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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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        30 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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        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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        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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        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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        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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        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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        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 ...
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        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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        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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        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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        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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        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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        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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