Questions tagged [differential-equations]
DO NOT USE THIS TAG just because the question contains a differential equation!
247
questions with no upvoted or accepted answers
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Can a "depressive soliton" wave exist? That is, can we have a trough without any crest? Why or why not?
I know that "soliton" waves can consist of a crest without a trough. One would expect the reverse to be true as well.
However, this Wikipedia excerpt says,
So for this nonlinear gravity ...
5
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0
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93
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Initial value formulation of Yang-Mills equation
In Wald Chapter 10, he discusses the initial value formalism of electromagnetism - how Maxwell's equations are actually a system of three equations plus an initial value constraint, and how we can ...
5
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203
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Minimum required initial conditions to uniquely solve geodesic equation
The geodesic equation is a 2nd order differential equation given as
$$\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d} \lambda^2 }+\Gamma^\alpha_{\beta\gamma}\frac{\mathrm{d} x^\beta}{\mathrm{d} \lambda }\...
5
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378
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Is there an intrinsic physical meaning for characteristic curves of a PDE?
For partial differential equations (such as those that govern many physical phenomena), there exist characteristic curves, along which the equations can be reduced to total derivatives and solved. The ...
5
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166
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on fundamental 2D conductivity equation boundary value problem
Consider the following homogeneous boundary value problem for a function/potential $u(x,y)$ on the infinite strip $[-\infty,\infty]\times[0,\pi/4]$ w/positive periodic coefficient/nductivity $\gamma(x+...
5
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327
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Is there a proof that the number of eigenstates is countable for a bound system?
When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number $k \in \mathbb R $ and $\mathbb R$ isn't countable but if you add a ...
5
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464
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Naive questions on the classical equations of motion from the Chern-Simons Lagrangian
Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
5
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2
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423
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Choosing initial condition for Hamilton-Jacobi PDE from initial $x$ and $p$
For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as
$$S= W(x) + W(y) - E*t$$
and treat the separate parts as constants and find $W(x)$, $W(y)$...
4
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85
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References for Exterior Differential Systems for physicists
Many problems in classical mechanics, classical and quantum field theory among others require the study of systems of partial differential equations where questions like existence and uniqueness ...
4
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89
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Does General Relativity satisfy the Homotopy (or “h”) Principle?
By this I mean in the standard second order (whether metric or tetrad/verbein-based) form of General Relativity. I've been reading about the homotopy principle of late (see Eliashberg's introduction ...
4
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43
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Potentials with continuous spectra except for discrete set of values
Section 18 of Landau & Lifschitz's Quantum Mechanics discusses how the Schrödinger equation with a potential that vanishes at spatial infinity can have a continuous spectrum, a discrete spectrum, ...
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102
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Quantum Particle in a Fractal Box
I was thinking about particle in a 2D box the other day, and I realize that it shapes actually affect its energy and wavefunction. Therefore I thought to myself, what if a particle is inside a fractal ...
4
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1
answer
316
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Modeling friction with Non-Lipschitz ODEs - Pendulum example
The main objective of this question is to figure out if the following differential equation have [finite-duration] solutions:
$$ \ddot{\theta}+0.021\,\operatorname{sgn}(\dot{\theta})\sqrt{|\dot{\theta}...
4
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0
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182
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Fokker-Planck equation for 2D SDE
Consider the following two-dimensional SDE
\begin{align*}
\mathrm{d}\mathbf{X}(t) &= {\mathbf{f}(\mathbf{X}(t))}\mathrm{d}t+\mathrm{d}\mathbf{W}(t)\\
\end{align*}
where $\mathbf{X}(t)=\begin{...
4
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0
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211
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Intuition about ADHM construction
I'm trying to understand reasons, why self-dual Yang-Mills equation can be reduced to algebraic equations. It's seem like a miracle.
In article Construction of Instanton and Monopole Solutions and ...
4
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56
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Master equation for a dynamical system on networks
I am trying to mathematically model the following idea that describes the dynamical evolution of a quantity over a graph.
Let us imagine we have a directed graph, with $n$ nodes and $m$ edges.
Each ...
