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2
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2answers
43 views

Interpretation of Eigenvalues and Eigenvectors of an hyperbolic conservation law $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
1
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0answers
54 views

What type of differential equation is this? [closed]

Need to find the general solution and characteristics, but I can't define type of this differential equation $ u_{ttx}=u_{tx}^3 $
3
votes
1answer
49 views

A model for constant temperature of water in a container

I put some water in a container with initial temperature $T_0$ in a room, and the room's initial temperature is $T_a$. Now the container is filled to the maximum, so any more water coming in will ...
0
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0answers
15 views

Is there a “natural” way to interpolate between a set of bound state wave functions?

Consider for example the Coulomb potential, $-Z/r$, for which there exist a set of bound states with energy $\epsilon_n := {-Z^2 \over 2 n^2}$ (in Hartree). If I want the "wavefunctions" for some ...
1
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0answers
26 views

Cooling of a surface due to fluid passing over it

I am working on a project that requires me to measure the cooling effect of a liquid flowing through a surface. In order for me to effectively calculate the cooling effect, the solution of the below ...
3
votes
1answer
75 views

Numerically solving a simple Schrodinger equation with fast Fourier transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
2
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0answers
53 views

Wronskian of complex second order linear differential equation

While studying certain analogue gravity models I came across a differential equation of the form: \begin{align} \frac{d^2y}{dz^2} + \omega^2 (z)~ y(z) = 0 \end{align} where $z$ is a complex variable ...
0
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0answers
21 views

Does nature prefer second order differential equations? [duplicate]

We all know Newton's second law: $m \ddot{x} = F(x)$ or equivalently Euler-Lagrange or Hamilton's equations. In quantum mechanics the Schrödinger equation is also a second order differential equation. ...
5
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0answers
139 views

on fundamental 2D conductivity equation boundary value problem

Consider the following homogeneous boundary value problem for a function/potential $u(x,y)$ on the infinite strip $[-\infty,\infty]\times[0,\pi/4]$ w/positive periodic coefficient/nductivity ...
1
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0answers
44 views

Contradictory argument to continous energy spectrum in quantum mechanics

The Schrodinger equation can be formulated as a regular Sturm-Liouville problem along with proper boundary conditions. Now the solution of regular Sturm Liouville problems with suitable boundary ...
0
votes
2answers
47 views

$Ae^{\mathrm{i}\omega t}$ assumption for oscillating systems (formal & intuitive)

When we obtain a system of ODE's for $n$ masses connected with springs (or otherwise obtained by small amplitudes approximation), the next steps are usually assuming a solution in form $Ae^{i\omega ...
1
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0answers
37 views

couple mode equation [closed]

Consider the nonlinear Schrodinger equation for the normalized wave envelope $\Psi(x,t)$, \begin{equation} i \frac{\partial \Psi}{\partial t}+\frac {1} {2}\frac{\partial^2\Psi}{\partial ...
2
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0answers
41 views

Transform QM radial equation to spherical Bessel equation

I'm currently learning about spherical potentials (ex. hydrogen and hydrogen-like systems) and am trying to work through the problem of a generic spherical potential well such as: $$V(r) = ...
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0answers
45 views

Existence of a solution for geodesic differential equations for a singular metric

In order to determine the geodesics, one must solve the following set of differential equations \begin{align} \frac{d^2 x^j}{ds^2} + {j\brace h\,\,k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0, \end{align} ...
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vote
0answers
14 views

Attractors in Duffing equation

The Duffing equation in its full form is $$\ddot{x} + \delta \dot{x} -ax + \beta x^3 = \gamma \cos(\omega t)$$ Now for specific values of the parameters several attractors exist (or not). Let's ...
2
votes
1answer
44 views

Difference between finite volume, characteristic method and plug flow models of a pipe

I have to model pipes (of a district heating network) with ODE's. My background is computer-science, so it is not that easy for me to understand different approaches. Finite volume approach. Method ...
0
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0answers
29 views

How to solve numerically or analytically this Partial Differential Equation?

