The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [differential-equations]

DO NOT USE THIS TAG just because the question contains a differential equation!

Filter by
Sorted by
Tagged with
0
votes
0answers
5 views

Derive Columb's Law from Gauss's Law by solving PDEs?

The standard way of deriving Columb's law is given in this post. I am just wondering if I can do it in another way. Let a point charge $q$ be located at the origin. The problem is then essentially ...
2
votes
1answer
19 views

Why are Cauchy boundary conditions an over-specification of boundary conditions for solving Poisson’s equation?

I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics: This rather surprising result [the fact that the potential within a charge-free volume is ...
0
votes
0answers
20 views

Cosmology - Demonstration for equation of the evolution of the density contrast

In a context of cosmology, I need help about a differential equation that I can't get to demonstrate: The growth of density fluctuations obeys a second order differential equation. At early enough ...
2
votes
0answers
52 views

Confusion on Quantum Harmonic Oscillator Eigenvalues

In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $\lambda_n$ for a Dirichlet ...
-1
votes
0answers
52 views

Solution to a differential equation in terms of Bessel function [closed]

I am looking after the solution of the following differential equation: $ \partial_y \partial_y \psi + \frac{\partial_y \psi}{y} + \left( \frac{\omega^2 }{y^2} - \frac{\Lambda}{y} \right) \psi = 0 $ ...
1
vote
0answers
27 views

Solving a PDE with Fourier transform [migrated]

[Homework problem:] In my assignment, I was given a problem where I had to solve a PDE using Fourier transform. It goes like this: Solve the PDE $$u_t=u_{xx}$$ subject to the initial conditions: $u(...
-4
votes
0answers
30 views

System of second order differential equations

Can someone give me some references to solve numerically system of second order differential equations using shooting method?
0
votes
0answers
33 views

Fractional differential equations and Physics [duplicate]

Are the "fractional differential equations" have any real significance in respect to physics? or are they just stilted math?
0
votes
1answer
44 views

Klein-Gordon equation with position-dependent mass [closed]

Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $m(x)$ is a known function.
0
votes
3answers
140 views

Numerical method for first-order non-linear differential equation

So I'm modeling a cycler on a 1000 m race track using the equation $$\frac{dv}{dt}=\frac{P}{m}\left(\frac{1}{v}\right)-\frac{k}{m}\left(v^2\right)-ug$$ where $P$ = power, $m$ = mass, $v$ = velocity of ...
2
votes
1answer
30 views

How to find the equation of motion of a particle in an electric and magnetic field?

I'm trying to solve a homework problem. The statement of the problem says: "When the negatively charged plate of a parallel plate capacitor is lit up by light of a certain wavelength, electrons are ...
9
votes
4answers
2k views

Is there a general way of solving the Maxwell equations?

Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics ...
-1
votes
1answer
41 views

Finding general equation for motion of a radioactive particle performing SHM [closed]

Let us assume we have a particle of initial mass $m_{0}$ such that a general time $t$: $$ m(t) = m_{0} e^{- \lambda t} $$ Now, let us say this particle is attached to a spring of spring constant $k$,...
1
vote
0answers
105 views

Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
4
votes
4answers
145 views

Radioactive Decay ($dN/dt = - \lambda N$)

The formula for radioactive decay, $$N = N_0 \ e^{- \lambda t}$$ is derived from the following assumption: $$\frac{dN}{dt} = - \lambda N.$$ I understand an assumption has been made that no external ...
0
votes
0answers
169 views

Physical interpretation of a PDE

I'm not a physicist and I would like to understand the physical meaning of the following equation: $$u_t (t,x)=-\Delta^2 u(t,x)+f(t,x).$$ This is a $4^{th}$ order parabolic equation similar to the ...
1
vote
2answers
51 views

Electromagnetic wave equation: can we ignore the constant of integration?

Suppose we obtain a solution for each of $\mathbf B$, $\mathbf E$ of maxwell equations in the vacuum ($\rho=0$). Clearly, for any constant vector $\mathbf k, \mathbf m$, $\mathbf {B+k}$ and $\mathbf{E+...
0
votes
2answers
56 views

Is separation of variables in the heat equation dimensionally consistent?

