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-2
votes
1answer
55 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
votes
0answers
31 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 ...
1
vote
1answer
61 views

Recurrence differential equations

We all know recurrence equations like e.q. Fibonacci relation $$F_{n} = F_{n-1} + F_{n+1}$$ In order to find general expression for any $n$, we can use generating function method $$G(x) = ...
2
votes
0answers
23 views

Time responses (position and speed) of system

This is a basic question regarding state space representation and differential equations. I want to find the time response of states $x_{1} = x$ and $x_{2} = \dot{x}$ of the following system: $$ ...
0
votes
1answer
66 views

Accuracy of differential equations [closed]

We use differential equations to model the world around us. For example, the logistic differential equation $$\frac{dx}{dt} = rP\left(1-\frac PK\right)$$ is used to model population. However, it ...
1
vote
3answers
60 views

General solution to the wave equation proving dependence on $x \pm vt$

I am trying to solve for a general solution to the wave function and demonstrate any solution has the form $f(x,t) = f_L (x+vt) + f_R (x-vt)$ I have used separation of variables f(x,t)=X(x)T(t) to ...
0
votes
1answer
92 views

How to solve highly oscillating differential equation [closed]

The equation looks like: $$x''(t)+bx'(t)+c x(t)+dx^3(t)=0.$$ This is the motion of a particle in a potential $cx^2/2+dx^4/4$ with friction force $bx'$. In my case, the friction term is very small and ...
0
votes
1answer
48 views

Modifying differential equations representing a projectile system to account for an arbitrary force

The following series of differential equations represents a projectile's path when solved (g=9.81): Here is some sample output from this system (with initial values x,y=0, v=1500, theta=1.33): I ...
2
votes
1answer
48 views

Could you give boundary conditions to the gravitational potential given the density distribution?

We´re doing a project that's all about solving differential equations with separation of variables. We´re trying to find the gravitational potential given the density distribution (that has azimuthal ...
0
votes
0answers
19 views

Galerkin-type weak formulation for electrokinetics

I am currently working on finite element simulations about electrokinetics. My solver (getdp) accepts directly galerkin-type weak formulation of equations. I am thus trying to write my equations in ...
2
votes
0answers
53 views

Is the diffusion coefficient time-dependent?

It is known that in the partial differential equation: $$u_t=au_{xx} $$ within the limits $0<x<1$ and $a>0$, the diffusion coefficient, arises in the mathematical modelling of a process of ...
1
vote
0answers
32 views

Physical explanation for dependence of initial conditions when solving differential equations using NDSolve

I'm checking some results in this [paper] and I'm currently having some issues with solving a set of differential equations (section 2 and 3.1-3.2 in the paper). I'm finding values to depend on an ...
2
votes
0answers
68 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol ...
3
votes
4answers
187 views

How do I correctly introduce time into this equation?

So, for the past few years it's been my goal to create an equation that would give me the position of an object in a gravitational field at time $t$, given it's initial position and velocity. At first ...
1
vote
1answer
122 views

Schrodinger equation, commutative operators, and Symmetry

When solving Schrodinger's equation in 3D with a spherical laplacian you reach a point at which you introduce a separation constant and can see that the same eigenvalue satisfies the radial and ...
5
votes
2answers
204 views

Poisson equation in 2D and 3D: geometrical reason for the difference

The Poisson equation in 3D shows a fundamental solution in 3D which decays with $\sim 1/r$, whilst in 2D it shows a much different decay $\sim -\ln r$. While in 3D not only the solution, but also its ...
1
vote
0answers
34 views

Laplace equation between circles [closed]

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$ with boundary conditions: $$f(a,0)=g(\theta)$$ $$\lim_{A\rightarrow\infty}f(A,\theta)=1$$ what would be a straightforward way to ...
1
vote
2answers
48 views

What dynamical system could this $\dot y = \alpha(y-\lambda), y\geq \lambda$ equation describe?

Just out of curiosity, can anyone identify electrical, mechanical, chemical, etc process that is governed by a differential equation of the form $$\dot y = \alpha(y-\lambda), y\geq \lambda$$ where ...
0
votes
0answers
100 views

General boundary condition for 1D heat equation

I'm studying from Numerical Solution of Partial Differential Equations by K.W.Morton and D.F.Mayers (Amazon link). I'm confused with general boundary conditions. Could someone give me a clue? For ...
3
votes
1answer
253 views

How to determine sign of coefficients in simple spring, damper, mass system?

