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Questions tagged [differential-equations]

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Spinning mass around Spring with gravity

I've seen similar posts, but none in which a mass rotates vertically while connected to a spring, with gravity acting on the mass. I want to know if the path of the mass can be described with ...
T.B.'s user avatar
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Tensor differential equation

How to solve the following differential equation with tensor indices? $\epsilon_{\mu\nu}\partial^{\gamma}\partial_{\gamma}f-2i\epsilon_{\mu\nu}p.\partial f+ip_{\mu}x^{\gamma}\epsilon_{\nu\gamma}+ip_{\...
Pratik Chatterjee's user avatar
1 vote
1 answer
35 views

How Does Frequency Change With Damping (Underdamped Harmonic Oscillators) [closed]

I'm studying harmonic oscillators and I'm trying to model a system where both the frequency and amplitude decay over time. This is throwing me off because frequency decay is much less intuitive than ...
Jeremy Kievit's user avatar
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0 answers
50 views

Given Green's function, can I find the corresponding operator? [migrated]

Green's function is the solution to the equation $L G(x;x') = \delta(x-x')$, where $L$ is a linear differential operator. Usually, we want to find the Green's function of a given $L$. Instead, if we ...
Sean's user avatar
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3 votes
1 answer
59 views

Decoupling Linearly Coupled Wave Equations with Potentials

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
Afraxad's user avatar
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Derivation of gyrocenter speed in gradient magnetic field, doubt in resolution differential equation with slow time variation force

I was reading Paul M. Bellan's book:"Fundamentals of Plasma Physics. Cambridge University Press, 2006.", the chapter "Drifts in slowly changing arbitrary fields", in particular the ...
Axel Togawa's user avatar
3 votes
2 answers
296 views

Solving the wave equation of a tensor $h_{\mu\nu} = (1/2) (e_\mu e_\nu + e_\nu e_\mu)$

It is known that the solution to the wave equation for a tensor $$ \square h_{\mu\nu} = 0 $$ is $$ h_{\mu\nu}(\vec{x}, t) = \int \frac{d^3k}{(2\pi)^3} \sum_{\lambda=+,\times} \left( \epsilon_{\mu\nu}^{...
Anon21's user avatar
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1 vote
1 answer
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General Solution to Maxwell's Equations with Duhamel's Principle

In one dimension, it is easy to prove that if two solutions $\{u_1, u_2\}$ are known to $\mathcal{L}u(t) = 0$ where $\mathcal{L} \equiv{a(t)\partial_t^2+b(t)\partial_t+c(t)}$, the general solution to ...
Cody Payne's user avatar
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1 answer
31 views

Is it possible that classical propagator be used as an integrating factor for solving differential equations?

I have two questions about the picture. 1) I think classical propagator itself is not function, is just an operator. And "(operator)(function)" is not that "(operator)X(function)". ...
user403049's user avatar
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1 answer
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General solution for the motion of a 1D particle with drag

Consider a particle of mass $m$, with equation of motion $$m\ddot{x}=-U'(x)-f(\dot{x})\dot{x}.$$ I am trying to find an equation of the form $v=g(x)$ so that we can reduce this to an integral of the ...
Don Al's user avatar
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Electromagnetic Field in a 3D Cavity with Lossy Boundary

I would like to find the electric and magnetic fields inside a cubic cavity with a lossy boundary (i.e. NOT a perfect conductor). I assume that the interior of the cavity is filled with a homogeneous ...
amrit 's user avatar
1 vote
0 answers
16 views

Bulk-to-bulk propagator in 3-point function in AdS-CFT correspondence. Trouble solving a PDE

I have encountered an issue in a PDE (A Green's function actually). I am solving it in $(d+1)$-dimensions and I use Poincare coordinates in AdS spacetime, meaning I have a dimension $z$ and I also ...
Βασίλης Γερμανίδης's user avatar
3 votes
2 answers
190 views

Epidemic spreading model

I'm studying a model in the field of complex systems regarding the epidemic spreading. The model is the susceptible-infected model, i.e., there is a population of N subjects and each of them can ...
Salmon's user avatar
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1 vote
1 answer
117 views

How is the wave equation derived or discovered? [duplicate]

I don't really understand where the fundamental or general wave equation $$\frac{\partial^2y}{\partial t^2} = v^2\frac{\partial^2y}{\partial x^2}$$ comes from. I understood the derivation of wave ...
Jack's user avatar
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1 vote
2 answers
63 views

Numerical solution of differential equations, e.g. the three-body problem

What forms of differential equations have numerical solutions with errors that go to zero with sufficient computational power? For example, suppose I want to solve a differential equation $E$ for a ...
Alex's user avatar
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Regarding to the asymptotic solution of quantum harmonic oscillator

In quantum mechanics, the radial equation of the SHO takes the form \begin{align} \frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0, \end{align} where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
Mr. Anomaly's user avatar
0 votes
2 answers
131 views

