Questions tagged [differential-equations]
DO NOT USE THIS TAG just because the question contains a differential equation!
792
questions
-1
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0
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19
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Solving a set of DE in any method, boundary conditions at $\rho\rightarrow -\infty$ [migrated]
I want to solve the following set:
$$
y'(\rho) = y(\rho)*g'(\rho)+a
$$
$$
f'(\rho) = -\frac{1}{a}*y(\rho)*f(\rho)
$$
$$
g'(\rho)=-y(\rho)*\left[\frac{1}{a}f(\rho)^2 +a\right]
$$
with the condition of $...
1
vote
1
answer
26
views
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 ...
0
votes
1
answer
22
views
Differential equation of an object dropped from certain height
I want to solve this problem -
A ball of mass 2kg is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experience a damping force of the form -2v, where v is ...
0
votes
0
answers
39
views
Manipulating a PDE to obtain an ODE by change of coordinates [closed]
I want to solve the wave equation for a scalar field in de Sitter spacetime and I am following this paper which performs this calculation. The wave equation is
$$-\frac{1}{a^2}\left(\partial_0^2 + 2 a ...
- 542
0
votes
0
answers
24
views
Wave equation on 2D semi-infinite plane
I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$\nabla^2 \Psi(x,z,t) - \...
- 21
2
votes
1
answer
81
views
Solving the Schrödinger equation [duplicate]
While solving Schrödinger solution we use separation of variables to separate time dependent and independent parts and then write the final solution as the product of the two solutions.
How can we be ...
1
vote
0
answers
52
views
Spin Connection, Killing Equation and Spinors under Diffeomorphism
Under a diffeomorphism $dx'^{\mu}=\frac{\partial x'^{\mu}}{\partial x^{\nu}}dx^{\nu}$ we have that the components $A_{\mu}$ of every 1-form $A=A_{\mu}dx^{\mu}$ transform as: $A'_{\mu}=\frac{\partial x^...
- 662
0
votes
1
answer
49
views
The reason why separation of variable works when solving laplace's equation in some cases
Given an appropriate situation, for example, a case where there are 2 grounded conductors (infinite sheet charges), one at $y=0$ and one at $y=a$, and a 3rd conductor (at $x=0$) perpendicular to both, ...
- 370
3
votes
0
answers
64
views
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 ...
- 41
0
votes
1
answer
59
views
Reference to understand this branch cut question
I am currently reading a physics paper in which the authors have complexified an ordinary differential equation (ODE). They mention the following statement in the paper:
"These branch points ...
0
votes
0
answers
25
views
Laplace equation for three semi-infinite conducting planes at potential V meet at the right angle to form a cubic corner
Let the planes be $x = 0$, $y = 0$ and $z = 0$ and they all have the same potential $V_0$. the question is to find the potential in the region $x > 0$, $y > 0$ and $z > 0$.
I know that from ...
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0
votes
1
answer
38
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References for harmonic oscillator with memory
I'm reading Neu's "Singular Perturbation in the Physical Sciences" and in problems 1.1 and 1.2 he defines systems that "have memory" as the the variant of the harmonic oscillator
$$...
1
vote
1
answer
49
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Phase-amplitude stochastic differential equations
In the book of $\textit{The Quantum World of Ultra-Cold Atoms and Light: Book 1 Foundations of Quantum Optics}$ by Peter Zoller and Crispin Gardiner on page 75, they derive the phase-amplitude ...
- 33
2
votes
0
answers
43
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Simplifying 2D Navier stokes equation over the top and bottom part of an airfoil - assumptions incompressible, steady, very high viscosity
I am trying to simplify the Navier-Stokes equations with my assumptions, to be able to solve them numerically:
I'm trying to model an airfoil flying through a very viscous fluid at relatively low ...
- 21
0
votes
0
answers
42
views
Finding the Green's function of the Wave Equation (with no time dependence)
Imagine you want to find a potential given the following equation:
$$
\square \phi (t, \mathbf r) = 4\pi \delta^3(\mathbf r) \\ \; \\
\phi(0, \mathbf r)= 0\\ \; \\
\partial_t\phi(0, \mathbf r)= 0
$$
...
- 1,053
4
votes
2
answers
154
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Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?
Many different physics techniques for approximately solving differential equations seem to follow the same basic pattern. One starts with some differential equation $Df(x) = s(x)$ (or $s(x) f(x)$), ...
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0
votes
1
answer
73
views
What is the general solution to Poisson's equation when source extends to infinity?
If the distribution of the source charges does not go to zero at infinity (as in the case of an infinite line charge), can we still write the most general solution of Poisson's equation $$\nabla^2\phi(...