For a system of the sort shown below: I have come to realize that I continuously make mistakes when it comes to determining the signs (or specifically the direction of the forces) of the ...
0
votes
1answer
71 views

Concentration distribution in a phase separated mixture. Can't get the correct ODEs and boundary conditions

I wish to compute the equilibrium concentration distribution of a binary mixture that has phase separated. I start with writing the free energy as a functional depending of the concentration. I use ...
5
votes
1answer
187 views

Integrating Factor Solution for Plasma Wave Equation

As part of a derivation in Bernstein '58 [1] a linear first-order (eqn. (9) in the image) appears: But the general solution I would usually take (as appears in Gradshteyn and Ryzhik and checked in ...
1
vote
1answer
54 views

Simulating Phase Space Evolution

I am interested in modeling the time evolution of phase-space $\rho(\vec{q},\vec{p},t)$. I have attempted to use Liouville's theorem $\partial_t\rho=-\sum_{i=1}^{3}(\partial_{q_i}\rho)\dot ...
5
votes
1answer
434 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
0
votes
1answer
37 views

Minimizing a damping constant in order to minimize the amplitude of oscillations

How can I determine the damping coefficient that minimizes the amplitude of vibrations? This is an extension of Coupled ODEs that model a quad rotor \begin{align} ...
1
vote
1answer
120 views

Heat equation with heat radiation and heat transfer

If I want to calculate steady temperature distribution on a one-dimensional stick, and I need to consider both the heat radiation and heat transfer, then my equation will be in the form: $$ ...
2
votes
0answers
46 views

Locus of a moving mass point

Two very small mass particles $m_1$, $m_2$ are connected by a $2l$ long, infinitely soft and inelastic thread without mass. The initial condition of the system before being freely released is as in ...
4
votes
1answer
182 views

field solutions for covariant derivative of vector field constrained to zero

Question: What do the solutions of $\nabla_\mu A^\nu = 0 $ look like? And is it possible for spacetime curvature to somehow restrict the solution to $A^\nu = 0$? Here is my current ...
17
votes
1answer
983 views

Does the heat equation violate causality?

I've ran across the idea that, besides simply writing partial differential equations in covariant form, they need to be hyperbolic with all characteristic speeds less than the speed of light. A ...
1
vote
1answer
174 views

Heat Equation with In-Depth Radiation Exact Solution [closed]

I am looking to solve the heat conduction equation in a semi-infinite solid with in-depth radiation on the domain $-\infty < x < 0$. The governing equation of this problem is: $$\rho c ...
8
votes
3answers
156 views

What physical phenomena are modelled by Chebyshev equation?

What physical phenomena are modeled by Chebyshev equation? The equation is below $$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y ~=~ 0 .$$ I could not find it in Wikipedia or in Google (at ...
3
votes
1answer
109 views

Non-deterministic particle system

This question is in the spirit of Norton's dome, an example of an apparently non-deterministic system in Newtonian mechanics. Under certain restrictions, the Picard–Lindelöf theorem guarantees the ...
1
vote
1answer
268 views

Describe Ising model dynamics in stochastic differential equation or stochastic process

The Ising model is described by the Hamiltonian $$ H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j, $$ and is treated extensively by equilibrium statistical ...
33
votes
6answers
863 views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 ...
2
votes
2answers
172 views

What happens to the position function when an oscillator is overdamped and does not have angular frequency?

My question is simple: What happens to the behavior of the position function, $x(t)$, when an oscillator is overdamped and $\omega$ does not exist? Here's the background on why I'm confused: For an ...
1
vote
1answer
52 views

Modeling wall's behaviour

Sorry if the quesiton is inconvenient, but I judged the physics forum would be the best place to go. My house is divided in two parts by a wall, and there's some tree pushing it, so the wall is about ...
2
votes
1answer
38 views

Simple modelling of seasonal variation of temperature?

I'm really curious about this: What is the simplest (or most simplified) differential equation that accounts for the variations of temperature throughout the year at some point on the northern ...
3
votes
2answers
134 views

Solving differential equations without approximations?

In physics, many problems start with a mathematical relationship of the physical phenomenon at hand, and then, in many occasion, always only leave whatever in the first order to get a nice and ...
3
votes
0answers
86 views

Non-linear Wave Equation - Numerical Methods

Motivation: I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where ...
11
votes
6answers
1k views

Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
2
votes
1answer
144 views

Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting ...
6
votes
2answers
584 views

WHY does the “order” of a differential equation = number of “energy storage” elements in a system?

OK. in all engineering courses there comes a point when they introduce you to systems theory and modeling of systems (for eg. via the impulse response) and then the Laplace transform. The modern ...
3
votes
3answers
242 views

Proper and rigourous derivation of Gaussian beam?

Gaussian beams are known solutions to the Paraxial Wave Equation: $$ \frac{\partial^2 \Psi(x,y,t)}{\partial^2 x} + \frac{\partial^2 \Psi(x,y,t)}{\partial^2 y} = 2ik\frac{\partial ...
4
votes
0answers
178 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
3
votes
1answer
192 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
1answer
305 views

Differential Equations for Block Diagram of Satellite Attitude Control System

I am trying to understand the procedure to setup differential equations from a block diagram. The enclosed example is about the attitude control of a satellite. The ultimate goal is to find a ...
0
votes
2answers
141 views

$R = dV/dI$ for varying temperature

I'm trying to do my prelab for an E&M course, and am asked if, for plotting $V$ vs $I$ with a varying temperature, I should expect a linear slope. I know that both $V$ and $I$ depend on $R$, and ...
5
votes
2answers
370 views

Time-dependent Schrödinger equation with $V=V(x,t)$

I was wondering about the following: If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} ...