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Questions tagged [differential-equations]

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

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1answer
51 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
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0answers
69 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
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1answer
122 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{\...
-1
votes
1answer
51 views

Diffusion equation in sphere with boundary conditions

I have the diffusion equation in a sphere of radius R given by: $\frac{\partial U(r,t)}{\partial t} = D[\frac{\partial^2 U(r,t)}{\partial r^2} + \frac{2}{r}\frac{\partial U(r,t)}{\partial r}]$ $D$ ...
0
votes
2answers
46 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
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1answer
29 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
72 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(\...
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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 ...
7
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1answer
189 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
110 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
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2answers
107 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
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1answer
32 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
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3answers
66 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}...
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2answers
57 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 ...
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0answers
13 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 $n_k$ ...
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0answers
24 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
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1answer
68 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
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1answer
42 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_{...
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0answers
18 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
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0answers
12 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
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2answers
157 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 ...
0
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0answers
14 views

Propagating sound waves on a 3-D semi bounded domain

So in my course notes, we are investigating a sound wave that is propagating through a rigid cylinder with radius $R$. The wave equation is given in cylindrical coordinates as $\dfrac{\partial^2 \Phi}...
1
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1answer
65 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
3
votes
1answer
56 views

Exact form of the damped wave equation

The undamped wave equation has the standard form \begin{equation*} \frac{\partial^2 \psi}{\partial t^2}=c^2\nabla^2\psi \end{equation*} while the damped wave equation is frequenly found written in ...
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0answers
44 views

General question about making differential equations dimensionless

Suppose you have a set of differential equations that you wish to normalize/make dimensionless. From what I've seen, you can usually use dimensional analysis to figure out a good choice of constants ...
0
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1answer
33 views

Application of linear constant coefficients ODE of the second order [closed]

I've asked this question in math forum. Apparently this question is not welcomed there. So maybe here I can get a proper response. Consider ODE in the form of $$y''+ay'+by=f(t)$$ where $a$ and $b$ are ...
1
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2answers
131 views

Can I apply the standard Runge Kutta 4th order method to the Langevin Equation?

If I have a Langevin Equation with an external force term (which may be time dependent), is it possible for me to apply the standard 4th order Runge Kutta algortihm to solve it numerically? Edit: I ...
5
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4answers
198 views

What are the boundary conditions for the Hydrogen Atom which cause the polar power series to need to terminate?

I am trying to solve the Hydrogen Atom, and I am stuck in the polar equation. To simplify, I have taken the special case in which $m=0$ to get the Legendre Equation: $$(1-x^2)P''(x)-2xP'(x)+AP(x)$$ $$(...
1
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0answers
55 views

Does the electrodynamics-like PDE $\epsilon^{ijk}\partial_j B_{kl}(x) = \delta^i_l\delta^{(3)}(x)$ have solutions?

Consider the following PDE in 3 dimensions $$ \epsilon^{ijk}\partial_j B_{kl}(x) = \delta^i_l\delta^{(3)}(x)$$ Does $B_{kl}(x)$ have a solution? (It can have any kind of singularity, e.g. it can have ...
2
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1answer
68 views

Formal Connection Between Symmetry and Gauss's Law

In the standard undergraduate treatment of E&M, Gauss's Law is loosely stated as "the electric flux through a closed surface is proportional to the enclosed charge". Equivalently, in differential ...
1
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0answers
22 views

diff equation for object sinking in water [closed]

I am trying to solve this problem but making a mistake somewhere. The problem is the following. An object weighing W =1000 lb sinks in water starting from rest. Two forces act on it, a buoyant ...
4
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2answers
171 views

Are characteristics the only solution to the advection equation in 1+1D?

I'm currently reading about fluid dynamics and the Riemann problem, and a very commonly used equation to introduce the topic is the 1+1D advection equation with constant coefficient $v$: $$ \frac{\...
2
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0answers
28 views

Is there a multiscale analysis for field theories?

Consider a (zero dimensional) Gaussian field theory described by the dynamical action $$S = \int_t \tilde{\phi}(t) \left[\partial_t \phi(t) + M(t) \phi(t)\right] - \gamma \tilde{\phi}(t)^2\, .$$ $\...
2
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3answers
75 views

Oscillator with decaying restoring force

Suppose a system that is described by the equation of motion: $$ \ddot{x} = -k\cdot x\cdot \exp\left(-\frac{t^2}{2\sigma^2}\right). $$ (For example a spring with decaying stiffness.) I'd like to ...
1
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1answer
33 views

Additive constant in Hamilton-Jacobi theory?