This may be a trivial question but is about the statement that the function $U(x,t) $ in the heat equation may be expressed in the form $X(x)\cdot T(t)$. It's that $X$ and $T$ both are functions ...
1
vote
1answer
61 views

Quantum Harmonic Oscillator- Solving the Differential Equation at a Limit

The eigenvalue equation for the quantum harmonic oscillator is $$\langle y | E\rangle '' +(2\epsilon-y^2)\langle y| E \rangle=0$$ where $\epsilon = \frac{E}{\hbar\omega}$ and $y=\sqrt{\frac{\hbar}{m\...
3
votes
2answers
231 views

Rayleigh scattering light intensity

Assume a single ray of unpolarized light propagating through an opaque medium (water). Due to absorption and Rayleigh scattering the intensity decreases and light gets scattered/diffused before hiting ...
0
votes
2answers
75 views

Why are the left- and right-hand sides of a differential equation with two separated variables equal to a constant?

While deriving the Time Independent Schrodinger Equation, my book mentioned this line. So time and position of a particle are two independent variables. If they are equal to one another for all ...
-1
votes
1answer
43 views

Can solutions of the Poisson's equation be written as linear combinations of Laplace's equation solutions? [closed]

Given that the Laplacian operator $\Delta$ acts on the space of functions(at least $C^2$), does the equation $\Delta\phi=0$, define a base of that space such that solutions of $\Delta\psi=f$ can be ...
1
vote
0answers
12 views

Boundary condition for a bulk-surface and bulk-bulk diffusion reaction system

Consider this simple example below and the corresponding geometries. I simplified these equations from the real system. Geometry 1 The first geometry is a sphere. Inside this sphere a species $b(t,...
0
votes
0answers
23 views

Extracting solution from driven SHM

I guess maybe I should rather ask at the math stack exchange? I have a simple harmonic undamped oscillator driven by a cosinusoidal force: $$ \ddot{x}+\omega_o^2x=f\cos(\Omega t).$$ I've managed to ...
0
votes
0answers
19 views

Difference between types of “reference” in second order ordinary differential equation

I'm looking at a second order ODE (wrt time $t$) from power systems defined as: $$ M \,\ddot{\delta}_i = P^m_i - f^e_i(\delta) - D \dot{\delta}_i \tag{1} $$ for $i \in 1, \ldots, N$ where $f^e_i(\...
1
vote
1answer
55 views

How is heat propagation a semi-deterministic process?

Arnol’d in his Ordinary Differential Equations, states that The propagation of heat is a semideterministic process: the future is determined by the present but the past is not. How is this so? If ...
2
votes
1answer
32 views

Boundary condition for partial reflection

I want to solve a wave equation for the wave $\psi(x,t)$. One boundary is moving, therefore I impose the velocity $$v(x=0)=v_a\cos(\omega t)$$ the other boundary is fixed, but reflecting. If the ...
3
votes
1answer
44 views

Spring constant of tuning fork

I was playing with a tuning fork and got to wondering how to find it's spring constant (assuming damped oscillation). I can find plenty of resources about materials for springs, but not a whole lot ...
0
votes
1answer
46 views

How to solve this boundary-value problem in electrostatics? [closed]

I was reading The Feynman Lectures on Physics, Vol. 2 and came across the following on page 7-1. There are a few problems for which Eq. (7.1) can be solved directly. For example, the problem of a ...
5
votes
0answers
53 views

Minimum required initial conditions to uniquely solve geodesic equation

The geodesic equation is a 2nd order differential equation given as $$\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d} \lambda^2 }+\Gamma^\alpha_{\beta\gamma}\frac{\mathrm{d} x^\beta}{\mathrm{d} \lambda }\...
0
votes
0answers
51 views

Symplectic time-integrator for Fokker-Planck equation

Is there a way to use a symplectic time-integrator for the numerical solution Fokker-Planck equation of the form: $$\partial_{t}p = -\nabla \cdot (\underline{v}p - D\nabla p)~?$$ where $p = p(\...
1
vote
0answers
30 views

Governing equations vs Transport equations

I posted it in computational-science SE site, and it was suggested I shift it here. This is a basic question. But I did not find any explanations for this. How are governing equations, like mass, ...
2
votes
1answer
51 views

The solution to the non-linear convection equation

The non-linear convection equation $$u_{t} +uu_{x}=0$$ admits implicit solutions of the form $$u=f(x-ut).$$ How does one interpret this solution intuitively? Is there an example of a solution of this ...
3
votes
0answers
52 views

Chua's Circuit: an inequality ensuring that the equilibrium is not stable

According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are $$\begin{aligned} C_{1} \frac{d v_{...
1
vote
1answer
53 views

Chua's circuit: Is $x=y=z=0$ a stable equilibrium?