- 10.7k
3
votes
3
answers
522
views
How does Kirchhoff's voltage law relate to the spatial derivative of voltage?
I'm reading this libretexts article on the basics of transmission line theory. In it, they include this circuit diagram as a model of a uniform transmission line:
They then say that applying ...
- 1,680
0
votes
0
answers
16
views
How can I form an advection-diffusion SDE to obtain the desired discretization?
Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
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2
votes
0
answers
45
views
Uniqueness of solutions of Maxwell equations [closed]
I have this exercise on my electromagnetism course :
Consider that there exist two pairs of fields E and B that satisfy Maxwell's equations, with the same boundary conditions and have the same ...
- 21
0
votes
0
answers
44
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Series expansion of a Differential equation form intersection theory
Hi i have the following Differential equation $\nabla_\omega \psi=\varphi$ where $\nabla_\omega(\psi)=d(\psi)+\omega(z)\wedge\psi$
With the local coordinates of $y=z-x_i$ the series expansions.
$$
\...
- 11
0
votes
1
answer
29
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Equations of motion for lever balance [duplicate]
What I am trying to do is to derive the equations of motion of a lever balance like the one in the picture
As can be seen the lever balance has achieved an static balance position, nevertheless it is ...
- 1
6
votes
7
answers
512
views
Is resonance a general property of second-order differential equations?
I have read at this site as an answer at a question about how antennas work but that is not important
The resonant frequency of an antenna is determined by its constitution. Mathematically speaking, ...
0
votes
0
answers
37
views
Neumann boundary condition for Maxwell equations?
For the Poisson equation, we have Dirichlet boundary condition, Neumann boundary condition and Robin boundary condition. But for the time-harmonic Maxwell equation, I have only seen two types of ...
- 1
13
votes
4
answers
2k
views
Why can't we run the laws of physics backwards and forwards in time infinitely?
So assuming we know all the laws of physics in differential equation form, and I have an estimate for the current large scale state of the universe (whatever standard assumptions/data cosmologists use ...
- 1,186
1
vote
1
answer
76
views
Need help finding Hamiltonian for equations of motion
I have the following equation of motion:
$$\ddot \theta+\dot\theta^2\theta+k^2\theta=0.\tag 1$$
This equation is from this question. I wanted to see if I could find a Hamiltonian for this equation but ...
0
votes
1
answer
40
views
Solution of the equation of motion for a free particle with time-varying mass
I cannot understand which kind of solution has this differential equation representing a free particle with time-varying mass:
$\ddot x + \frac{\dot m(t)}{m(t)}\dot x=0$
I would like to find the ...
- 21
4
votes
0
answers
76
views
Does General Relativity satisfy the Homotopy (or “h”) Principle?
By this I mean in the standard second order (whether metric or tetrad/verbein-based) form of General Relativity. I've been reading about the homotopy principle of late (see Eliashberg's introduction ...
- 2,757
1
vote
3
answers
77
views
Why we taking $ a = A \sin \phi$ and $b = A \cos\phi$ in place of constants in the Linear Harmonic Oscillator eq.?
The General Physical Solution of motion of the linear harmonic oscillator, $d^2x/dt^2 + \omega^2 x(t)= 0 $ is $$ x = a \cos \omega t + b \sin \omega t$$ where $a, b$ are two arbitrary real constants. ...
- 11
0
votes
0
answers
45
views
What's the difference between the mass equilibrium equation and the diffusion equation?
Was wondering whether the mass equilibrium ODE and the diffusion equation PDE stem from the same physical concepts (at least when the mass represents Chemical Pollution):
In my own words, the "...
- 113
1
vote
0
answers
54
views
Diffusion from an instantaneous spherical source with a continuous spherical sink
I assume this is a solved problem, but I cannot find the solution in some common sources, so I am asking here. Suppose you have diffusion of a molecule (diffusion coefficient $D$) from an ...
- 11
0
votes
1
answer
179
views
Poisson equation and surface charge distribution
Poisson equation is given by $\nabla^2V=\frac{\rho}{\epsilon_0}$. Here $\rho$ indicates a volumic charge distribution, which is known in the region $\Omega$ where we solve the Poisson equation.
Is it ...
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1
vote
1
answer
84
views
How to set up differential equation for gravitational system?
Consider the following system
Here, two immovable objects with mass $M$ are positioned $2d$ distance apart (in an empty universe). Meanwhile, an object of mass $m$ is placed somewhere above them on ...
- 159
1
vote
0
answers
58
views
Solitons and traveling waves for a Schrödinger type equation
I am a mathematician and not a physicist.
