# Questions tagged [differential-equations]

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

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### A variant to the Stokes system and Navier-Stokes equation

The linearization of the Navier-Stokes equation (in a smooth bounded domain in dimensions 2 or 3) is the following non-stationary Stokes system
$$v_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$
whose $W_p^...

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**2**answers

96 views

### Hypergeometric equation in a particular case

I have a question to make in relation to the solution of the hypergeometric differential equation. Let us consider the aforesaid equation,
\begin{equation}
y(1-y)h'' + [c-(1+a+b)y]h' -abh=0,
\end{...

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60 views

### How we can do the derivative for this equation w.r.t.to time t>0

Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...

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27 views

### Differential equations (euler, taylor series and runge-kutta) [closed]

Can anyone help me, the problem are..
(1) first, use the Euler method to solve the following differential equations:
dy/dx = xy + y; y(0) = 2 to determine y(3) with h = 1
(2) second, use Taylor(...

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**1**answer

126 views

### A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf

Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.

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12 views

### Calculate $\Phi ^{*}\omega $ for a given $\omega$ [migrated]

$\Phi : \mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$
$(x,y,z) \rightarrow (xy,yz^{2},z^{3})$
Calculate $\Phi ^{*}\omega $ for :
i) $\omega= xdx\wedge dz - dx\wedge dy$
ii) $\omega= xdx\wedge dy \...

**3**

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78 views

### What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?

I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations.
Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the ...

**3**

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**1**answer

122 views

### An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background
Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...

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48 views

### Flow lines of a real analytic vector field convergent to a point

Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...

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33 views

### Are Kolmogorov equations degenerate elliptic?

Let $\mu : \mathbb{R}^d \rightarrow \mathbb{R}^d $ and $\sigma: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times d} $ be smooth and Lipschitz continuous. Furthermore, let $\varphi : \mathbb{R}^d \...

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**1**answer

108 views

### Solving Fractional Laplacian Equations with Boundary Condition

I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:
$r^{+}(\nabla^s) v = f$
where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ ...

**0**

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53 views

### Signal sets that arises from a detection theory problem

Consider the discrete time signal $y(t)=\frac{d x(t)}{dt}$ where $x(t)\in[0,1]$ where $t\in\mathbb Z$ (differential is just subtraction of consecutive samples).
Suppose we make two signals $w(t)$ and ...

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21 views

### Reference request for non-standard singular perturbation theory

Though I am by no means a professional mathematician, I am meekly posting this question on here because (a) its merely a reference request (b) the other site said to 'wait' and (c) the experts may ...

**5**

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**1**answer

257 views

### Explicit solution of a Hamiltonian system

It is well-known that the following Hamiltonian system
\begin{eqnarray}
\left\{\begin{array}{rcl}
\frac{dx}{dt}&=&y,\\
\frac{dy}{dt}&=&x(-1+x^2),
\end{array}\right.
\end{eqnarray}
...

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50 views

### State of art of hyperfunction theory in solving partial differential equations

What are the advantages of 'representing distribution(or more generalized functions) as boundary value of holomorphic functions', and their use in solving pde?