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Questions tagged [differential-equations]

DO NOT USE THIS TAG just because the question contains a differential equation!

3
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0answers
23 views

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 ...
2
votes
0answers
50 views

Number of e-folds in phase space for the inflaton field

In José Gálvez's paper "Observational constraints on Constant Roll Inflation" it's said that, using the equation of motion for the inflaton field \begin{equation} \ddot{\phi}+3H\dot{\phi}+\frac{\text{...
3
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0answers
36 views

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 ...
0
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0answers
12 views

Splitting coupled ODEs of complex variable into phase and amplitude ODEs

I have derived a system of two coupled ODEs of the following form: $\frac{d}{dt}z_1 = - Az_2 + B|z_1|^2z_1$ $\frac{d}{dt}z_2 = - Az_1 + B|z_2|^2z_2$ where $A,B \in \mathbb{R}$ and $z_{1,2}(t) \in \...
1
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0answers
48 views

What numerical methods are being used here to solve the differential equation?

I am having trouble following and understanding the numerical approach to solving for $h(r,t)$ in the following text: How is the integral force balance (10) used to calculate the new pressure ...
0
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0answers
19 views

Elastic wave in isotropic medium

The 2D elastic wave equation is given by: $$ \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= C_{11} \frac{\partial^2 u}{\partial x^2} + C_{44} \frac{\partial^2 u}{\partial y^2} + (C_{12} ...
1
vote
1answer
47 views

Why are all solutions to this system of pendulum differential equations a linear combination of the two given solutions?

I am currently trying to do a lab report for a coupled pendulums experiment in which we find the following linear system of second order differential equations (describing the position as a function ...
2
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0answers
20 views

Physics of a catenary with sinusoidal motion at one end

If we have a catenary line with a mass per unit length of m with one end fixed and the other end moving in sinusoidal motion like $A.sin(\omega.t)$, how would I ...
1
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0answers
41 views

Fourier transform of the wave equation

In string theory, as the one dimensional string propagates in time, it sweeps out a two-dimensional surface known as the string worldsheet. The spacetime coordinates are taken to be functions $X = X(x,...
0
votes
1answer
12 views

Can't find coefficient for my IVP

In my problem I have to set up an IVP and model freefall with air resistance before the bungee starts being pulled on. Beta being my airresistance coefficient. I have: $$ mx'' + \beta x' = f(t) = 0$$ ...
0
votes
3answers
92 views

In the Schrodinger Equation for the Hydrogen Atom does $\frac{{\partial}f(x)}{{\partial}x}$ equal $f(x)\frac{\partial}{{\partial}x}$? [closed]

I was looking at the Schrodinger Equation for the Hydrogen Atom, and saw it in the form $$\left(-\frac{\hbar^2}{2{\mu}r^2}\left(\frac{\partial}{{\partial}r}\left(r^2\frac{\partial}{{\partial}r}\right)+...
3
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0answers
73 views

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?...
0
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0answers
19 views

Rigid body in a vector field

If I let a particle with coordinates $x(t)$ move in a vector field $F$ the equation I have to solve is $x'(t) = F(x(t))$, right? But if, instead of a particle, I have a rigid body, which equation I ...
0
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0answers
16 views

Bessel differential equation from integral [migrated]

It is a relatively well-known fact that $$\int_{0}^{2\pi}e^{-ikr\cos\theta}d\theta=2\pi J_{0}(kr),$$ where $J_{0}$ is the Bessel function of the first kind and order zero. I'm trying to show that this ...
1
vote
0answers
99 views

Transient solution system of differential equations obtained from master equation

I have to solve the following equation (or at least obtain an approximate estimate) for the diagonal terms of the density matrix. We consider that the initial state is a coherent state $\rho_{n,n}(0)=...
0
votes
1answer
28 views

Is it possible to have discontinuities in the phase portrait of a dynamical system? If yes what does it really mean?

I've been using Mathematica to draw the phase portrait of a system and I got some jumps along the trajectory. I have a deviation term which might be the reason of this but is it possible to have them ...
1
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1answer
52 views

Uniqueness Theorem and the 1D Infinite Square Well

Consider the 1D infinite square well problem: $$\frac{d^2\psi (x)}{dx^2} = -k^2\psi (x)\tag{1}$$ along with the boundary conditions $\psi (0) = \psi (L) = 0$. This seems to be a well posed problem ...
-1
votes
0answers
17 views

Method to solve 2nd order system of coupled ODEs in MATLAB ode45?