:D I'm modeling a problem of ecology with PDEs, So I gotta solve numerically this Reaction-Diffusion Partial Differential Equation $$ \frac{\partial u(t,x,y)}{\partial t}=D\Big( ...
0
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0answers
16 views

IBP Identities to solve differential equation

I am wondering if anyone has experience in using IBP( Integration by parts) identities in the evaluation of Feynman diagrams via differential equations? My question is that I can't seem to understand ...
0
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0answers
44 views

Physical background to the ODE $y'(x) + \frac{1}{x} = y(x)$

Most books on asymptotic methods start with a discussion on the ODE $y'(x) + \frac{1}{x} = y(x)$, which has solution $$y(x) = \int_0^{\infty} \frac{e^{-xt}}{1+t} \,dt. $$ A discussion on the ...
7
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0answers
64 views

Can one classify partial differential equations according to the causality properties of their solutions (and if yes, then how)?

Recently, I bumped into this interesting comment by Valter Moretti which made me wonder about the following, more general question (to which I suspect the answer is affirmative): Can we easily tell, ...
0
votes
1answer
15 views

Solving a first order non linear differential equation in the specific case of the orbit of a particle moving under a central conservative force

Hello it's my first time posting in here and I hope someone in here can help me as my teachers quite dislike questions that aren't specifically in the curriculum. I'm reading About classical mechanics ...
0
votes
1answer
53 views

Differential equation for velocity regarded as a function of distance [closed]

Given a differential equation for velocity, $dv/dt + v = 1$, as well as its solution, is it possible to derive a differential equation for velocity with respect to distance? I found a solution to the ...
2
votes
1answer
138 views

Peskin Schroeder and the general solution to Callan-Symanzik Equation

I have a couple of questions regarding Peskin and Schroeder's derivation of the solution to the Callan-Symanzik equation. First of all, they claim that using ...
3
votes
0answers
64 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
1
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0answers
31 views

Steady-state solution of Fokker-Planck DE

I have this differential equation: $$\frac{\partial f}{\partial t} = \frac{1}{\tau_s v^2} \frac{\partial}{\partial v}(v^3+v_c^3)f + S$$ It is a Fokker-Planck equation that describes collisional ...
0
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0answers
69 views

Lang-Kobayashi rate equation derivation

The Lang-Kobayashi rate equations of a semiconductor laser experiencing feedback are as follows: \begin{align*} \frac{d}{dt}\left(E(t)e^{i\omega t}\right) &= \left[\omega_N(n) + \frac{1}{2}(G(n) ...
2
votes
1answer
58 views

Massless limit to massive scalar in AdS space

I was trying to solve massive scalar wave equation in AdS spacetime (or rather in BTZ). I noticed few funny things : The $m\to 0$ limit to the solution is subtle. One of the two independent ...
0
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0answers
36 views

How can I get the boundary and initial conditions of the convection–diffusion equation consistant?

I want to solve the 1D convection–diffusion equation. The boundary conditions are a flux in from the bottom and a flux out on the top. Furthermore I want no concentration inside at the beginning. I ...
0
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2answers
110 views

Diffusion of gas into vaccum

I'm interested in solving the diffusion equation for gas in vacuum. I have a general question and a more specific questions. What I know: The Diffusion Equation: For density function ...
4
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0answers
93 views

Is there a proof that the number of eigenstates is countable for a bound system?

When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number $k \in \mathbb R $ and $\mathbb R$ isn't countable but if you add a ...
2
votes
1answer
53 views

Wave equations - how to get a real solution from imaginary roots

Im trying to follow the derivation on how to solve Laplace equation used in my fluid dynamics course. We are trying to solve for the velocity potential in potential theory. So far we have this: ...
0
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0answers
94 views

Spring-mass system with variable stifness $m \ddot{x}+k(x)x=0$, time period is known, stifness is unknown

This question is somewhat related to my previous question: What is the time period of an oscillator with varying spring constant? In that question, time period of mass-spring system with variable ...
0
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0answers
38 views

What is the scale factor of a hyperbolic universe?

I wanted to derive the solution to this question from the Friedmann equations myself but I ran into some trouble. I was working in natural units where $c=G=1$, then, for brevity, I defined ...
5
votes
1answer
181 views

Why is the wave equation so pervasive?

