Questions tagged [differential-equations]

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System PDE's non-linear [closed]

I am trying to solve the following system of PDE's. The idea behind this is that I have two media A,B and a collection of molecules. At medium A they can actively move at +x direction, whereas at ...
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1answer
57 views

Solving Schrodinger equation for the hydrogen atom

In University Physics with Modern Physics by Hugh Freedman in Chapter 41.3 they go about solving the Schrodinger equation for the hydrogen atom. At one step they say to substitute the following ...
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Dynamical system fixed point by perturbation?

Suppose I have an non-linear autonomous system : $$ \dot{x}_i(t) = f(x_i(t)) + \lambda g(x_i(t)) $$ I am interested in finding its fixed points. I want to know if the following method of ...
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41 views

Is the linear combination of stationary states exhaustive? [duplicate]

After solving the Schrodinger Equation and getting the stationary states (separable solutions), we say the the general solution of the Schrodinger Equation is just the linear combination of these ...
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2answers
73 views

Can't the heat equation be inverted?

I have heard people say that the heat equation isn't invertible because it smooths out irregularities that can not be recovered by backwards time evolution. But is this so? I will now argue that it ...
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1answer
57 views

Differential equation [closed]

I am trying to solve the following differential equation; $$\frac{d^2 x}{d t^2}=-\omega^2 x \delta(t-t^\prime).$$ I know this is of the form $$x(t)= A \sin(\omega t) + B \cos(\omega t).$$ However this ...
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2answers
152 views

David Hilbert's Prediction of Schrödinger Equation

In (Constance Reid 1996, Hilbert, p. 182): according to Condon: " ... when [Born and Heisenberg and the Gottingen theoretical physicists] first discovered matrix mechanics they were having, of ...
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52 views

Cauchy problem of classical Maxwell equations in Minkowski spacetime

I'm wondering a bit about the classical Maxwell equations in flat spacetime and their Cauchy problem. For the following, I suppose that the sources are already given and don't react to their own ...
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1answer
53 views

Angular Frequency of a charged particle moving in a Magnetic Field

I am trying to solve the Differential Equation in Mathematica but it is giving me error. How to find it's solution in Mathematica or in general. $$y'''=\frac{\omega^2E}{B}-\omega^2y'$$ The above ...
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1answer
33 views

Looking for a good book on Differential Equations [duplicate]

I know many of you are tired of book recommendation posts and questions. But I am self learning Theoretical Physics, and I am having a hard time choosing a book to learn differential equations (ODEs). ...
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Derivation of MTZ Black Hole

I am trying to derive from scratch the MTZ Black Hole: https://arxiv.org/abs/hep-th/0406111 I have obtained equations (2.3) and (2.4) in terms of the metric functions and the scalar field. The metric ...
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1answer
26 views

Uncertainty from variable to derivative in data fitting

If I have a system of differential equations, say $x' = f(x,p)$ and a set of data $(t_n,x_n)$, where $p$ is the set of constant parameters. I can then use a fitting method like least squared to obtain ...
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30 views

Have water first order system dynamics or second order dynamics? [closed]

This is two transfer functions. One first order and the other is on second order $$G_1(s) = \frac{K}{1 + Ts}$$ $$G_2(s) = \frac{K\omega^2}{\omega^2 + 2\zeta \omega s + s^2}$$ The know characteristic ...
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31 views

Solving the Lane-Emden equation via Chebyshev differentiation matrices

Problem So I'm trying to learn spectral methods but I can't quite proceed for some reason. In particular, I have been trying to solve the Lane-Emden equation (which I know how to solve via ...
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52 views

How to obtain a solution for the following IBVP?

I am trying to solve the following advection-diffusion equation for transient flow conditions for radial flow. The governing equation is as follows. $$\frac{\partial T}{\partial t} = \frac{\partial^2 ...
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31 views

Velocity waves in spherical geometries

I am currently working on velocity waves in spherical geometries: I am considering a 1D many-particle system confined on a circle with a global drift leading to rotations, similar to this simulation ...
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1answer
73 views

How can I interpret the equation of the orbit in Schwarzschild metric?