I came across a non--linear PDE whose linear part is a Schrödinger equation (i.e. a dispersive equation) and we know that this equation has a solution for $x\...
- 11
4
votes
0
answers
29
views
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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1
vote
0
answers
33
views
Solution for forced harmonic motion with non-constant frequency
Is there any integral form of the solution for the equation below?
$$
\ddot{y}+\omega^2(t) y = f(t)
$$
where it's basically the equation for forced harmonic motion with non-constant frequency.
If $\...
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0
votes
0
answers
24
views
How to find a modelling function?
I have a list of boundary conditions, inequalities and general facts about a real world scenario involving probability distributions that evolve according to some parameters, but I'm struggling to ...
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2
votes
2
answers
166
views
Closure and Completeness of basis functions
Consider the brief attached discussion on closure and completeness (used, I think, in the physics sense) of basis functions from Zangwill's Modern Electrodynamics.
Zangwill demonstrates that closure ...
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1
vote
1
answer
98
views
Uniqueness of Poisson equation solution with dielectrics
Let's suppose we have an isotropic homogeneous dielectric $D$ (with given relative constant $\epsilon_r$) surrounded by the void.
Inside and outside $D$ we can obviously write the Poisson equations:
$\...
- 107
0
votes
0
answers
70
views
Neumann boundary conditions and interface conditions for electric field
Let's suppose we have an isotropic homogeneous dielectric $D$ in the void, and a charge distribution $\rho(\vec{r})$. We want to find the potential $V(\vec{r})$ in the space. We have then to solve the ...
- 107
2
votes
1
answer
49
views
Existence Of Phase Flow Provided Potential Energy is Positive
I am reading through Arnold's "Mathematical Methods Of Classical Mechanics". In the section 4D on p. 21 concerning Phase Flow there is a question that reads as follows:
Show that if ...
- 47
2
votes
1
answer
71
views
When equations of the three-body problem reduce to 6 order, why it has no closed-form solution? [closed]
I know that Sundman had given a series of power expansion solutions to the three-body problem.
But I also find that the three-body problem doesn't have closed-from solution which means it can't be ...
- 29
0
votes
2
answers
80
views
Schwarzschild Metric - Why only four Geodesic equations considering nine non zero Christoffel symbols?
When employing the Schwarzschild metric, I understand there are nine non-vanishing Christoffel symbols:
$ \Gamma^t_{rt} = -\Gamma^r_{rr} = \frac{r_{\rm s}}{2r(r - r_{\rm s})} \\[3pt]
...
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1
vote
1
answer
139
views
Why the Green's function is zero on the Dirichlet boundary surface?
Given a differential equation
$$
Lu(x) = f(x),
$$
where $L$ is the differential operator, and $f$ is a given function of $x$, the general solution of $u(x)$ is
$$
u(x) = \int G(x, s) f(s)\, ds
$$
$G(x,...
- 237
2
votes
2
answers
201
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Fundamental solutions and initial conditions (for d'Alembert operator)
I would like to understand the plane wave solution for the (3+1-dimensional) d'Alembert operator
$$
\square = \nabla^2 - \frac 1{c^2}\frac{\partial^2}{\partial t^2}\tag{1}
$$
in terms of its ...
- 199
0
votes
0
answers
26
views
Why we need hypersurface to solve initial value problem in general relativity? [duplicate]
Almost all in general relativity book to tackle the initial value problem there we need the concept of hypersurface and another concept like future or past domain of dependence.
My question is what is ...
0
votes
0
answers
18
views
Using repelling and attracting forces, how to achieve a grid/square layout of nodes instead of triangular layout
I am trying to figure out what kind of forces, however artificial, would result in a layout of objects in a grid:
I am trying to use d3 to draw grids - there may be more than one solution. I ...
- 111
0
votes
1
answer
135
views
Stability of Euler-Cromer method
Euler method doesn't perform well in the context of oscillatory problems like the harmonic oscillator; the amplitude of the oscillation gets bigger with time, which clearly contradicts theory as no ...
10
votes
1
answer
763
views
Why is the Yukawa potential equal to Green's function for free space?
I recently saw the computation for Green's function for the Helmholtz equation
$$-\Delta f + \kappa^2 f=0$$
in free space. The computation is done using the Fourier transform, and it turns out that in ...
- 333
1
vote
1
answer
80
views
Boundary conditions of a heat PDE [closed]
I have recently read a paper on heating a metal bar. The heat equation is used to analyze the system but as I can't find the right boundary conditions used to solve the heat equation. I'm hoping that ...
- 225