The homogenous wave equation can be expressed in covariant form as $$ \Box^2 \varphi = 0 $$ where $\Box^2$ is the D'Alembert operator and $\varphi$ is some physical field. The acoustic wave ...
2
votes
1answer
81 views

When to use separation of variables in E&M? [closed]

I'd really like to know if there is a fast way to recognize if separation of variables is the most appropriate way to go about solving a problem. Are there any kind of guidelines for when to use ...
6
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2answers
133 views

What is Method of Characteristics?

I am a final year student of BS Mechanical Engineering and method of characteristics is not a part of our curriculum. In-fact I heard of it first time after finally picking my FYP. My final year ...
0
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0answers
109 views

1D drift-diffusion equation with single absorbing boundary

If we have just the simple diffusion equation (in 1D): $$ \frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{\partial x^2} $$ with an absorbing boundary at x=0 and initial condition ...
0
votes
2answers
47 views

Generalised velocities enough to be deterministic in Lagrangian mechanics?

In classical determinism we need to know $2n$ quantities of our system and the equation of motion to predict it's future. In Lagrangian mechanics this is equivalent to knowing $q$ and $\dot q$, the ...
0
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0answers
54 views

Example of a physically motivated jerk equation

When I say jerk equation, I mean a differential equation of the form: $\frac{d^{3}x}{dt^{3}} = f\big( x(t), \frac{dx}{dt}, \frac{d^{2}x}{dt^{2}} \big) $ I am doing some work in dynamical systems, ...
0
votes
1answer
53 views

Help recognizing partial differential equation

I would be very grateful if someone could tell me something about the following partial differential equation: $$ \frac{\partial U}{\partial t} = K * (\frac{\partial^2 U}{\partial r^2} + ...
0
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0answers
27 views

Initial conditions for second order ODE with complex stiffness

I tried this on Math Stack Exchange. I'm trying to find initial conditions to ensure systems of the form stay bounded $$\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}.$$ For ...
0
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0answers
38 views

Derivation of the Stuart Landau time dependent amplitude time evolution equation for Hopf or Pitchfork bifurcations

I am studying the Hopf / Pitchfork bifurcations, the ordinary differential equation of which is: $$\dot x = x \ (\rho - x^2)$$ Which is a cubic order equation which describes the time evolution of ...
2
votes
0answers
70 views

Non-Linear O.D.E

I have reached a set of ODE as \begin{align} &\ddot{\vec{a}}(t)+\omega_0^2\frac{\cos{(b(t))}\sin{(a(t))}}{a(t)}\vec{a}(t)=0\\ &\ddot{b}(t)+\omega_0^2\cos{(a(t))}\sin{(b(t))}=0 \end{align} ...
0
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0answers
50 views

Energy Oscillations in a One Dimensional Crystal?

Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially interested in ...
0
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0answers
36 views

What discrete form of the wave equation do you need to use to make a wave simulation?

I'm working my way through these blog posts about the wave equation. All has made sense up until now. The wave equation is $$ \frac{\partial^2h}{\partial t^2} = c^2 \frac{\partial^2h}{\partial x^2} ...
0
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1answer
37 views

What is meant by “method of approximate numerical method” or “method of digital computer” for solving the differential equation of resistive force?

I was reading "motion against resistive forces" in Newtonian Mechanics by A.P. French; here is the excerpt: [...] In general, the resistive force $\mathbf{R}$ is is a function of speed, so that ...
-2
votes
1answer
54 views

Do delay differential equations (DDEs) ever describe real-world phenomena? [closed]

I've recently become interested in DDEs, but I don't know much about them. A DDE has the form $$\begin{align*}\dot{x}(t) = f(t, x(t - \tau)) && \tau > 0\end{align*}$$ My understanding ...
3
votes
2answers
99 views

Numerical modelling of a step function in time in a hydrodynamic system. (Runge Kutta fourth order)

So I'm trying to model a hydrodynamic system that introduces a sudden "jump" in the value of a function at a specific time. The system is solved with a Runge-Kutta fourth order method. I have a ...
2
votes
1answer
110 views

Causality and natural modeling of physical systems using integral forms [closed]

I posed a closely related question here but it received a tumbleweeds award. So I thought I would post it from a different angle to see if I can illicit at least some thoughtful comments if not ...
0
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0answers
30 views

What are maximally dissipative boundary conditions?

I ran into this term when reading about the initial boundary value problem in general relativity. They seem to be relevant when you need to impose boundary conditions on a timelike boundary, for ...