Given the standard geodesic equation: $$\frac{d^2 x^\mu}{d\lambda^2}+\Gamma ^\mu _{\sigma \rho}\frac{d x^\sigma}{d \lambda}\frac{d x^\rho}{d \lambda}=0$$ we want to apply it to the Schwarzschild ...
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1answer
132 views

Advection-diffusion with periodic boundary conditions

Context: Consider the advection-diffusion equation with periodic boundary conditions (PBC) over a flat square domain $L \times L$. The scalar density $\rho $ is transported by a prescribed field $\...
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Coupled solid-fluid heat transfer over a rectangular plate heated from the bottom (Boundary value problem)

I have the two-dimensional temperature Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another fluid equation (which is one-dimensional). The Laplacian is defined over $x\in[0,L], y\in[0,l]$. On ...
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2answers
44 views

Heat equation with a source and homogenous boundary and initial conditions

I am trying to solve the following: $$\frac{\partial^2u}{\partial x^2}-3\frac{\partial u}{\partial t}=-9$$ $$u(0,t)=0=u(\pi,t)$$ $$u(x,0)=0$$ So solving the homogenous case first by separation of ...
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2answers
72 views

Writing universal law of gravitation as a function of position with respect to time

I wasn't sure whether this question belongs here or on the maths page. Anyway I'm new here, and this may be a stupid question, so please forgive my ignorance. Given that the force of gravity can be ...
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2answers
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Time evolution of the operators vs. the expectation values

The time evolution of a quantum mechanical operator $A$ (without explicit time dependence) is given by the Heisenberg equation $$ \frac{d}{dt}A = \frac{i}{\hbar} \left[H,A\right] \tag{1}$$ where $H$ ...
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1answer
108 views

Analytical Solution to Heat Equations

I have the steady state equation with an internal source as $$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2k\frac{\partial T}{\partial r}\right)+Q=0$$ which has the analytic solution $$T(r)=-\frac{...
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2answers
434 views

Getting zero as solution to the 1D wave equation

I was solving an example of 1D wave equation with given BC and IC by separation of variables and Fourier series. $$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2} $$ $$BC: u(0,t)=...
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2answers
125 views

Solutions to wave equation in 1+1D

This might seem like a silly doubt but I am confused about this. For what kind of waves is the wave equation in 1+1D satisfied? $$\frac{\partial^2 f(x,t)}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f(...
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58 views

“The state-space for a single particle in classical space is 6-dimensional” — Is this wrong?

The general argument is as follows. By Newton's second law $\mathbf F=m\ddot{\mathbf{x}}$. Now it is said that this is a second-order ODE and hence requires $\mathbf x(0)$ and $\mathbf{\dot{x}}(0)$ as ...
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Looking for Chaotic region in Duffing oscillator

In many engineering and physics fields, there are lots of models can be converted into the following forced Duffing equation: $$\ddot{x} -\omega_{0}x+D \dot{x} +\gamma\dot{x}^3+\beta x^3 = F \cos(\...
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32 views

Multi-pole expansion for non-Green solution

In electrostatic, we use the Green function to describe the multipole expansion coefficient as: $ \int \rho G = monopole + dipole + quadrupole + ... $ This is possible, because we can find a green ...
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Second order linear filter and coefficients sign

Courses on automatics define usually the second order transfer function as $$ H(s) = \frac{1}{s^2/w_n^2 + 2z/w_n s + 1} $$ (see here or here for instance ) However, it assumes that the coefficents $...
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4answers
75 views

Differential Equations in a Discharging RC Circuit in Parallel

Please consider the following RC circuit as context: Assume that the circuit has been connected for a long time. If switch S has been opened at $t=0,$ the differential equation used to solve for the ...
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1answer
33 views

Differential Equation - Why do beams break?

I've never studied physics before and I'm a Maths major. But I do have a physics-related differential equation question which I need help on. Given the beam deflection equation: $$F(x) = EI \frac{d^...
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2answers
96 views

Solving a differential equation [closed]

($x$ here refers to position) $\frac{d^{2} x}{dt^{2}} + ω^{2}x$ = 0 After solving the above differential equation, we get $x(t) = Ae^{iωt} + Be^{−iωt}$ where $A$ and $B$ are some constants. My ...
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How can a wave function that is both solution of classical wave equation and solution of Schrödinger equation be written?