In Hamilton-Jacobi theory Hamilton's principal function S is a function of n+1 constants , But we take one of the n+1 constants as an additive constant . I don't get this step?
1
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1answer
61 views

Why is the Laplace Transform essentially never used when dealing with problems involving resonance?

Both the Laplace Transform and the Fourier Transform can be applied to a PDE, for example the wave equation, and used to derive a solution to the equation. But I never see the Laplace Transform used ...
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0answers
41 views

Solving Laplace equation (polar coordinates) with Dirichlet boundary condition $T(\rho=a,\varphi)=\phi_0 + \phi_1 \cos(\rho \varphi)$

Problem: Solve Laplace's equation for an infinitely long cylinder with radius $a$ with the Dirichlet boundary condition $T(\rho=a,\varphi)=\phi_0 + \phi_1 \cos(\rho \varphi)$ My approach: Problem ...
2
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2answers
62 views

What is the most useful to learn out of complex analysis and differential equations for undergraduate studies in physics? [closed]

Next year I'm planning to start on my bachelor's in physics, however, I have already started taking some undergraduate courses in mathematics and next semester I will have to choose between complex ...
1
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1answer
39 views

Numerically Modeling Coupled Oscillators Point Masses

I seek to model the motion of two coupled oscillating point masses as shown below: Note that x1(t) models the leftmost point mass and x2(t) is the motion of the rightmost point mass. I would like to ...
3
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2answers
74 views

What is the difference between solutions to 2nd order homogeneous ODE?

I’m studying Vibrations, and we have two forms to the 2nd order homogeneous ODE: $$mx ̈+kx ̇=0$$ $$x(t)=C_1 e^{iw_n t}+C_2 e^{-iw_n t}$$ and $$x(t)=A\cos(w_n t)+B\sin(w_n t)$$ Even though I can use ...
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0answers
56 views

Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
2
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1answer
107 views

Is *every* planar/2D system integrable?

Consider the generic following planar/2D system: $$\begin{cases} \frac{dx}{dt} = A(x,y)\\ \\ \frac{dy}{dt} = B(x,y), \end{cases}$$ where $A,B$ are two functions. Reading Classical Mechanics by ...
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0answers
25 views

Conjugate points in spacetime [Proposition 4.4.2 in Hawking & Ellis]

In Hawking & Ellis The Large Scale Structure of Space-time page 98, they proved a proposition 4.4.2 which is show in the attached figures and . I do not understand the statement underlined in ...
0
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1answer
55 views

About the quadratures method

in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75 it says: "Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures..."...
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1answer
120 views

How to solve the wave equation? [closed]

The solution of classical wave equation $$\nabla^2\textbf{A}=\frac{1}{c^2}\frac{\partial^2\textbf{A}}{\partial t^2}$$ is $$\textbf{A}(r,t)=\textbf{A}_0(e^{i(\textbf{k}\cdot\textbf{r}-wt)}+e^{-i(\...
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1answer
25 views

Propagation speed of solutions to sourced wave equation

The unsourced wave equation in one dimension reads \begin{equation} u_{tt} \left(x,t\right) = c^2 u_{xx} \left(x,t\right) . \end{equation} (Here, subscripts indicate derivatives with respect to the ...
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1answer
67 views

Help with 2nd Order Coupled ODEs [closed]

I am trying to solve a 2nd Order Coupled ODE for A "Looping Pendulum" numerically with MATLAB. Can anyone please recommend how? (which ODE solver to use) Thanks!
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14answers
2k views

How can the solutions to equations of motion be unique if it seems the same state can be arrived at through different histories?

Let's assume we have a container, a jar, a can or whatever, which has a hole at its end. If there were water inside, via a differential equation we could calculate the time by which the container is ...
1
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2answers
114 views

Derivation of wave equation and Newton's 2nd Law

I am a mathematics student studying the derivation of the wave equation. In the derivations I have been reading, Newton's 2nd law is applied to a string, I'm having trouble making sense of it. Newton'...