I have the equations of motion from Lagrangian analysis of the double pendulum, but I don't know how to solve them analytically using MATLAB's ode45. Does anyone know the method for this?
0
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0answers
17 views

How to evaluate the Feynman gauge photon propogator in coordinate space?

In coordinate space the Feynman gauge looks like: $$\frac{g_{\mu\nu}}{\partial^2} .$$ Without fourier transforming, how does this operator act on currents? In general for inverse derivative operators ...
0
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0answers
17 views

Compression of air layer by parallel planes

When object is falling and close to the ground "liquid pillow" appears under it. Object is of cylindrical form. The differential equation that describes the motion is: $$m\frac{dV}{dt} = mg - \frac{...
0
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0answers
27 views

Boundary conditions of spun string

Problem: Consider a string with mass per unit length $\rho$ and length $L$. It is spun about one end, with angular velocity $\omega$ , such that the motion is in a plane (we neglect gravity). Let $x$...
0
votes
1answer
23 views

How to verify generally that the formula for electric potential is a solution to Poisson's equation?

$$V(\vec{a})=\frac{1}{4 \pi \epsilon_0}\int_\tau \frac{\rho(\vec{r})}{l}d\tau$$ This is the formula for the potential at a general point $\vec{a}$. Note that $l$ in this formula is the magnitude of ...
2
votes
0answers
20 views

Is there a physical circumstance where the flow of a system changes from a nodal source/sink to a saddle?

Phase portraits for sinks can often like eddies in water, so I would assume there is some fluid or aerodynamic system where these curves appear. But, the only difference between a saddle and a nodal ...
0
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0answers
45 views

Why does it matter that the propagator is related to the Green's function for the Schrodinger equation?

If $L = i \hbar \hat{H} - \dfrac{d}{dt}$, then $ L \psi(x,t) = 0$ is the Schrodinger equation. It is well known that we can solve the Schrodinger equation with initial condition $\psi(x,0) = f(x)$ ...
0
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0answers
37 views

Transient wave equation - solving second order differential equation

I have a second order differential equation to solve - however I am struggling to know how best to do this. It is to resolve for head pressure of a transient wave in a pipe (H). Here is what I have ...
2
votes
2answers
64 views

Flow around a rock in a river: which differential equation?

I'm a canoist, so I know that when I go with my kayak behind a rock in a river, I feel a current that is opposite to the river current. I'm also I student mathematician, so I would like to see this ...
3
votes
1answer
69 views

Diffusion equation with time-dependent boundary condition

I was trying to solve this 1D diffusion problem \begin{equation} \dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi} \end{equation} with ...
0
votes
3answers
51 views

Inserting an arbitrary phase in the equation for driven damped oscillations

In Classical Mechanics by Taylor, we find the solution to the differential equation of a damped oscillator with a sinusoidal driving force: $$\ddot{x} + 2\beta\dot{x} + \omega_0^2x = f_0\cos\left(\...
-2
votes
1answer
39 views

How to get the constants in the link I provided? [closed]

http://www.scholarpedia.org/article/Ostrogradsky%27s_theorem_on_Hamiltonian_instability Can someone explain how he got the frequency (equation 15) and 4 constants (right after equation 15)?
0
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1answer
90 views

How are the differential forms for Maxwell's Equations used?

I am currently reading up on Maxwell's Equations (specifically Ampere's Circuital Law- with Maxwell's Addition) for a presentation on differential equations. I chose the topic ignorant of how the ...
1
vote
0answers
75 views

Does the “O” in this google doodle for Olga Ladyzhenskaya have anything to do with her work, or is it completely fanciful? [closed]

Google Doodle celebrates mathematician Olga Ladyzhenskaya She was famous for fluid dynamics and partial differential equations, both of which are beyond my pay grade. And she worked on the Navier-...
1
vote
1answer
131 views

How is the 1D transient heat conduction equation derived? [closed]

From my book: $$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$ with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\...
0
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2answers
48 views

Question about vector field on a manifold [closed]

Arnold defined a vector field on a manifold M is a map from M to the tangent space of M (which has all derivations, roughly). In his ODE book, he talks about $\dot{x}(t) = v(x(t))$ for a vector field ...
1
vote
1answer
33 views

