51
views
How to solve a differential equation that includes the expectation of a derivative?
Here $x$ is a real variable, its derivative $\dot{x}(t)$ is a stochastic variable but satisfies $\mathbb{E}(\dot{x}(t))=ax(t)+b$, where $\mathbb{E}(\cdot)$ represents expectation, and $a, b$ are ...
5
votes
1
answer
286
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 51
2
votes
0
answers
69
views
Differential equations where we are tryng to find a smooth space(maybe with additional structure)
Is there literature on the differential equation $TM=N$ where $M$ and $N$ are two smooth spaces (maybe with additional structure) and $T$ stands for the tangent functor?
- 67
3
votes
1
answer
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views
Frobenius antecedent of a differential module
Let $K$ be an ultrametric complete field of mixed characteristic, and let $F_{\rho}$ be the completion of $K(t)$ with respect to $\rho$-Gauss norm. After Christol and Dwork, for a differential module $...
0
votes
1
answer
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Any research on ODE $\frac{1}{\sin ^{N-2} \theta} \frac{\partial}{\partial \theta}(\sin ^{N-2} \theta \frac{\partial u}{\partial \theta})=f(u)$?
I want to know if there is any research on ODE $$\frac{1}{\sin ^{N-2} \theta} \frac{\partial}{\partial \theta}(\sin ^{N-2} \theta \frac{\partial u}{\partial \theta})=f(u)$$
with any boundary condition?...
- 755
2
votes
0
answers
46
views
Random solutions to non-Lipschitz ODEs, optimal transport, and general solutions to the continuity equation
I am reading Cedric Villani’s book “Optimal Transport: old and new” and I am stuck on one paragraph (see page 26/27 in this book). He speaks about random solutions to an ODE and I simply cannot figure ...
- 21
2
votes
1
answer
74
views
Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
- 31
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vote
0
answers
176
views
Solving the Moutard PDE
I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE
$$h_{uv} = q\,...
- 203
3
votes
1
answer
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Deriving differential equation from difference of PDE solutions
This is an edited cross-post from Math SE because after several days it's received no good answer. I think it's less appropriate for a general QA Math site and is likely better for Overflow with ...
- 33
3
votes
1
answer
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Analytic solutions to analytic differential equations
Let $U \subseteq \mathbb R^{n+2}$ be an open set for some $n \geq 0$, and let $f: U \to \mathbb R$ be an analytic function. Then we say the equation $f(x,y,y',\ldots,y^{(n)})=0$ is an analytic ...
- 31