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4
votes
1answer
229 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 ...
12
votes
3answers
321 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
4answers
138 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
0answers
144 views

Solving the equation of relativistic motion

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant ...
2
votes
0answers
42 views

Regular initial data

I have a very basic question. What exactly is meant by "regular" initial data in general relativity? Does it mean smooth? at least $C^{2}$? All literature on the subject just uses this term without ...
0
votes
0answers
107 views

What's the physical interpretation of constants in Laplace equation and diffusion equation?

What's the physical interpretation of constants in wave equation and diffusion equation? $$u_{tt}=c^2u_{xx},$$ $$u_{t}=ku_{xx}.$$ Please introduce some reference about mathematical modeling of ...
1
vote
0answers
142 views

Problems related to Green's function? [closed]

My teacher told me to do a research studying some physics problems that has connection with Green's function on solving differential equations (with programmed numerical solutions) in my final year ...
2
votes
2answers
83 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 ...
9
votes
2answers
878 views

What is the mathematical reason for topological edge states?

There are many free fermion systems that possess topological edge/boundary states. Examples include quantum Hall insulators and topological insulators. No matter chiral or non-chiral, 2D or 3D, ...
5
votes
1answer
931 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?
0
votes
0answers
65 views

coordinate change differential equation polar

I noticed that v [in step (2.5)] is not the same as the terms from the first formula, even if they are related.. I tried to understand how did he reach to this ...
0
votes
1answer
136 views

Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation

A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic ...
2
votes
1answer
128 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 ...
3
votes
3answers
341 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 ...
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 ...
0
votes
3answers
449 views

Solving the diffusion equation

I am trying to clarify the relation between random walk and diffusion, and the source book proposes the following which I can't get. Starting from the diffusion equation $$ \frac{\partial C}{\partial ...
5
votes
1answer
225 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
241 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 ...
6
votes
2answers
307 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 ...
2
votes
1answer
255 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
88 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
415 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 ...
6
votes
1answer
380 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 ...
12
votes
2answers
76 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
1answer
112 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
410 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 ...
16
votes
6answers
634 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 ...
3
votes
2answers
329 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: ...
0
votes
1answer
209 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
3answers
273 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
349 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
290 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
277 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
86 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 ...