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2
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1answer
123 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
1
vote
1answer
113 views

Heat Equation with In-Depth Radiation Exact Solution

I am looking to solve the heat conduction equation in a semi-infinite solid with in-depth radiation on the domain $-\infty < x < 0$. The governing equation of this problem is: $$\rho c ...
0
votes
1answer
52 views

Concentration distribution in a phase separated mixture. Can't get the correct ODEs and boundary conditions

I wish to compute the equilibrium concentration distribution of a binary mixture that has phase separated. I start with writing the free energy as a functional depending of the concentration. I use ...
5
votes
0answers
62 views

What physical phenomena are modelled by Chebyshev equation?

What physical phenomena are modeled by Chebyshev equation? The equation is below $$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y ~=~ 0 .$$ I could not find it in Wikipedia or in Google (at ...
4
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0answers
104 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
3
votes
0answers
58 views

Non-linear Wave Equation - Numerical Methods

Motivation: I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where ...
3
votes
0answers
95 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving Schrodinger equation, such as tasks about particle in Morse potential, Poschl-Teller potential and many others, we usually find an approximations (lets call them as ...
2
votes
0answers
23 views

What is the relationship between the Boltzmann Transport Equation and the Navier-Stokes Equation?

What is the relationship between the Boltzmann transport equation and the Navier-Stokes equation? Can the Navier-Stokes equation be derived from a moment of the Boltzmann transport equation?
2
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0answers
34 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol ...
2
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0answers
86 views

Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting ...
2
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0answers
44 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
158 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 ...
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0answers
19 views

Physical explanation for dependence of initial conditions when solving differential equations using NDSolve

I'm checking some results in this [paper] and I'm currently having some issues with solving a set of differential equations (section 2 and 3.1-3.2 in the paper). I'm finding values to depend on an ...
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0answers
41 views

Is there a second order differential wave equation that only allows a finite set of discrete eigenvalues?

I tried constructing a second order differential wave equation that only allows a finite set of discrete eigenvalues by using the power series expansion such as \begin{align} A_{j+2} = ...
1
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0answers
226 views

A general solution to continuity equation

Let us write the standard continuity equation $$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{\jmath} = 0.$$ Should the relation $\vec{\jmath} = \rho \vec{v}$ be considered as a general ...
1
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0answers
64 views

Derivation of cylindrical line heat source problem?

I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is ...
1
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0answers
72 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
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0answers
33 views

General boundary condition for 1D heat equation

I'm studying from Numerical Solution of Partial Differential Equations by K.W.Morton and D.F.Mayers (Amazon link). I'm confused with general boundary conditions. Could someone give me a clue? For ...
0
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0answers
19 views

Locus of a moving mass point

Two very small mass particles $m_1$, $m_2$ are connected by a $2l$ long, infinitely soft and inelastic thread without mass. The initial condition of the system before being freely released is as in ...
0
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0answers
27 views

Calculate the potential induced by beam

This exercise is from Plasma Physics and Fusion Energy by Freidberg. A cylindrical conducting vacuum chamber of radius $r=a$ is filled with a uniform plasma of density $n_0$ and temperature ...
0
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0answers
36 views

characterising basins of attractions for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
0
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0answers
76 views

Compatibility between solutions of explicit Maxwell equations vs. wave equation?

When trying to solve for the allowed propagation frequencies in a cylindrical waveguide, I approached the problem by solving the wave equation for all three components of $\bar{E}$, and subsequently ...
0
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0answers
30 views

Looking for Solutions to Symmetric Potential

I'm a little confused on the basic method of finding a separable solution to a give potential distribution. If we have a symmetric potential, say it hits zero and $-a$ and $a$, constituting two sides ...
0
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0answers
37 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
0
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0answers
284 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
0
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0answers
82 views

Boundary conditions for 2D helical waveguide

I'm interested in looking at standing wave solutions for the wave equation on a 2D annulus, with the twist that the annulus is "streched" in to a helix in 3D, but so that the rings themselves are ...
0
votes
0answers
124 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 ...
0
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0answers
90 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 ...