Rescaling time in differential equations

On a scientific paper, I found the following equations about a compass gait (one leg behaves like an inverted pendulum, the other one as a simple pendulum; $\theta$ and $\phi$ are time-dependent): $$ \...
Federica Guidotti's user avatar
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46 views

Proof that separation of variables leads to a complete basis of wave function in spherical coordinates [duplicate]

In griffith's introduction to quantum mechanics (chapter 4), there is an analysis of the stationary states of a particle given a potential function $V(r)$ that only depends on the radial distance $r$, ...
user56834's user avatar
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1 vote
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A missing link in the logical chain about the Aharonov-Bohm effect

The usual treatment of the Aharonov-Bohm effect (which appeared already in Aharonov and Bohm's original paper) takes two particular local solutions of the Schrödinger equation, $\psi_1$ and $\psi_2$. ...
mma's user avatar
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1 vote
3 answers
138 views

Why we take only the real part of a solution as the actual motion?

I am taking Analytical Mechanics, and in Goldstein's book, chapter 6 (page 241) about linear oscillations, he says the following: "... $\eta_i=Ca_ie^{-i\omega t}$ (6.11) ... It is understood of ...
A24601's user avatar
  • 13
1 vote
2 answers
76 views

Static solution to an implicitly dynamic problem - heat equation

Heat equation This is the heat equation: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} $ ...
Megidd's user avatar
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1 answer
88 views

Green's function solution in 2D for the potential of solenoids in the Lorenz gauge

My main goal is to find a general expression for the potential in the Lorenz gauge of some solenoidal (not necessarily circular) current density using the Green's function. I assume that the current ...
Arceon's user avatar
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0 answers
31 views

Need help identifying missing boundary condition in charge transport problem

The system I am considering consists of a conducting liquid sandwiched between two electrodes. The electrodes supply a constant flux of electrons $J_{ext}$. In the liquid there are also uncharged ...
Ornate's user avatar
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0 answers
26 views

Can anyone please give some explanation in terms of the frequency domain of the time evolution?

This might be a silly question. But I was puzzled for a long time, even some comments are greatly appreciated. Is it possible to claim that "All the time domain evolution can be thought of ...
MathArt's user avatar
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1 answer
39 views

Physical interpretation of the semilinear Dirichlet problem

As a non-physicist student, I’ve been trying to gain some intuition on the problem I’m studying, namely \begin{equation}\begin{cases} -\Delta u + f(u) = h, \; \text{in } D \subset \mathbb{R}^n\\ u=0, \...
Spida's user avatar
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1 vote
0 answers
52 views

A problem from Goldstein's Classical Mechanics (2nd ed., page 15) [duplicate]

There is a footnote on Goldstein's Classical Mechanics (2nd ed., page 15) which says the following: In principle, an integrating factor can always be found for a first-order differential equation of ...
Dazhong Ren's user avatar
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0 answers
54 views

Differential equation with step function

I want to solve the equations of motion for a system with a unit step function. Are there any methods that can be used to solve these? As a toy model, I picked a sliding mass bouncing off a spring. ...
EraserDriver's user avatar
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0 answers
16 views

Logistic Growth and the Use of Changing Units

I am having a hard time understanding how this unit change is used to get rid of the need to use specific values for $ N $ and $ \mu $. Could somebody explain? I just can't figure it out. Help would ...
j.primus's user avatar
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1 answer
85 views

Solving divergence and curl equations numerically

I've recently come to learn about Jefimenko's general solution for Maxwell's equations as well as the FDTD method in electromagnetic optics, and that has got me thinking whether I myself can solve ...
Lagrangiano's user avatar
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1 vote
0 answers
16 views

Is this PDE describing the variable length cable dynamics right? [closed]

I am solving a variable length cable-Mass dynamics and having some problem. Here is the problem description: I noticed that this post is off-topic, while I found a related paper about this question, ...
Y Randolf's user avatar
0 votes
1 answer
83 views

Method of characteristics with coupled ODEs

I am having trouble following the derivation in this paper https://arxiv.org/abs/1810.07775 using the method of characteristics. By using the method of characteristics, they derive the following ODEs ...
Idieh's user avatar
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0 answers
13 views

Movement of a mass on a spring in damped SHM

Suppose a ball is connected to a spring attached to a wall, and they are in space, i.e. assume no gravity. The ball is put into a fluid with Stoke's drag and oscillates backwards and forwards relative ...
Yitian Chen's user avatar
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1 answer
89 views

Dirichlet’s Theorem and Solutions to Laplace Equation in Cartesian Coordinates

I have been reading Introduction to Electrodynamics - Griffiths about solving Laplace equation in cartesian coordinates, and in that book, I saw this statement: The functions $\sin(n\pi y/a)$ are ...
Sanjay's user avatar
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0 votes
1 answer
43 views

What are streamlines and pathlines

I asked a question earlier about having a vector field and a starting point, and then making a parametric that starts at the starting point and the derivative at any point in the parametric equals the ...
GIORGI GOGIBERIDZE's user avatar
2 votes
3 answers
66 views

