Questions tagged [differential-equations]

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How the eigenvalue problem was solved?

In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem I checked the solution's manual, When I tried to integrate it, the answer I got is $$ \psi(x)=Ce^{x^2/2\lambda} $$ Can you ...
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3 answers
112 views

Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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3 votes
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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}...
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31 votes
3 answers
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Why aren't Runge-Kutta methods used for molecular dynamics simulations?

One of the most used schemes for solving ordinary differential equations numerically is the fourth-order Runge-Kutta method. Why isn't it used to integrate the equation of motion of particles in ...
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How differential wave equation is in general? [closed]

How the differential wave equation is in general? While deriving the equation it is assumed that wave maintains constant shape and constant velocity. Then how can it be a general wave equation as the ...
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Equilibrium and constitutive partial and algebraic equations describing stresses and deformation of an axisymmetric elastic thin shell over a hole

I want to design a vacuum table to clamp down a very thin plate and I want to know the stresses and deformations due to the atmospheric pressure. Consider the simplified model below: ...
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2 answers
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Confused about the solution to the pendulum differential equation

So I’ve learned how to derive the exact solution to the pendulum differential equation (in respect to its period), $\ddot{\theta} + \frac{g}{l}\sin\theta=0$, where $g$ is gravitational acceleration ...
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6 votes
3 answers
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What are the differential equations that model a self-propagating gravitational wave in space-time?

Light is a self-propagating wave, but it's very complicated. Imagine, if you will, a wave in space-time that by assumption was self-propagating like light, except that it was a gravitational wave. ...
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1 vote
2 answers
70 views

When setting up 2nd order first derivative approximations in a finite differencing scheme, why are these equations equivalent?

In approximating a first derivative term (assuming $\delta z$ is the distance between two spatial grid points) using a finite differencing scheme I came up with these basic equations: $$\phi \frac{\...
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Approximation for two level system differential equations

I am currently reading the book "the quantum theory of light" link: http://rplab.ru/~as/2000%20-%20R.Loudon%20-%20The%20Quantum%20Theory%20of%20Light%20-%203rd%20ed%20Oxford%20Science%...
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Continuity equation for the estimation of cosmological dimensionless parameters

I want to set up a system of equations to find the dimensionless density parameters $\Omega_i$ as a function of $N=\ln(a)$, from the continuity equation: $$\dot{\rho_i}+3H(p_i+\rho_i)=0$$ where the ...
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4 answers
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What does is really mean to say that a 3-body problem is not solvable? [duplicate]

What does it really mean to say that a three-body problem (the Sun, the earth, and the moon) is not solvable? Why is it not possible to solve the differential equations on a computer with adequate ...
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1 answer
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Unique solutions to divergence equation?

A very common problem in physics is to search for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $$ \nabla \cdot f = g $$ for some given source density $g: \mathbb{R}^n \rightarrow \...
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Metric tensor from hyperbolic PDE

It is clear that when a differential equation is composed of the second partial derivatives only, it could be written in the form $$ g^{\mu\nu} \frac{\partial^2 \psi}{\partial x^\mu \partial x^\nu} = ...
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Analytical solution of the Fokker-Planck equation for a damped harmonic oscillator

I was having some trouble while trying to understand the solution for the Fokker-Planck equation for a damped harmonic oscillator, as given in chapter-3 of the textbook "Statistical Methods for ...
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Spheroidal (elliptical) waves: a boundary problem

I'm trying to solve a boundary problem. I have known EM waves, which "bounce" of a charged, rotating oblate spheroid (ellipsoid). Now the formulation of the boundary problem is not so hard, ...
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Physical Interpretation of Large-Time Decay Estimates of Solutions to Navier-Stokes

It is well known (see for example Hoff-Zumbrun (1995)) that solutions to the compressible Navier-Stokes equation converge in $L^p$ spaces to the heat kernel. Formally, to keep things simple, we can ...
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How to deal with a BiGlobal linear (in)stability analysis (LSA) by spectral method?

