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3
votes
2answers
2k 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 ...
5
votes
1answer
339 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 ...
2
votes
4answers
373 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
1answer
336 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
92 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 ...
0
votes
1answer
591 views

Temperature Vs. Volume of Water

Here and here it states that water is at its highest density around $4^\circ$ Celsius. I know very little physics and a Google search has left me without an answer. I am teaching an ODE class in the ...
10
votes
4answers
3k 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 ...
6
votes
1answer
405 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 ...
1
vote
1answer
121 views

equivalence of wave equations

I wonder if the following 2 PDEs are equivalent: $$\frac{\partial^2}{\partial t^2}\psi(\vec{r},t)-c(\vec{r})^2\nabla^2\psi(\vec{r},t)=s(\vec{r})\delta'(t)$$ subjects to zero initial conditions ...
8
votes
1answer
593 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 ...
0
votes
1answer
285 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 ...
13
votes
3answers
368 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 ...
2
votes
3answers
300 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
1answer
452 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 ...
3
votes
2answers
347 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 - $v^\alpha$, ...
2
votes
3answers
356 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 ...
0
votes
0answers
92 views

deriving differential equation of a cart moved by a motor

This is homework and I'm having some trouble getting started. How do I go from whats given to the form that they ask for? Normally in something like this I would try to balance torque, but I'm not ...
3
votes
2answers
428 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: ...
12
votes
2answers
88 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 ...
1
vote
2answers
314 views

What does transport equation represent in terms of physical quantities?

In my math course we're taught to solve PDE (partial derivative equations) like transport equation: $$ c\frac{\partial u}{\partial x} +\frac{\partial u}{\partial t}~=~0. $$ If $u(x,t)$ is the ...
16
votes
6answers
926 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 ...