# Tagged Questions

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

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### Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
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### First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation $$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$ where $A_0, A_1, A_2, B$ are some constants, ...
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### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question: Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as ...
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### Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
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### Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
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### Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
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### Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
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### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as \begin{align} \mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega \end{align} where the ...
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### acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial y_i}\left(...
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### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
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### About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper: A concave–convex elliptic problem involving the fractional Laplacian - C. Brandle, E. Colorado, A. de Pablo and U. Sánchez. At the Pgs 41, 42, the ...
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### Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B. Wonderfulness of roller comes from this property ...
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### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...
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### Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...
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### Solution to Helmholtz equation with non-circular boundary

Let $D$ be an homogeneous 2D domain with non-circular boundary $\partial D$. I am trying to solve the Helmholtz equation $$\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)$$ in which ...
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### 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 ...
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### BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better? The BDF2 method requires the ...
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### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension. The first one I know is the Peano existence theorem. I ...
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### Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...
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### Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote: ...
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### General description of surface with zero gaussian curvature

Let suppose function $g(s,t)$ satisfies partial differential equations: $g_{ss} g_{tt} - g_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution ...
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### Shooting method for non-oscillatory solutions

I am having some trouble with using the shooting method : I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I ...