The differential-equations tag has no wiki summary.
2
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2answers
57 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 ...
0
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2answers
108 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
48 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 ...
-2
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0answers
42 views
Klein-Gordon equation (boundary value problem) [closed]
Could some help me with this question. One of my friends ask me but I have no idea about it I am pure mathematician
This the equation $u_{tt}-u_{xx}+\frac{3}{4}u-\frac{3}{2}u^3=0$,
Here the ...
1
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0answers
43 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
395 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
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0answers
30 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
177 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
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0answers
44 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 ...
2
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0answers
91 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
50 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
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1answer
77 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
94 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
113 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
286 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
214 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
$$ \frac{\partial C}{\partial ...
4
votes
1answer
75 views
What formulas should I use to realistically model the diffusion of a drop of ink in a water?
I am a mathematician and am originally from the math side of stackexchange. I want to model the behaviour of a drop of ink diffusing in water. I dont want to simply use the diffusion equation ...
2
votes
4answers
119 views
Why is linear independence of harmonic oscillator solutions important?
The equation of motion for the harmonic oscillator (mass on spring model)
$$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$
with $\omega_0^2 = D/m$, $D$ and $m$ being the force constant of the spring and the ...
6
votes
2answers
214 views
What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?
I have a problem with one of my study questions for an oral exam:
The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
1
vote
1answer
105 views
What does it mean that Einstein's equations are hyperbolic-elliptical?
I says on Wolfram MathWorld that Einstein's field equations are a set of "16 coupled hyperbolic-elliptic nonlinear partial differential equations".
What does it mean that the equations are ...
0
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0answers
33 views
Approximating a first order ODE when the Hessian is available [closed]
I'm attempting to numerically approximate a simple ODE, I'm using it to describe the motion of a gradient descent search, but it could easily have physical interpretation. In particular,
$$
x'(t) = ...
2
votes
1answer
55 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
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0answers
63 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
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1answer
113 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
273 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 ...
5
votes
3answers
389 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
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2answers
50 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
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0answers
116 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 ...
7
votes
1answer
210 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
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6answers
304 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
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2answers
190 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
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1answer
132 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
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0answers
47 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
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3answers
208 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 ...
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0answers
62 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
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1answer
171 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
185 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
173 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
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0answers
56 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 ...