Trouble following a chapter on harmonic oscillators (classical mechanics 5th edition)

I'm following Classical Mechanics, 5th Edition by Tom W.B. Kibble and Frank H. Berkshire. I'm following it since I'm interested in studying physics (although, am doing it at home myself). I've worked ...
0
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0answers
98 views

Solving the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space

How to solve the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space (e.g. $d$ dimensional sphere or hyperboloid)? I was thinking in the following line : I know how to solve $\square f(\...
0
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0answers
13 views

An energy/model based controller to minimize overshoot and response time of a mass spring system

Consider a very simple system: $$m a + k x = F \, , \tag{1}$$ where $m$ is mass, $a = \ddot{x}$ is acceleration, $k$ is the spring's elasticity, $x$ is position of the mass and $F$ is the force from ...
11
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4answers
2k views

Why can all solutions to the simple harmonic motion equation be written in terms of sines and cosines?

The defining property of SHM (simple harmonic motion) is that the force experienced at any value of displacement from the mean position is directly proportional to it and is directed towards the mean ...
9
votes
1answer
200 views

Motivating classical wave equation PDE

I'm teaching a geometry course covering spectral problems, using eigenvalues of the Laplace operator for shape analysis ("Can you hear the shape of a drum?"). I thought I'd cover where the wave ...
0
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2answers
115 views

Symmetries of a differential equation, its solutions and hydrogen atom

A symmetry of a differential equation need not be shared by its solutions. However, under that symmetry, the one solution goes to another. For example, consider the time-independent Schrodinger ...
1
vote
2answers
123 views

Show Solution to Hamilton's Equations are Given by Circular Paths

I am asked to compute Hamilton's equations and check that the solutions to said equations are circular paths, centered at the origin, with angular velocity $\xi \in R^3$. The Hamiltonian is given by $...
0
votes
1answer
33 views

Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
-1
votes
3answers
70 views

Solving ODE equation for classical field [closed]

I would like to solve the following homogeneous, ODE: $$\left[\frac{d^2}{dt^2} + m^2\right]\phi(t) + \frac{1}{6}\lambda \phi^3(t)=0.$$ I know the solution is $$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
-1
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2answers
64 views

Differential Equations for Physicists

I find differential equations in physics to be quite challenging so I'm looking for a book to help me master them. I'm familiar with solving ordinary differential equations via seperation of ...
0
votes
0answers
16 views

Is Smoluchowski equation for aggregation conservative?

The Smoluchowski equation for the aggregation with no breakage can be written as $$\frac{dn_k}{dt}=\frac{1}{2}\Big(\sum_{i+j=k} n_i \beta_{ij}n_j\Big) - n_k \sum_{i=1}^N \beta_{ki} n_i$$ where $n_k$ ...
0
votes
0answers
28 views

Converting this navier stokes solution into a incompressible solution?

I have an equation for a non-viscous compressible fluid with density, pressure and velocity given by: $$ \begin{align} \rho(x, y, z) &= \frac{3B}{a^2 + x^2 + y^2 + z^2} \\ p(x, y, z) &= \frac{...
0
votes
1answer
96 views

How can the analytical solution of the diffusion equation be used for a series of $N$ positions?

Given the exact solution to the diffusion equation: $$C(x,t) = \frac{1}{\sqrt{4 \pi D t}} \exp\left[-\frac{x^2}{4 D t}\right]$$ I am unsure as how it can be applied to a 1D series, as this equation ...
-1
votes
1answer
45 views

Non-linear optics - solve differential equations coupled with the finite difference method [closed]

I have these three differential equations in which I need to solve numerically: $$ \frac{dn_0}{dt}= -n_0(t)W_{01}(t) + n_1(t)K_{10} $$ $$ \frac{dn_1}{dt}= -n_1(t)W_{12}(t) - n_1(t)K_{10} + n_2(t)K_{...
0
votes
0answers
24 views

Computational solution of exponential decaying wavefunction tail

As I am going through some (quite simple) computational physics exercices I have a question concerning one exercise that involves solving the radial Schrodinger equation. This is done with the ...
0
votes
0answers
15 views

Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
2
votes
2answers
164 views

The mean of Langevin equation

I have a very basic question regarding the mean of the Langevin equation. So we have an equation of the form: $$\dot{v}(t)=-\beta v(t)+ \xi (t)$$ Where $\xi (t)$ is a Gaussian white noise with an ...