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-2
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0answers
19 views

Solving IVP $\frac{dE}{dx}= -\gamma\frac{dx}{dt}-\alpha$ [on hold]

I need to solve the differential equation $$\frac{dE}{dx}= -\gamma\frac{dx}{dt}-\alpha$$ with the initial value $$v(t=0)=v_O$$ I tried writing everything in terms of $v$ which gives me: ...
0
votes
0answers
21 views

Looking for an ODE problem involving a tower and structural vibration [on hold]

I am a teaching assistant of an ODE course (my assignment is just grading according to the grading guideline...) and the Prof. asked me to help design a undergrad level ode problem involving a tower ...
3
votes
0answers
32 views

Is there an analytic solution for this Fokker-Planck equation? [migrated]

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
0
votes
0answers
42 views

Heat equation with with time varying boundary condition [on hold]

I am trying to solve the heat equation: $$\frac{\partial U}{\partial t}=k \frac{\partial^2 U}{\partial x^2}$$ with the following boundary conditions: \begin{align} U(x,0)&=0 \qquad t<0\\ ...
0
votes
1answer
25 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
20 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 ...
1
vote
2answers
122 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 ...
0
votes
0answers
9 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 ...
1
vote
0answers
27 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 ...
2
votes
1answer
133 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
122 views

Heat Equation with In-Depth Radiation Exact Solution

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 ...
2
votes
0answers
27 views

What is the relationship between the Boltzmann Transport Equation and the Navier-Stokes Equation?

What is the relationship between the Boltzmann transport equation and the Navier-Stokes equation? Can the Navier-Stokes equation be derived from a moment of the Boltzmann transport equation?
1
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0answers
23 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
39 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
164 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 ...
0
votes
1answer
279 views

Solution of a partial differential heat equation with derivative and boundary conditions

I want to solve the following partial different equation. Find $u(x, t)$, satisfying $u_t = u_{xx}$ , $u(x, 0) = x − x^2$ , $u(0, t) = T_0$ , $u_x (1, t) = 0$ and $|u|$ is bounded. Using separation ...
2
votes
1answer
90 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
78 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 ...
2
votes
0answers
31 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 ...
0
votes
0answers
44 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 ...
1
vote
2answers
39 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 ...
3
votes
1answer
67 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
54 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 ...
3
votes
1answer
126 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
154 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 ...
6
votes
1answer
305 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 ...
1
vote
1answer
32 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 ...
4
votes
1answer
110 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 ...
0
votes
1answer
33 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
67 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: $$ ...
1
vote
0answers
24 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 ...
0
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0answers
31 views

Calculate the potential induced by beam

This exercise is from Plasma Physics and Fusion Energy by Freidberg. A cylindrical conducting vacuum chamber of radius $r=a$ is filled with a uniform plasma of density $n_0$ and temperature ...
16
votes
1answer
843 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
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2answers
542 views

Geometrical interpretation of complex eigenvectors in a system of differential equations

Let's consider a system of differential equations in the form $$\dot{X} = M X$$ in two dimensions ($X = (x(t), y(t))$). In the case that $M$ has real values, it is easy to give a geometric ...
5
votes
0answers
71 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 ...
2
votes
0answers
88 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 ...
8
votes
1answer
467 views

Relation between Black-Scholes equation and 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 ...
17
votes
5answers
816 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, ...
30
votes
6answers
737 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 ...
3
votes
1answer
77 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 ...
2
votes
1answer
83 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 ...
4
votes
1answer
243 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 ...
2
votes
2answers
102 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
50 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 ...
1
vote
1answer
30 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
3answers
107 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 ...
6
votes
2answers
309 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
0answers
61 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 ...
2
votes
1answer
163 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 ...
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 ...