Do wee need to solve the classical wave equation and Schrödinger equation together? Schrödinger equation has first time derivative while classical wave equation has the second time derivative. In ...
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2answers
93 views

Formulations of the Einstein equations

The Einstein equations can be written as (here I am following the notation of Wald's book on General Relativity) \begin{equation} \partial_{\alpha}\Gamma^{\alpha}_{\mu\nu} - \partial_{\mu}\...
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1answer
32 views

Physical interpretation of the heat equation with variable coefficient

I would want to know what is the physical interpretation of the heat equation with variable coefficients such that: $$u_{t}-\frac{1}{1+t^2}u_{xx}=0$$ well, I think I got it, it means that the ...
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49 views

Problem with Numerical Solution for motion of a particle around Black Hole using Fourth order Runge Kutta method

I have been trying to plot the variation of radial distance of a particle orbiting a Black Hole. The variation of radial distance from Black Hole is given by second order differential equation, $$ \...
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2answers
50 views

Radioactive decay differential equations [closed]

I am trying to form a differential equation between two different isotopes, Uranium-238 and Thorium-234. The rate of decay of an isotope is proportional to the amount present. So that: $$ \frac{dx}{...
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1answer
39 views

Find velocity of small satellite [closed]

Consider a small satellite of mass $m_0$ and initial velocity $v_0 >0$ that is far away from any external forces. It entered a dust cloud containing particles at rest that cling into the satellite'...
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1answer
36 views

Oscillation on Angled Rails (Diff Equation)

This problem was taken from David Morin's Introduction to Classical Mechanics My attempt at solving the problem: First, I labeled all the relevant forces acting only on one of the particles of mass $...
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0answers
68 views

Solving an NMR Schrodinger equation

Fairly standard differential equations question. I always struggled with this as an undergrad because I skipped the diff. eq. math course. It has obviously come back to bite me. I am trying to solve ...
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1answer
74 views

Derivation of London equation $\vec{\nabla}\times\vec{j}=-\frac{n_se^2}{m}\vec{B}$

London's first equation $$\frac{d}{dt}\vec{j}=\frac{n_se^2}{m}\vec{E}$$ where $j=-en_s\vec{v}_s$, $n_s$ is the number density of electrons that contribute to the supercurrent and $\vec{v}_s$ is their ...
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1answer
28 views

Solutions to Stokes flow with no external force and known pressure

I have a problem that is isomorphic to the Stokes problem, but with the external force zero and the pressure known. Specifically, I am trying to find solutions/methods in 3D to solve $$\nabla^{2}\...
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27 views

Delta functions for surface of non trivial shapes

Say I'm trying to solve Laplace's equation on an object that isn't a sphere or a trivially characterized surface. I will want to solve: $$\nabla^2G\left(\mathbf{r},\mathbf{r}'\right)=-4\pi\delta\left(...
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3answers
155 views

How do Maxwell's equations uniquely determine ${\bf E}$ and ${\bf B}$ despite no. of equations exceeding no. of unknowns?

Maxwell's equations in free space are given by $${\bf\nabla}\cdot\textbf{E}=0,~~{\bf\nabla}\cdot\textbf{B}=0$$ and $${\bf\nabla}\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t},~~{\bf\nabla}\...
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49 views

How to prove that $u(r)=k \dfrac{1}{r}$ is the only solution for the integral equation $\displaystyle\int_{V'}\rho'\ u(r)\ dV' = constant$?

Consider a hollow spherical charge with density $\rho'$ continuously varying only with respect to distance from the center $O$. $V'=$ yellow volume $k \in \mathbb {R}$ $\forall$ point $P$ inside ...
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1answer
38 views

Solving the continuity equation for a given mass-density

Given a smooth and time-dependent mass density $\rho(t)>0$ which is spatially constant in 3-dimensional Euclidean space. Let's further assume the continuity equation $$\dot\rho+\text{div}(\vec{j})=...
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1answer
75 views

Are numerical relativity solutions actual solutions to the Einstein Field Equations?

Perhaps this requires a more advanced knowledge of differential equations, but are solutions in numerical relativity (for example, the merger of two black holes) actual solutions to the Einstein Field ...
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1answer
80 views

Solving non-linear ODE with cosine term [closed]

I want to find an equation of motion for non-uniform circular motion under the inlfuence of gravity (like, say, a pendulum) of the form r(t)=R cos⁡(θ(t)) i + R sin⁡(θ(t)) j where $\theta (t) $ is only ...
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1answer
170 views

Is there a link between the logistic differential equation and Fermi-Dirac statistics?

I was working out some statistical problems and I could not fail to notice that Fermi-Dirac distribution, $$f_{\rm Fermi-Dirac}(E)=\frac{N_{\rm sites}}{1+e^{\beta(E-\mu)}},$$ looks like the kind of ...
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0answers
37 views

Solution of Raman-Nath equation [closed]

In this paper [1] the authors claim that the solution of the differential equation $$ \frac{\partial \varphi_n}{\partial z}+\frac{\nu}{2L}(\varphi_{n-1}-\varphi_{n+1})=-i\frac{nQ}{L}\alpha \varphi_{n}...

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