Questions tagged [differential-equations]
DO NOT USE THIS TAG just because the question contains a differential equation!
928 questions
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Mass lifting off a harmonic oscillator
Consider this setup:
A classic, harmonic oscillator made of a spring with spring constant $k$ and a mass $m_1$ that oscillates vertically. $m_1$ is formed like a horizontal plate, and on the plate ...
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Partial Differential Equations (PDEs) in 2+1 Spacetime with Gradient of the Time Derivative
I have a set of 2 variables $f_1,f_2$, on the Domain of 1+1 spacetime $\{t,x\}$ and a set of PDEs with multiple terms of mixed 2nd-order partial-differentials.
$$\partial_t{f_1} = F_1(f_1,f_2, \...
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The continuum limit of antiferromagnetic magnons
There is a lot of setup needed to ask this question, and numerous steps of which I'm not 100% sure, but my main question is contained in the last paragraph.
Consider an antiferromagnetic quantum spin ...
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Is it possible to construct Smooth solutions for the unsimplified version of the viscid and incompressible Navier-Stokes eqs.? Smooth is essential
There are many exact solutions to the simplified Navier-Stokes equations. However smooth and 3d solutions do still remain elusive.
Is there a way to construct an exact 3d smooth solution generator of ...
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Ensuring the stability and validity of a numerical solution for the Fokker-Planck equation
I am developing a finite-difference numerical scheme for solving the Fokker–Planck equation. The scheme is validated by comparing its solutions with histograms constructed from trajectories of the ...
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Well-posedness of initial value problem in chaotic systems?
Quoting Wald from his seminal textbook on general relativity (Chapter 10):
First, in an appropriate sense, "small changes" in initial data should produce only correspondingly "small ...
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Ignore weight in simple harmonic oscillation ODE [closed]
In Kreyszig's Advanced Engineering Mathematics, he introduces ODEs for simple harmonic oscillations by combining:
$F=-kx$
and
$F=ma=mx''$
So we get the homogenous linear second order differential ...
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Noise term in ODE
Imagine that I am in a lab measuring a certain force that is time dependent, e.g. there is a spring subjected to changes in temperature, which results in a time-dependent stiffness,
$$F(t)=k(t)\delta,$...
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How are the modal matrix elements derived in this PDE projection?
I'm trying to replicate some results I found in a research paper ("Newtonian and Variational Formulations of the Vibrations of Plates With Active Constrained Layer Damping" by Chul H. Park ...
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Why can’t we reduce some PDE to ODE despite the symmetry?
There is an old physics joke called “a spherical cow in vacuum”, which means it’s much easier to solve a PDE by assuming a spherical symmetry. When a system is spherically symmetric, we can use a ...
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Electromagnetic wave equation vs. Maxwell equations (possible solution vs. derivation)
One can take Maxwell equations in an empty space and then derive the (classical) wave equations for both $\vec{E}$ and $\vec{B}$ fields. Examples are given in almost every book or at the Wikipedia (...
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Modelling of oxygen content in dynamic tank
I have a closed tank with an outlet at the bottom where I want to control the water level and oxygen concentration.
For the level control I have a inlet pump where I can control the speed. For the ...
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Globally hyperbolicity of manifold with boundary
On this wikipedia page on globally hyperbolic manifolds, it is stated that a manifold with boundary is considered globally hyperbolic if its interior is globally hyperbolic. I have skimmed through ...
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Phase curves for a non-$\mathcal{C}^2$ potential
Following an exercise in my book, I have drawn the phase curves in the $(x,\dot{x})$ plane for the one-dimensional potential
$$\begin{cases}
(x+1)^2, x<-\frac{1}{2} \\
-x^2+\frac{1}{2}, -\frac{1}{...
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Radial perturbation of a particle in a fluid vortex
$\textbf{Purpose:}$ My goal is to prove,
\begin{align*}
\min \left\{ \int_{0}^{T} \mathcal{L}\left ( \gamma (t),\dot{\gamma}(t),t \right )dt \right\} \leftrightarrow \displaystyle \lim_{t \to \...
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What is initial condition on Cauchy surface?
I am reading definition of Domain of dependence which is defined for Cauchy surface. We say if initial conditions are given on Cauchy surface then we can predict what can happened on the entire ...
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Frobenius method for fourth-order differential equation
I am trying to reproduce some results from a paper
https://iopscience.iop.org/article/10.1209/epl/i1998-00235-7
The authors solved a 4th order partial differential equation
$$\nabla^4 u+(2-\lambda)\...
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Nonlinear dynamics: analytical solutions to a sinusoidally forced
I asked this question on math stack exchange but I wanted to repeat it here, since I was studying a physical system when I came across the following differential equation:
$$ \ddot{\theta}+\alpha \dot{...
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Diffusion from an annulus
I'm wondering if the following problem can be solved using the method of images (I'm familiar with how this works for straight boundaries, but not circular boundaries, as in this problem). Suppose we ...
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In the undriven $RLC$ circuit, why would one conjecture the solution with $Ae^{Zt}$ when we perfectly know that critical case has $te^{-\alpha t}$?
There's something I don't quite get in this video (MIT $8.02$ course titled "Electricity and Magnetism", video number $208$, taught by Pr. Peter Dourmashkin).
Professor conjectures the ...
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Conjecturing the homogeneous solution of a $2^{nd}$-order constant coefficients ODE - why conjecturing a $3$ D.O.F. solution in this course?
In this video (MIT $8.02$ course titled "Electricity and Magnetism", video number $208$, taught by Pr. Peter Dourmashkin) professor solves the undriven $RLC$ circuit ODE ($2^{nd}$-order ...
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Driven $RLC$ circuit why is the permanent signal sinusoidal and why two degrees of freedom (amplitude and phase)?
In a second-order ODE with constant coefficients, with a sinusoidal RHS term (such as the ODE of the driven $RLC$ circuit), how do we know:
that the particular solution (i.e. the permanent signal) is ...
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Can the turbulent viscosity in Large Eddy Simulation (LES) be taken out of the derivative for compressible flows?
I am working on Large Eddy Simulation (LES) for compressible flows and using the Smagorinsky model to compute the turbulent viscosity $\nu_t$.
In the filtered Navier-Stokes equations for compressible ...
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Order parameter of vortex anti-vortex pair
I am working on a project and I am having trouble finding a source which actually calculates the order parameter from the Landau-Ginzburg equations for a vortex anti-vortex pair (superfluid or SC). ...
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Density-Dependent Diffusion Dispersion Relation
I'm studying a model which incorporates the following PDE
$$\frac{\partial u}{\partial t}=uf(u)+D(1-\alpha f(u))\frac{d^{2}u}{dx^{2}}-D\alpha\frac{df(u)}{dx}\frac{du}{dx}$$
With zero flux boundary ...
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