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      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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      29 views

      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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      votes
      2answers
      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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      votes
      0answers
      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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      votes
      0answers
      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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      vote
      1answer
      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 \...
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      votes
      0answers
      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
      votes
      1answer
      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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      votes
      0answers
      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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      vote
      1answer
      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 $[0,1]$, $r^{+}$ ...
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      0answers
      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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      0answers
      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 ...
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      votes
      1answer
      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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      votes
      0answers
      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?

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