I know some basic process of using Chebyshev spectral method to solve a two dimensional Poisson problem. Discritize the compotation domain by Chebyshev-Gauss-Lobatto points: $\pmb{x}=\cos\frac{j\pi}{...
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1 answer
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Kepler problem in cartesian coordinates

I'm trying to solve the Kepler problem in Cartesian coordinates, that is, I want to show that the trajectory is an ellipse using Cartesian coordinates instead of using polar coordinates, as is usually ...
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Analytical solution to Poisson's equation for gravity

I am studying Poisson's equation for gravity. $$\nabla^2 \varphi = 4\pi G\rho$$ I have read that it is solved analytically using some Green's function, to give the well known formula of potential $$ \...
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Obtaining the Green's function for a 2D Poisson equation ( in polar coordinates)

I am trying to solve the following BVP within an annular region of radii $r_1$, and $r_2$ : $$ \begin{cases} \nabla^2u=f\\ u(r_1) = p\\ u(r_2) = q \end{cases} $$ If we define an auxiliary problem in ...
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3 votes
1 answer
47 views

An intuitive reason for the fourth derivative in the beam equation?

The appearance of the second derivative (or Laplacian in higher dimensions) in the diffusion equation ($u_t=u_{xx}$) and the wave equation ($u_{tt}=u_{xx}$) seems intuitive to me. The quantity simply ...
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Is it possible to built a variational principle for this first-order system?

Imagine there is a mechanical system described in unitary units by the equation: $$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$ such it has a finite duration solution: $$x(t) = \frac{...
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2 votes
1 answer
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Second linearly independent solution of Airy Differential equation

The Airy differential equation is $$ \frac{d^2y}{dx^2}=xy. $$ After Fourier transforming the equation, we get $$ y=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\left(kx+\frac{k^3}{3}\right)}dk. $$ Here $k$...
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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{\...
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Numerical solution of the 2D linear partial differential equation of first order

It is an actual geophysical problem where we study the liquid flow. We measure at each 2D grid point and each time interval three components of the liquid velocity and we want to compute how the ...
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7 votes
1 answer
338 views

Can I see separation of variables as a tensor product?

Can I see separation of variables as a tensor product? For example, in a radial potential, the separation of variables brings to the solution $R(r)\Theta(\theta) \Phi (\phi)$. This sounds like an ...
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Relation between equations of the form "Derivative" $f=0$

I'm currently taking an introductory course in QFT, and I've noticed that lots of equations in physics take the form of "Derivative" of a funcition equal 0. Some examples being the wave ...
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Algorithm for solving Poisson's equation numerically

I need an algorithm to solve Poisson's equation for gravitational potential. $$ \nabla^2\phi = 4\pi G\rho $$ where, $\phi$ is Gravitational Potential. I am trying PDE for the first time so, I need ...
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Model of a box hit by a Force or Given an Initial Speed

What would be a good way to model a box in a horizontal plane that is hit with a certain blunt force in order to move it some distance. What would be a good mathematical model given that the box is ...
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0 votes
1 answer
68 views

Simple harmonic oscillator in a rocket which accelerates upward [closed]

I'm working my way through a textbook that deals with differential equations. Here's a problem that I need some help to solve: Suppose a spring with a constant 4.5 kg/sec² is attached to a body with ...
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24 views

Why inverse flow of separable Hamiltonian with even kinetic energy can be written like this?

Why is it true that the inverse of the flow of a separable Hamiltonian with even kinetic energy can be written as $\phi_N \circ \varphi_t \circ \phi_N$ where $\varphi_t$ is the flow of the Hamiltonian ...
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3 votes
1 answer
190 views

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}...
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1 vote
1 answer
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Application of Helmholtz equation

I have the Helmholtz equation $$\nabla^2f = -k^2f $$ I am trying to solve it as a second order differential equation using a numerical method. However, I am unable to find an application of it, other ...
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1 vote
1 answer
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Expression for Steady state of Forced vibration [duplicate]

In my book under the topic Steady state of the forced oscillator, they started with the equation: $$\frac{d^2x}{dt^2}+γ\frac{dx}{dt}+ω_0^2x=fe^{jωt}$$ I know the equation for damped oscillation but it ...
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1 vote
1 answer
85 views

Integrating Hamilton's equations: is there any difference between these two integration methods?

