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26
votes
6answers
606 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 ...
17
votes
5answers
1k views

Any physical example of an “explosive” differential equation $ y' = ky^2$?

I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends ...
16
votes
6answers
724 views

Applications of delay differential equations

Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional ...
16
votes
5answers
754 views

Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, ...
13
votes
3answers
332 views

Chemical reaction as state transition?

When considering diffusion of chemicals, the reaction part is business of chemical kinetics, where the relevant characteristics of different substances come from collision theory together with some ...
12
votes
2answers
701 views

First-order wave equation: Why is its presence not common?

The (one-dimensional) wave equation is the second-order linear partial differential equation $$\frac{\partial^2 f}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\tag{second order PDE}$$ ...
12
votes
2answers
81 views

Numerical Analysis of Elliptic PDEs

I am looking for an elementary reference regarding issues of stability in numerical analysis of non-linear elliptic PDEs, particularly using the finite difference method (but something more ...
10
votes
4answers
2k views

What is the physical meaning/concept behind Legendre polynomials?

In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential ...
9
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 ...
8
votes
1answer
461 views

Modification of Newton's Law of Cooling

Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I ...
7
votes
1answer
297 views

Black & Scholes and the Quantum Mechanics

I am interested in the link between the Black & Scholes equation and quantum mechanics. I start from the Black & Scholes PDE $$ \frac{\partial C}{\partial t} = -\frac{1}{2}\sigma^2 S^2 ...
6
votes
1answer
389 views

Diffeomorphisms and boundary conditions

I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism which preserve the boundary conditions in the same paper. I found this ...
6
votes
2answers
158 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 ...
6
votes
2answers
333 views

What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
5
votes
2answers
283 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} ...
5
votes
1answer
250 views

What formulas should I use to realistically model the diffusion of a drop of ink in a water?

I am a mathematician and am originally from the math side of stackexchange. I want to model the behaviour of a drop of ink diffusing in water. I dont want to simply use the diffusion equation ...
5
votes
1answer
120 views

Is the mathematical form of the acoustic diffusion equation present in other fields of physics?

We are working in the field of High Performance Computing and we have developed a very efficient parallel implementation for solving the Acoustic Diffusion Equation as described below: $$ ...
5
votes
1answer
1k views

What type of PDE are Navier-Stokes equations, and Schrödinger equation?

What type of PDE are Navier-Stokes equations, and Schrödinger equation? I mean, are they parabolic, hyperbolic, elliptic PDEs?
4
votes
3answers
258 views

Heat equation with strange boundary condition

I've attempted a few different solutions to this math methods problem from an old qualifying exam, but I can't seem to hack it. The setup for the problem is that the temperature sand in the Australian ...
4
votes
1answer
240 views

Solving the differential equation of a beam under moving load using green functions

i started working on this paper and i didnt understand one part of it , the problem is : Solve this equation using green functions : $$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu ...
4
votes
2answers
158 views

What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?

The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d ...
4
votes
1answer
110 views

What exactly is the meaning of weak formulations and what is its purpose?

What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it ...
4
votes
1answer
184 views

Research problems in application of Lie groups to differential equations [closed]

Are there any open problems in physics involving Lie groups and differential equations for a phd theses. Some applications are say, Noether's theorem in classical or quantum field theory. But I am ...
3
votes
3answers
73 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 ...
3
votes
3answers
95 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
3answers
371 views

Bessel vs. modified Bessel in radial equation of hydrogen

I am trying to understand the difference between Bessel functions and modified Bessel functions (simply googling is yielding complicated, non-intuitive answers). I was under the impression that one ...
3
votes
1answer
52 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 ...
3
votes
2answers
305 views

Does air resistance ever slow a particle down to zero velocity?

If a particle moves in a place with air resistance (but no other forces), will it ever reach a zero velocity in finite time? The air resistance is proportional to some power of velocity - $v^\alpha$, ...
3
votes
2answers
355 views

Boundary conditions for crystals

As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example: $\psi(x)=\psi(x+L)$ where $L$ is the length of the 1D crystal. My question is: ...
3
votes
0answers
45 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 ...
3
votes
0answers
76 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
68 views

solution of pendulum equation [closed]

I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: ...
3
votes
0answers
88 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving Schrodinger equation, such as tasks about particle in Morse potential, Poschl-Teller potential and many others, we usually find an approximations (lets call them as ...
2
votes
3answers
283 views

How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$

How do I integrate the following? $$\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$$ where $C$ is a constant. I'm supposed to get a Gaussian function out of the above by integrating but don't ...
2
votes
2answers
80 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 ...
2
votes
3answers
306 views

Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms?

I was always wondering about the acausal nature of solutions obtained by Fourier transforms in the case of inhomogeneous equations. The solution usually revolves around the integration of the ...
2
votes
2answers
329 views

What does it mean to “solve an equation”?

I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and what are we solving for? Of course, we can ...
2
votes
2answers
1k views

Greens function in EM with boundary conditions confusion

So I thought I was understanding Green's functions, but now I am unsure. I'll start by explaining (briefly) what I think I know then ask the question. Background Greens are a way of solving ...
2
votes
4answers
277 views

Why is linear independence of harmonic oscillator solutions important?

The equation of motion for the harmonic oscillator (mass on spring model) $$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$ with $\omega_0^2 = D/m$, $D$ and $m$ being the force constant of the spring and the ...
2
votes
2answers
71 views

How do you integrate an expression over a variable in the limit of an integral?

I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: ...
2
votes
1answer
82 views

Solving 2nd-order ODEs

I was reading this PDF REF per request: ...
2
votes
4answers
147 views

Suggestions of a non-linear example for a small research project on numerical solution of ODEs?

I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project ...
2
votes
1answer
90 views

Solution of motion in hamiltonian formalism

I have these canonical equations: $$\dot p = - \alpha pq$$ $$ \dot q =\frac{1}{2} \alpha q^2$$ I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$. I thought to simply ...
2
votes
2answers
201 views

Stick and slip motion: mass and spring inside a box model

I am trying to determine a set of differential equation which can describe the motion of a mechanical system as below. Here, at the bottom we have a plate, and a box on top of it. Inside the box, ...
2
votes
1answer
131 views

Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
2
votes
1answer
272 views

What does it mean that Einstein's equations are hyperbolic-elliptical?

I says on Wolfram MathWorld that Einstein's field equations are a set of "16 coupled hyperbolic-elliptic nonlinear partial differential equations". What does it mean that the equations are ...
2
votes
1answer
77 views

product solutions for PDEs, physical motivation

Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by ...
2
votes
2answers
86 views

What is the derivation for the exponential energy relation and where does it apply?

Very often when people state a relaxation time $\tau_\text{kin-kin}, \tau_\text{rot-kin}$,, etc. they think of a context where the energy relaxation goes as $\propto\text e^{-t/\tau}$. Related is an ...
2
votes
1answer
373 views

A differential equation of Buckling Rod

I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the diff. eq. of hard clamped on both ends rod. And the force compresses the rod at both ...
2
votes
1answer
94 views

Kirchhoff current loop in Resistor Diode Ladder network

I am looking for an approach on how to apply Kirchhoff current / voltage law in the infinitely long diode ladder network. Can anyone help me with this ? I am looking for 1D differential equation or ...