Electric Potential Due to Point Charge in a Plane

In general physics courses, we are told that the electric potential a distance $r$ away from a point charge of magnitude $Q$ is given by $$ V = \frac{Q}{4\pi \epsilon r} \tag{1}. $$ Using this ...
Zachary Candelaria's user avatar
0 votes
2 answers
52 views

Implications of time reversibility in the initial conditions

I read somewhere the following statement: In a partial differential equation (PDE) in order to get the property of time reversibility in your solutions, you need to set the initial condition $u'=0$ (...
Penilin Lot's user avatar
2 votes
2 answers
157 views

Linearity of Lindblad equation in the Heisenberg picture

I am interested in solving the dual (adjoint) Lindblad master equation for a time-dependent operator $O(t)$ as follows \begin{equation} \dot{O}(t) = i[H, O(t)]+\sum_{\alpha\in I} L_\alpha ^\dagger O(t)...
GSLAM's user avatar
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0 votes
4 answers
87 views

Separation of variables: When can we say that a function of $x$ and $t$ (for example) is a function of $x$ times a function of $t$?

I've seen this a lot in physics so far. For example in the stationary state solution to the Schrodinger equation for hydrogenic atoms, it's commonly approached using $\psi(r,\theta,\phi)=R(r)\Theta(\...
JBatswani's user avatar
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0 answers
50 views

How to properly discretize and solve the Liouville equation?

Consider some dynamical system $\dot{\textbf{X}}(\textbf{x},t)=F(\textbf{X})$ where $\textbf{X}$ is discretized along a 1-dimensional spatial coordinate $\textbf{x}=(x_1,\dots,x_N)^T$. Let $\rho(\...
thespaceman's user avatar
1 vote
1 answer
39 views

Radioactive Decay Chain for the special case of equal decay constants (parent-daughter)

I have been trying to obtain an analytic solution for a daughter radionuclide's activity (or just the number of daughter atoms), as a function of $t\geq0$, resulting from the decay of a parent ...
Username134's user avatar
0 votes
0 answers
71 views

Self-similar solution of the second kind

I have a problem trying to understand the procedure for using self-similar solution of the second kind. More specifically, I was reading about an equation of this form, $$\partial_t{d} + \frac{1}{r} \...
Waxler's user avatar
  • 109
1 vote
3 answers
116 views

Charge Distribution and Stability in a Conductive Solid Sphere

My friend came to me with a simple question: What is the charge distribution on a conductive solid sphere? Of course, I answered: 'Since the solid sphere is conductive, the electric potential would be ...
Danny Wen's user avatar
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0 votes
0 answers
22 views

Memristor Differential Equation Help

I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
Seb's user avatar
  • 1
2 votes
0 answers
45 views

Finding condition for Adiabaticity

I have a differential equation describing a resonator that looks like this: $$ \frac{d\alpha(t)}{dt} = [j a - b]\alpha(t) + \sqrt b e^{jct}$$ where I can solve it putting: $$\alpha(t) = \alpha e^{jct}$...
SiPh's user avatar
  • 21
-3 votes
1 answer
58 views

A least principle action and human behaviour [closed]

Does anyone have an idea how the authors of this paper https://www.nature.com/articles/s41598-021-81722-6#additional-information solved the equation 10? Thank you in advance
physics22's user avatar
-1 votes
1 answer
48 views

Time taken for a rocket to travel upwards [closed]

My doubt is a rather silly, simple one but i cant seem to understand what's wrong. Let's assume a rocket is moving up with a constant acceleration of a, is moving strictly vertically(no gravity turns, ...
Star Gazer's user avatar
1 vote
1 answer
102 views

Klein-Gordon equations in quintessence

I'm studying the quintessence model from a dynamical system point of view. I denoted the scalar field for the dark energy as $\phi$, so I have the following Klein-Gordon equation for the field $$\...
Guillermo García Sáez's user avatar
1 vote
0 answers
48 views

Physical meaning of some values

I'm working with the paraxial wave equation $$2ik_0\frac{\partial u}{\partial z}+\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\,-\infty<x<\infty,\,-\infty<y<\infty,\,...
Guillermo García Sáez's user avatar
0 votes
1 answer
103 views

Solutions of Laplace's equation for stream functions in cylindrical coordinates [closed]

I was reading Fluid Mechanics by Richard Fitzpatrick. Somewhere in the book, he tried to solve inviscid flow past a semi-infinite wedge https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node76....
Lohrasb's user avatar
  • 119
0 votes
1 answer
42 views

Convergence of harmonic oscillator to the free particle and the issue of asymptotic freedom in QM

For any $\epsilon >0$, consider the following harmonic oscillator with $m= \hbar =1$: \begin{equation} \frac{\partial}{\partial t}\psi(x,t)= -\frac{1}{2}\Delta \psi+ \frac{\epsilon}{2}x^2 \psi \end{...
Keith's user avatar
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