According to this wikipedia page, the differential equations satisfied by several physical quantities in Chua's circuit is (What each letter represents doesn't matter that much in this question) For ...
1
vote
1answer
56 views

Old unsolved question on greens function

So I was looking up Kf Riley’s 3rd edition, and bump into a problem about greens function. I went online and googled and notice other people had the same problem and no one really could answer: ...
1
vote
1answer
30 views

Proving that motion of an $n$ dimensional oscillator can be written as a linear combination of “sine waves”

Here is a related question which might provide some context: LINK. Let's consider an oscillator with equation of motion in $n$ dimensions: $$ \frac{d^2}{dt^2} \vec{x} = K \vec{x}. $$ Given that $\...
-3
votes
1answer
224 views

How do we find the equation for the gyrating motion of a particle in a uniform magnetic field and a non-uniform Electric field? [closed]

Considering the gyrating motion is not negligible and also retaining the guiding center drift, how do we get the trajectories x(t),y(t),z(t) of the particle? In this case is the variation in the ...
3
votes
1answer
80 views

Choosing initial condition for Hamilton-Jacobi PDE from initial $x$ and $p$

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
1
vote
1answer
36 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
1
vote
0answers
69 views

How to solve these coupled differential equations?

I am trying to solve for wavefunctions of 2D tilted Dirac systems, the Hamiltonian for which is: $$\hat H = v_{x}\sigma_{x}\hat p_{x}+v_{y}\sigma_{y}\hat p_{y}+I_{2}(v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat ...
0
votes
0answers
47 views

More than two linearly independent solutions to the (linear second order) radial wave equation?

I'm puzzled by the following radial wave equation: $$ \left(\frac{\hbar^2}{2m_r}\left(-\frac{d^2}{dr^2} -\frac{2}{r}\frac{d}{dr} + \frac{l(l+1)}{r^2}\right) + V(r)\right)R_{nl}(r) = ER_{nl}(r)$$ ...
1
vote
1answer
104 views

Modification of the Verlet algorithms for the pendulum problem

I'm trying to write a program to integrate the motion equations of the pendulum in the damped and forced case, that is, following this equation: $$ \frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin(\theta)-\mu\...
0
votes
0answers
29 views

Symmetries of solution

I have a system of coupled nonlinear differential-difference equations as model of particles with harmonic interaction in some potential, of the form: $$ \dot{x}_{i}=x_{i+1}+x_{i-1}-2x_{i}-\sin(2\pi ...
3
votes
0answers
39 views

Master equation for a dynamical system on networks

I am trying to mathematically model the following idea that describes the dynamical evolution of a quantity over a graph. Let us imagine we have a directed graph, with $n$ nodes and $m$ edges. Each ...
3
votes
0answers
41 views

Initial Condition in Spaghetti Cracking

In this Paper B. Audoly, S. Neukirch - Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half on Page 2 (bottom), the author argues that using an integral of motion, the ...
1
vote
0answers
48 views

What numerical methods are being used here to solve the differential equation?

I am having trouble following and understanding the numerical approach to solving for $h(r,t)$ in the following text: How is the integral force balance (10) used to calculate the new pressure ...
0
votes
0answers
27 views

Elastic wave in isotropic medium

The 2D elastic wave equation is given by: $$ \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= C_{11} \frac{\partial^2 u}{\partial x^2} + C_{44} \frac{\partial^2 u}{\partial y^2} + (C_{12} ...
1
vote
1answer
52 views

Why are all solutions to this system of pendulum differential equations a linear combination of the two given solutions?

I am currently trying to do a lab report for a coupled pendulums experiment in which we find the following linear system of second order differential equations (describing the position as a function ...
2
votes
0answers
26 views

Physics of a catenary with sinusoidal motion at one end

If we have a catenary line with a mass per unit length of m with one end fixed and the other end moving in sinusoidal motion like $A.sin(\omega.t)$, how would I ...