4
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184
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Non-linear Wave Equation - Numerical Methods
Motivation:
I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form
$$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$
where $'=\frac{...
3
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0
answers
56
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Observed frequencies of a drum versus mathematically predicted frequencies
I built a small circular drum in order to compare the actual resonant frequencies versus the resonant frequencies predicted by modeling it as a circular membrane obeying the wave equation with fixed ...
3
votes
0
answers
79
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Lie symmetries of differential equation and ladder operators
There is literature on the lie symmetries of quantum harmonic oscillator differential equation. The generators satisfy certain lie algebra.
On the other hand, we have ladder operator method. The ...
3
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0
answers
64
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Rationale for Continuity of Azimuthal Equation
To solve Laplace's equation $\nabla^2\psi =0$ or the Schrödinger equation: $$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi =E\psi$$ in spherical coordinates, we often separate variables as follows: $$\psi(r,\...
3
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84
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About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form
$$\frac d{dt}\vec x(t)=\vec y(t)$$
$$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$
(or equivalently $\frac {d^2} {dt^2}...
3
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0
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108
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Equation of motion for a coil around a mass, attached with a spring driven by magnetic field?
First of all, thanks for reading and hello! First time poster on Physics Stack Exchange. I'm an electronics engineer (so forgive my poor maths below!) and tend to hang out there.
I have a questions ...
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75
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Why is the space $L^2(a,b)$ preferred over the space $C([a,b])$ of all continuous functions on $[a,b]$?
This question might be better asked on the Math.SE site but I feel it could be well placed here as well.
My textbook (Sturm-Liouville Theory and its Applications , Al Gwaiz) defines the vector space $...
3
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267
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Solving Laplace equation on a triangle with mixed boundary conditions
From sources [Ref: 1 to 5], one learns that there is a class of boundary conditions (see Fig 1) on a triangle (in this case the 30-60-90 triangle) that allow for closed form solution for eigensystems ...
3
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135
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Initial value problem on $\mathcal{I}^-$ for Maxwell fields
In the paper "Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity" by Ashtekar and Streubel the authors state the following:
Fix, as in § 2(a), a conformal completion $(...
3
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0
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82
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Chua's Circuit: an inequality ensuring that the equilibrium is not stable
According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are
$$\begin{aligned} C_{1} \frac{d v_{...
3
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56
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Initial Condition in Spaghetti Cracking
In this Paper B. Audoly, S. Neukirch - Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half on Page 2 (bottom), the author argues that using an integral of motion, the ...
3
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145
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Characteristics of the Navier-Stokes equations as a set of PDE's
I am not entirely sure if I should ask this question here or not, but here goes: can anyone suggest any reference (book, article, etc.) about the Navier-Stokes equations from a mathematical point view?...
3
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1k
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Relationship between Green's function and impulse response
In my field, electrical engineering, we frequently study linear time-invariant systems of the following form:
$$
a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
3
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0
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67
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Wave solutions on a quotient manifold
Is there a general rule for, given the solution of the wave equation $\phi$ on a manifold $M$, what the solution will be like for the quotient of the manifold by some group $\Gamma$? In particular ...
3
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120
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Linear KDV eq. asymptotics
The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq.
$$
u_t+u_{xxx}=0
$$
My first step was to take a Fourier transform of the equation, find that the dispersion ...
2
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0
answers
43
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Finding condition for Adiabaticity
I have a differential equation describing a resonator that looks like this:
$$ \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}$...
2
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139
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Renormalization group equation and method of characteristics
All of this question refers to ref. 1. The equation are numbered alike.
The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
2
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0
answers
62
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Green's identity for arbitrary differential operators
If we have a scalar field $\psi$ that satisfies an equation $\nabla^\mu \nabla_\mu \psi = \rho$ where $\rho$ is some known source we can use Green's identity to express it as
\begin{equation}
\psi (x)...
2
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1
answer
160
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Why does the eigenvalues of an angular frequency matrix are the natural frequency? (INTUITION)
lest say we have a system of differential equations of some coupled oscillator such that:
$$\overrightarrow a = [w^2]\overrightarrow x$$
if we find the eigenvalues of $[w^2] = \lambda$ why those ...
