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Random walk of a polymer in an annular space - filling in some steps

In Muthukumar, M., 2003. Polymer escape through a nanopore, J. Chem. Phys., 118, 5174–5184, the author solves the diffusion equation for the probability $P(\vec{r},\vec{r_0},N)$ that the ends of a ...
3
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1answer
44 views

Derivative with respect to a difference of independent variables

I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used. The equation looks like this $$ ...
-4
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0answers
39 views

Derivative of an integral help [on hold]

OK, I lied a bit. It's not JUST the derivative of an integral. It's the derivative of a cosine of an integral. Solving the problem of the motion of a simple pendulum under a gravitational field using ...
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0answers
21 views

Mathieu equation nonstable solutions

This israther mathematical question, but it is connected with some physics. Let's have Mathieu equation: $$ \tag 1 y''(t)+ (a -2q\cos(2t))y(t) = 0 $$ Suppose domain of parameters $a, q$ values, where ...
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0answers
17 views

Stability of fix-point of a system of 3 non linear first order ODE, when one of the eigenvalues of Jacobian is zero

I have been working on a mean-field solution for am open quantum system model, to compare with the numerical solution of the exact solution. I have solved the system for steady state, but am now ...
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1answer
78 views

Are all diffusion-like processes described as wave-like in relativity-compatible formulations?

Citing from Wikipedia's article on relativistic heat conduction: For most of the last century, it was recognized that Fourier equation (and its more general Fick's law of diffusion) is in ...
6
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1answer
193 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 ...
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1answer
65 views

Hamiltonian from a differential equation

In my differential equations course an example is given from the Lotka-Volterra system of equations: $$ x'=x-xy$$ $$y'=-\gamma y+xy.\tag{1}$$ This is then transformed by the substitution: $q=\ln x, ...
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23 views

Are time-$t$ maps of a Hamiltonian system with 1 degree of freedom typically twist?

If we take a typical Hamiltonian system $H(q,p)$ with one degree of freedom, and look at its time-$1$ map $(q(0),p(0)) \mapsto (q(t),p(t))$, will it generically satisfy the twist property, e.g. ...
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3answers
1k views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
6
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1answer
201 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 ...
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20 views

Solving traveling wave using the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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1answer
15 views

Equations involved in freezer burn

Assuming a sphere with a given percentage of moist uniformly distributed through the sphere. It's surrounded by air with no humidity. How could I model the sublimation of the sphere's moist into the ...
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0answers
8 views

Modeling stilts with equations

I'm trying to make a set of stilts 20 ft in the air that someone could wear to run/jog with, but I was advised that I should first start by modeling the system (user + environment + stilts) in the ...
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0answers
29 views

Linear DEs without separation of variables in physics?

I'm looking for examples of real world physics problems that require solving linear (or linearisable) differential equations (DEs) that aren't separable in the variables. Most (admittedly) simple, ...
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2answers
143 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
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3answers
176 views

Maxwell's equations - underdetermined - uniqueness

Maxwell's equations can be seen as two dynamical equations (the two curl equations), and two constraint equations (the two divergence equations). So we have 6 unknowns ($E_x,E_y,E_z,B_x,B_y,B_z$). ...
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7answers
4k views

Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and ...
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0answers
15 views

Modeling: State variables and algebraic variables

I am new to modeling of differential-algebraic systems. I dont understand the coherence between: state-variables, differential variables and algebraic variables. Standard form of the system: ...
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0answers
21 views

Which formulas would tell me the gradient of an electromagnetic field at an arbitrary distance from a pole? [duplicate]

I'm a newbie to physics and was wondering where I can read about electromagnetic gradients. From what I understand (and my intuition) electromagnetic fields create force gradients around its poles. ...
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0answers
16 views

Linear viscoelastic differential operators

I am starting with differential operators: $P = \sum_{i=0}^{N}p_i \cfrac{d^i}{dt^i}$ $Q = \sum_{i=0}^{N}q_i \cfrac{d^i}{dt^i}$ $p_i$ and $q_i$ are functions of time only. $K$ is a constant that ...
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1answer
89 views

Can $Ae^{-bt^2}\sin(kx-\omega t)$ be considered a wave?

The damped wave PDE can have an exponential term, but the argument for the exponential term cannot be quadratic, AFAIK. $Ae^{-bt^2}\sin(kx-\omega t)$ So this isn't a solution for the damped wave ...
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1answer
48 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: $$ ...
3
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1answer
77 views

Existence and Uniqueness of Newton's Laws

I'm reading Arnold's book on classical mechanics. This is kind of a dumb question, but I'm having problems understanding his explanation for existence and uniqueness of Newton's laws. On page $8$ he ...
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0answers
23 views

Lets consider a cube with side $2$, which is cooling in an environment. Find its temperature at any point at any time: $u(x,y,z,t)$

Lets consider a cube with side $2$, which has an initial temperature of $1$°K and it is cooling in an environment of temperature $0$°K. Find its temperature at any point at any time: $u(x,y,z,t)$. ...
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1answer
66 views

Monodromy matrix and differential equations

What is the significance of monodromy matrix in the context of differential equations? I have seen some papers(1,2,3 etc) in CFT which use the monodromy method to compute conformal blocks at large ...
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1answer
29 views

How to extract heat transfer model parameters from empirical data?

I have made a simple model of heat transfer between ambient and a silicon chip (module) from which I can read its internal temperature $T_m$. I do not need fancy equations and an approximate model ...
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0answers
33 views

car dashboard problem

I stumbled upon this question while I was driving my car. On my dashboard I have fuel gauge and engine temperature gauge next to each other, look at the pic: http://i.stack.imgur.com/aDgKj.png Fuel ...
4
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1answer
158 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving the Schrödinger equation, such as tasks about a particle in a Morse potential, a Poschl-Teller potential and many others, we usually find approximations (lets call ...
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0answers
37 views

Question about a solution of a partial differential equation by separation of variables

I'm trying to understand this text: http://www.ekayasolutions.com/UCDMath/HeatCondSphere.pdf But I'm having problem with this part: Whe have to solve: \begin{equation} \dfrac{\partial ...
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1answer
46 views

Differential derivation based upon time and space confusion

I have been doing a lot of derivations recently involving heat transfer. I was attempting to derive heat accumulation in a differential element based upon inflow and outflow as well as thermodynamic ...
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2answers
79 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) ...
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0answers
55 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
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1answer
71 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 ...
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0answers
17 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 ...
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0answers
28 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
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1answer
96 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
66 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 ...
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0answers
141 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 ...
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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. ...
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50 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 ...
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2answers
49 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 ...
2
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0answers
43 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
48 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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17 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 ...
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6answers
895 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
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1answer
56 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 ...
5
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1answer
214 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 ...
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35 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( ...
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0answers
24 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 ...