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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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      How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

      I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
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      2answers
      223 views

      Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$

      Consider the ode $$ f''(t)-e^{-2t} f(t)=0. $$ What is the general behaviour of $|f|$ for large $t$s? Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(...
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      138 views

      Solution of nonlinear second-order ODE $y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0$

      Is there any way of solving the following second-order ODE $$y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0,$$ where $a$ and $b$ are some constant? If we know that one solution exists, how would it ...
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      0answers
      40 views

      Second order non-instantaneous impulsive evolution equations

      The first order linear non-instantaneous impulsive evolution equations is given as; $u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$ $u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=...
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      0answers
      44 views

      Annihilator of the of the generating function not holonomic

      The following is a generating function in $x,h$ with infinite parameters $q_1,q_2\ldots,$ and $w_1, w_2,\ldots$. $$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
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      1answer
      97 views

      Holder inequality with respect to convex function

      Given $a, T>0$, by Holder's inequality we have $$ \int_T^{T+a}f(s)ds\leq \left(\int_0^{T+a}|f(s)|^2ds\right)^{1/2}\cdot\sqrt{a}. $$ Do we have similar result if we replace $|x|^2$ by some convex ...
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      0answers
      57 views

      Keeping track of limit cycles via certain second order differential operator

      Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
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      22answers
      5k views

      Which high-degree derivatives play an essential role?

      Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
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      17 views

      Inequality for integro-differential system

      Reading a paper about homogenization of stochastic coefficients [A. Gloria et al., Invent. math. (2015)], I found the following lemma that gives an estimate of the solution for a given ODE system. The ...
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      31 views

      Limits of the wave equation with piecewise constant propagation speed

      This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty. Consider a wave equation $$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...
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      51 views

      $L^1$ estimate of the heat equation

      The classical heat equation read $\partial_tu-\Delta u=0$ in $\Bbb R^3$. If $u$ is sufficintly smooth, then we can multiply both sides by $|u|^{p-2}u$ to get the $L^p$ estimate, where $1<p<\...
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      0answers
      275 views

      On Solving a Fourth-Order Non-Linear PDE

      I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity at the leading edge of a flat plate when fluid flows past it. The ...
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      44 views

      Lax Pair for Kadomtsev-Petviashvili Equation (2-dim KdV)

      Is there a simple way to show that the pair \begin{equation}\begin{cases}\alpha \psi_y+\psi_{xx}+u\psi=0 \\\psi_t + 4\psi_{xxx}+6u\psi_x+3u_x\psi -3\alpha (\partial_x^{-1}u_y)\psi=0\end{cases}\end{...
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      25 views

      Underspecified Riccatti-type ODE

      I came across the following Ricatti-type ODE in my reading $$ \begin{aligned} \partial_t \psi(t,x) &= \Psi(\psi(t,x)),\\ \psi(0,x)&=x,\\ \Psi(x)&\triangleq \partial_t\psi(t,x)|_{t=0^+}. \...
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      Riccati equations for dealing with stiffness of ODEs

      Early texts [Forman Acton, Numerical Methods that Work; Numerical Recipes, first edition] suggested that for integrating a stiff linear ODE it was very helpful to integrate the associated Riccati ...

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