2
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0
answers
71
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Simplifying 2D Navier stokes equation over the top and bottom part of an airfoil - assumptions incompressible, steady, very high viscosity
I am trying to simplify the Navier-Stokes equations with my assumptions, to be able to solve them numerically:
I'm trying to model an airfoil flying through a very viscous fluid at relatively low ...
2
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0
answers
78
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Problem analyzing Dirac equation in an arbitrary coordinate system using geometric calculus
This question was partially inspired from the Dirac equation in spherical coordinates.
For simplicity, let’s suppose I have a two variable partial differential equation initially written in a ...
2
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0
answers
88
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Numerical calculation of spherical photon orbits near Kerr black hole
I'm using RK4 method for solving differential equation
$$\frac{d\theta}{d\phi}=\pm(2-2r)\frac{\sqrt{Q-\left(\frac{\Phi^2}{\sin^2\theta}-1\right)\cos^2}\theta}{2r+\left(r^2+\cos^2\theta-2r\right)\frac{\...
2
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0
answers
242
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Deriving the form of the metric on the Poincare half-plane in polar coordinates
This question relates specifically to problem 3.11 from Sean Carroll's book Spacetime and Geometry.
After having proven in the text that the Poincare half-plane is a maximally symmetric space and that ...
2
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90
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Solving for fluid velocity field given positions as a function of time
I have a distribution function that describes the number of particles as a function of time and their mass, $n(M,t)$, which I can calculate analytically for all $M,t$. I want to know the evolution of ...
2
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421
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What is the problem of higher-order time derivatives with causality?
I've heard that equations of motion with third- or higher-order time derivatives have problems with causality, but can't seem to find any proof or reasoning for this. Could anyone please help me? I ...
2
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35
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Boundary condition for $\Box\vec{E}(t,\vec{x})=0$ that preserves scale-invariance
In short, this is a question about the symmetry of a differential equation preserved by its boundary condition.
In free space, the vector wave equation satisfied by the electric and the magnetic field ...
2
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40
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Find an optimal shape analytically
This is better described with an example: Imagine we posses our current understanding of physics, but we do not know how to make an airplane because we do not know about airfoils. To make this more ...
2
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103
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How to solve general wave equation and dispersion relation using Fourier series?
In this paper (open access), the authors used Fourier series with most general wave equation to find the dispersion relation. I am presenting some main equations as snippets to depict their solution. ...
2
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0
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59
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Boundary conditions for equation of motion of a chain under small vertical motion of its support
I would like to find the boundary conditions for the of motion of a chain under small vertical motion of its support endpoint. I also assume displacement of the chain in the vertical $y$ direction is ...
2
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0
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105
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What does it mean for resistance, as it appears in Ohm's law, to be an integral, evaluated over the body as a whole?
In literature I read:
The three linear flux laws mentioned are:
As seen, a correspondence exist between the hydraulic conductivity $K$, thermal conductivity $\lambda$, and electrical conductivity $\...
2
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40
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What is the intuition behind a high Froude number causing instability in uniform flow?
Uniform flow in an inclined plane becomes unstable for high froude numbers.
I can follow the spectral analysis, but I am curious why I would expect high froude numbers to cause instability. What is ...
2
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0
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126
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Cauchy problem of classical Maxwell equations in Minkowski spacetime
I'm wondering a bit about the classical Maxwell equations in flat spacetime and their Cauchy problem. For the following, I suppose that the sources are already given and don't react to their own ...
2
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0
answers
35
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Velocity waves in spherical geometries
I am currently working on velocity waves in spherical geometries: I am considering a 1D many-particle system confined on a circle with a global drift leading to rotations, similar to this simulation ...
2
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81
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"The state-space for a single particle in classical space is 6-dimensional" -- Is this wrong?
The general argument is as follows. By Newton's second law $\mathbf F=m\ddot{\mathbf{x}}$. Now it is said that this is a second-order ODE and hence requires $\mathbf x(0)$ and $\mathbf{\dot{x}}(0)$ as ...