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Questions tagged [differential-equations]

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

-4
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0answers
17 views

Physical interpretation of a singular pde [on hold]

Can you kindly have a look at the following question attached in the link below: https://math.stackexchange.com/questions/3058490/physical-interpretation-of-a-singular-pde?noredirect=1#...
0
votes
0answers
20 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
46 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
36 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
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0answers
15 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 ...
1
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0answers
10 views

General distributional solution of the Airy Equation [migrated]

How can I prove that the Airy equation $$ \frac{d^2u}{dx^2}-xu = 0 $$ has at least two linear independent solutions?
0
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0answers
10 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
143 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
13 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}...
0
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0answers
39 views

Why are partial derivatives necessary when deriving the equation for a vibrating string? [migrated]

AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation. From purely a physical phenomena to math mapping POV: Why ...
1
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1answer
52 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
51 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
43 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
votes
1answer
31 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
vote
2answers
88 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
votes
4answers
186 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: $$(1-x^2)P''(x)-2xP'(x)+AP(x)$$ $$(...
0
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0answers
25 views

Finding the Disturbance Flow with Faxen's Law

Suppose I want to know the disturbance flow from a sphere in a solution, but I want to use Faxen's Law to find it. In particular, lets say I have a generic background flow: $\mathbf{u}^{\infty}\mathbf{...
1
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0answers
50 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
votes
1answer
58 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
21 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
votes
2answers
170 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
votes
0answers
26 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
votes
3answers
73 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
vote
1answer
29 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
vote
1answer
53 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 ...
1
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0answers
34 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
votes
2answers
55 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
33 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
votes
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 ...
2
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0answers
49 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
votes
1answer
93 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
22 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
52 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
104 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(\...
-1
votes
1answer
23 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 ...
-1
votes
1answer
55 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!
8
votes
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
90 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'...
2
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0answers
65 views

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 ...
2
votes
2answers
49 views

Finding dispersion relations

I was wondering if there is a general (theoretical, not experimental) method for finding the dispersion relation for waves in a medium, say given the equation governing purturbations in the medium? ...
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0answers
41 views

Why the dynamic nonlinear differential equations of plate are said to be of eight order?

In Chia Chuen-yuan's book Nolinear Analysis of Plates, on the begining of section 1.8, the author said that "any system of nonlinear differential equations is of eight order". I can only figure out ...
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1answer
97 views

How to understand Duhamel's principle?

I have difficulty about the explanation of Duhamel's principle on my book. Here is what's written on my book: Take wave equation as an example. Consider the equation: \begin{cases} \frac{\partial^2u}{...
3
votes
2answers
292 views

General solution of Poisson's equation [closed]

How to find general solution of Poisson's equation in electrostatics. $$ \nabla^2V=-\frac{\rho}{\epsilon_0} $$ Where, V = electric potential ρ = charge density around any point εₒ = absolute ...
2
votes
0answers
41 views

Convergence to electrostatic equilibrium in a conductor

I am interested in proving mathematically that a conductor always converges to equilibrium in the surface. We model it as follows: We have an ohmic conductor, which means that it is a $3$-manifold $M ...
0
votes
1answer
57 views

How can I split the Klein-Gordon equation into first order ODEs?

The Klein Gordon eqution in conformal time without perturbations is: I want to solve the equation numerically, but to do so I need to split it into 2 first order differential equations. What would it ...
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0answers
33 views

System of 2 pipes of different sizes with very high pressure

Hello everyone! Let's imagine that we have a closed system of 2 gas-filled pipes, which are connected as it shown on the picture, with next initial conditions: P1(~200 bar), V1, T, D1(diameter) - ...
59
votes
3answers
13k views

Why do I see a saddle in this picture of a computer screen?

I am not entirely sure this is an appropriate question for PSE, however since many of you have such diverse backgrounds, I'll give it a shot. I have noticed that when one takes a picture of a ...
0
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0answers
51 views

Stationary solutions: Fokker-Planck

I've a question about the stationary solutions of the FP equation. I know that for a differential stochastic equation like $$\frac{dx}{dt} = a(x) + \sqrt{2c}\eta $$ the FP equation is: $$\frac{\...
1
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1answer
72 views

What physically determines Bessel functions' orders?

I would like a simple explanation of what, in a physical problem, determines the order of the Bessel functions that describe the solutions. What are dimensional(?) parameters of the system that induce ...
0
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3answers
147 views

Hyperbolic harmonic oscillator

The classical harmonic oscillator can be associated to the differential equation: $$y''+\omega^2y=0$$ and solutions $$y=A\cos(\omega t)+B\sin(\omega t)$$ or $$y=A\cos(\omega t+\delta)$$ The harmonic ...