DO NOT USE THIS TAG just because the question contains a differential equation!
0
votes
2answers
122 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 ...
1
vote
2answers
50 views
Does spatial coupling prohibit resonances due to an external source field?
The harmonic oscillator coupled to a sinodial external source
$$\tfrac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$
has the solution
$$x(t)=x(0)\cos(\omega_0 t)+C ...
1
vote
0answers
84 views
Approximate solution for an ODE [migrated]
I've been working on a problem of Newton's gravity on different geometries. On a, physically meaningful, geometry I ended up with the following nasty first ODE:
...
2
votes
0answers
35 views
Regular initial data
I have a very basic question. What exactly is meant by "regular" initial data in general relativity? Does it mean smooth? at least $C^{2}$?
All literature on the subject just uses this term without ...
-2
votes
0answers
73 views
Velocity, Wave Equation, Differential Equations [closed]
Suppose you have a differential equation of the form:
$$
\frac{\partial^2 u}{\partial z^2} = C \frac{\partial^2 u}{\partial t^2} + D \frac{\partial u}{\partial t}$$
Is it possible to find the ...
0
votes
0answers
48 views
What's the physical interpretation of constants in Laplace equation and diffusion equation?
What's the physical interpretation of constants in wave equation and diffusion equation?
$$u_{tt}=c^2u_{xx},$$
$$u_{t}=ku_{xx}.$$
Please introduce some reference about mathematical modeling of ...
4
votes
1answer
101 views
Solving the differential equation of a beam under moving load using green functions
i started working on this paper and i didnt understand one part of it , the problem is :
Solve this equation using green functions :
$$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu ...
1
vote
0answers
52 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
58 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 ...
1
vote
0answers
51 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 ...
8
votes
2answers
431 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, ...
0
votes
0answers
32 views
Examples in physics modelled with a linear ODE [closed]
I have just come across the modeling of an harmonic oscillator that comes out as a linear ordinary differential equation.
As this equations are heavily studied in ODE courses i wonder if there are ...
5
votes
1answer
207 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
46 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 ...
3
votes
0answers
98 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
$$
\frac{\partial C}{\partial t} = -\frac{1}{2}\sigma^2 S^2 ...
0
votes
1answer
57 views
Hyperbolic, parabolic, elliptical PDE related to under-, critical- and overdamped in harmonic osciallation
A damped harmonic oscillator has three cases for the damping: underdamped, critically damped and overdamped. With partial differential equations, I know the hyperbolic wave equation, the parabolic ...
0
votes
1answer
84 views
FWHM in resonance amplitude square derivation
Consider a linear harmonic oscillator subject to a periodic force:
$$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$
The solution tends to:
$$A \cos (\omega t - \delta)$$
where:
...
2
votes
1answer
99 views
Does a constant of motion always imply a Hamiltonian formulation?
If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
3
votes
3answers
126 views
Bessel vs. modified Bessel in radial equation of hydrogen
I am trying to understand the difference between Bessel functions and modified Bessel functions (simply googling is yielding complicated, non-intuitive answers). I was under the impression that one ...
2
votes
2answers
306 views
Greens function in EM with boundary conditions confusion
So I thought I was understanding Green's functions, but now I am unsure. I'll start by explaining (briefly) what I think I know then ask the question.
Background
Greens are a way of solving ...
0
votes
3answers
228 views
Solving the diffusion equation
I am trying to clarify the relation between random walk and diffusion, and the source book proposes the following which I can't get. Starting from the diffusion equation
$$ \frac{\partial C}{\partial ...
4
votes
1answer
92 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
123 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
217 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 ...
1
vote
1answer
114 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 ...
0
votes
0answers
37 views
Approximating a first order ODE when the Hessian is available [closed]
I'm attempting to numerically approximate a simple ODE, I'm using it to describe the motion of a gradient descent search, but it could easily have physical interpretation. In particular,
$$
x'(t) = ...
2
votes
1answer
61 views
Solution of motion in hamiltonian formalism
I have these canonical equations:
$$\dot p = - \alpha pq$$
$$ \dot q =\frac{1}{2} \alpha q^2$$
I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$.
I thought to simply ...
2
votes
0answers
66 views
Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?
Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
0
votes
1answer
128 views
Temperature Vs. Volume of Water
Here and here it states that water is at its highest density around $4^\circ$ Celsius. I know very little physics and a Google search has left me without an answer. I am teaching an ODE class in the ...
6
votes
1answer
274 views
Diffeomorphisms and boundary conditions
I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism which preserve the boundary conditions in the same paper.
I found this ...
6
votes
3answers
452 views
What is the physical meaning/concept behind Legendre polynomials?
In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential ...
12
votes
2answers
52 views
Numerical Analysis of Elliptic PDEs
I am looking for an elementary reference regarding issues of stability in numerical analysis of non-linear elliptic PDEs, particularly using the finite difference method (but something more ...
1
vote
1answer
98 views
equivalence of wave equations
I wonder if the following 2 PDEs are equivalent:
$$\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)$$ subjects to zero initial conditions ...
0
votes
0answers
121 views
How can I find the solution to this wave equation? [closed]
$$\dfrac {\partial ^{2}y} {\partial x^{2}}=\dfrac {\mu } {To}\left( \dfrac {\partial ^{2}y} {\partial t^{2}}\right)$$
General form given by $y(x,t) = f(x)\cdot cos(\omega t )$.
I can't understand ...
8
votes
1answer
219 views
Modification of Newton's Law of Cooling
Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I ...
16
votes
6answers
320 views
Applications of delay differential equations
Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional ...
3
votes
2answers
195 views
Boundary conditions for crystals
As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example:
$\psi(x)=\psi(x+L)$
where $L$ is the length of the 1D crystal.
My question is:
...
0
votes
1answer
138 views
Solution of a partial differential heat equation with derivative and boundary conditions
I want to solve the following partial different equation.
Find $u(x, t)$, satisfying $u_t = u_{xx}$ , $u(x, 0) = x − x^2$ , $u(0, t) = T_0$ , $u_x (1, t) = 0$ and $|u|$ is bounded.
Using separation ...
0
votes
0answers
48 views
Cauchy Problem in Convex Neighborhood
While reading the reference
Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available here,
there is something that I don't quite understand.
...
2
votes
3answers
214 views
How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$
How do I integrate the following?
$$\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 ...
1
vote
0answers
63 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
1answer
174 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
189 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
175 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
59 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 ...