Consider a general set of Hamilton's equations $$ \begin{align} \dot{q}(q, p) &= \partial_p H(q, p) \\ \dot{p}(q, p) &= -\partial_q H(q, p) \end{align} $$ A first-order integrator one could ...
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1 vote
1 answer
41 views

Hand damping of a vibrating string or membrane

Problem is the following: If I have a guitar string or a drum membrane which is vibrating (and thus creating sound), when I place my hand or finger on it looses energy quickly, and eventually ...
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1 vote
1 answer
63 views

Derivative of operator with respect to parameters

From Shankar's QM book pg. 56: For an operator $\theta(\lambda)$ that depends on a parameter $\lambda$ defined by $$\theta(\lambda)=e^{\lambda\Omega}$$ where $\Omega$ is also a constant operator, we ...
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5 votes
0 answers
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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 ...
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1 vote
0 answers
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What areas of research use linear algebra the most? [closed]

I'm in my 4th semester as a physics and recently added a minor in math. I'm in my second linear algebra course, and am finding it extremely interesting and fun to understand. To complete my minor, I ...
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1 vote
1 answer
100 views

Analytic solution to time-dependent Schrödinger equation

Is there a known analytical solution for the following Schrödinger equation $$i \partial_t \psi=-\frac{1}{2}\partial^2_{x} \psi + \psi x.$$
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3 votes
2 answers
110 views

How to obtain Green function for the Helmholtz equation?

all. I am following Jackson's Classical Electrodynamics. At Chapter 6.4, the book introduces how to obtain Green functions for the wave equation and the Helmholtz equation. I have a problem in fully ...
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2 votes
3 answers
191 views

Logistic population equation and exponential model

In Verhulst's model of population growth and control, Let $K$ represent the carrying capacity for a particular organism in a given environment, and let $r$ be a real number that represents the growth ...
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2 votes
2 answers
163 views

Differential equation of a series $RLC$ circuit driven by a DC voltage source?

From math below it seems no oscillations are possible and the steady state reaches instantly. I know this is wrong but I'm new to differential equations and don't see my mistake. Summary: For the ...
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0 votes
0 answers
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Wave equation - three questions

there are three question that I've never thought about the homogeneous wave Equation. For the Cylindrical wave, here it is possible to find the derivation (see paragraph 5.9.3). Is the cylindrical ...
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4 votes
7 answers
191 views

Speed of heat through an object

According to the Heat equation (the PDE), heat can travel infinitely fast, which doesn't seem right to me. So I was wondering, at what speed does heat actually propogate through an object? For example,...
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Inhomogeneous Wave Equation

In some literature that I am consulting this is called inhomogeneous wave: $$\sigma=\sigma_{0}e^{-i\omega t}e^{-ikx}e^{\gamma x}$$ where $\gamma$ is an attenuation parameter. I imagine the term ...
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-1 votes
1 answer
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Intuition for group property of the flow of differential equations

Consider the IVP $$\left\lbrace\begin{aligned}x' &= f(t, x), \\ x(0) &= x_0, \end{aligned}\right.$$ with complete flow $\phi(t, x_0)$. If $f$ is smooth, then the flow has the group property, ...
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4 votes
4 answers
294 views

Question about the Wave equation

I have a question. I was looking for the Wave equation (first Eq. of this wikipedia page). I saw for the first time a version of this equation during an Acoustic course, where we obtained it for the ...
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1 vote
1 answer
48 views

Hamilton-Jacobi Theory: Can we take 2 additive constants?

I am thinking continuously regarding the additive constant in Hamilton-Jacobi theory. But I didn't get a good idea. Why only one additive constant, can we take 2 or 3 additive constants? $$S=S'+\...
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