What is Separation of variables: Definition and 171 Discussions
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
When studying the hydrogen atom, given that the potential depends only on the distance and not an any angle, we can do a separation of variables of the wavefunction as the product between a function depending only on the distance between particles (protons and electrons) and a spherical...
I am going through this,
I noted that, i shall have a separation of variables, that leads to
$$\left[\int \dfrac{1}{y(y-1)} dy\right]= \int \dfrac {1}{6} dt$$
and using partial fraction, i then have,
$$\left[\int -\dfrac{1}{y} dy - \int \dfrac{1}{y-1} dy\right] = \int \dfrac {1}{6} dt$$...
Imagine you have a plane wall with constant thermal conductivity, that is the intermediate between two thermal reservoirs:
The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the...
I did a change of variable $$\theta(r,z) = T(r,z)-T_{\infty}$$ which resulted in
$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$
$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$...
I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems.
I list them below:
1.Cartesian coordinates...
Problem:
Solution:
When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials.
The example problem:
This is what I did:
$$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...
using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me...
Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##.
Suppose I have a partial differential equation :
$$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$
where k = constant
Then I do a separation of...
I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...
So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is:
1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system:
-Here I struggle a bit with actually...
Given a probability density distribution ##P(\vec{x})##, for what named distributions is the following true:
\begin{equation}
\begin{split}
P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n)
\end{split}
\end{equation}
I'm having troubles setting up this problem. I know we are to use boundary conditions to determine An and Bn since in this case (a<r<b) neither can be set to 0. I don't know how the given potentials translate into boundary conditions, especially the V3 disk.
If the boundary condition is not provided in the form of electric potential, how do we solve such problem?
In this case, I want to use ##V = - \int \vec{E} \cdot{d\vec{l}}##, but I don't know how to choose an appropriate reference point.
I was solving the van't Hoff equation over an interval ##[T_1 , T_2]##:
The van't Hoff equation
##
\frac{\mathrm{d} \ln K}{\mathrm{d} T} = \frac{\Delta_r H^{\circ}}{RT^2}
##
which can be solved with separation of variables:
##
d \ln K = \frac{\Delta_rH^\circ}{RT^2}dT
##
##\Updownarrow##...
I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...
Is it possible to use separation of variables on this equation?
au_{xx} + bu_{yy} + c u_{xy} = u + k
Where u is a function of x and y, abck are constant.
I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed.
Thank you.
Consider the static field configuration shown in the image. There are three layers: 0 = vacuum, 1 = magneto-optic fluid and 2 = covering shell. Each of these layers have their own permittivity and permeability (ε_i,μ_i) (isotrope). A uniform electric field H_0 = H_0/sqrt(2) * (e_x + e_y) is...
Good Morning
I have a very vague memory of having read (about 40 years ago) that there are only 11 coordinate systems in which the field equations of physics can be separated.
I can no longer be sure if my memory has failed me. But this issue has been in my head for all these years. (Gotta do...
Hey there!
I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I...
I have a PDE which I have solved numerically using a guass-seidel method, but I want to compare it to the analytical solution. I have used separation of variables to get the general solution, but I need help applying it.
The PDE is
(1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω...
Attempted rewriting acceleration, a, in terms of dv/dt and then separating variables to integrate. This didn’t work...
So then I remembered that my gamma factor is also a function of v (!), but then I think I would be required to play around with integration by parts, which seems overly...
When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t).
What is the justification for this?
While separating variables in the Schrodinger Equation for hydrogen atom, why are we taking separation constant to be l(l+1) instead of just l^2 or -l^2, is it just to make the angular equation in the form of Associated Legendre Equation or is there a deeper meaning to it?
Within a cylinder with length ##\tau \in [0,2\pi]##, radius ##\rho \in [0,1]## and angular range ##\phi \in [0,2\pi]##, we have the following equation for the dynamics of a variable ##K##:
$$\left( - \frac{1}{\cosh^{2} \rho}\frac{\partial^{2}}{\partial\tau^{2}} + (\tanh\rho +...
Homework Statement Homework Equations
If i solve the wave equation using separation of variable and laplace tranform. Will i get the same answer ?
The Attempt at a Solution
Homework Statement
Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V0(y) when x=0 iv) V=0 when x app infinity.
I understand and follow this problem (separating vars and eliminated constants) until the potential
is found to be V(x,y) = Ce^(-kx)*sin(ky)
Condition ii...
Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation:
##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...
I've solving some separation of variables exercises, and I have a doubt when it comes to the Laplacian
$$
u_{xx} +u_{yy} =0
$$
I usually have a rectangle as boundary conditions, so I use the principe of superposition and arrive to
$$
\dfrac{X''(x)}{X(x)} = - \dfrac{Y''(y)}{Y(y)} = - \lambda
$$...
Hello! (Wave)
I want to check if the method of separation of variables can be used for the replacement of the following given partial differential equations from a pair of ordinary differential equations. If so, I want to find the equations.- $[p(x) u_x]_x-r(x) u_{tt}=0$
- $u_{xx}+(x+y)...
hello there
Im trying to do a derivation of tsiolkovsky's rocket equation, but i got stuck at the step when i have to use separation of variables (marked with red in the pic), i used maple to solve it, so i could get on with it, but i want to understand what is happening to solve this, so can...
Homework Statement
Let us look at a 3-dimensional potential box. Show, that the wave function in this situation can be written as the product of 3 single-argument functions.
Homework Equations
The 3D Schrödinger equation:
\begin{equation}
-\frac{\hbar^2}{2m} \left( \frac{\partial^2...
Homework Statement
Find the general solution to the differential equation:
Homework Equations
Separation of variables for solving 1st order separable differential equation.
The Attempt at a Solution
Using separation of variables, I can write:
My questions are:
1) Am I correct to...
I was overlooking page 47 of "The Physics of the Manhattan Project" 2.2 Critical Mass: Diffusion Theory, and author Bruce Cameron Reed reported that:
Can anyone explain how Bruce Cameron Reed got from (2.18) to (2.19)
I tried plugging ## N(r,t) = N(r) N(t) ## into (2.18) to get (2.19), but it...
<< Mentor Note -- thread moved from the technical math forums >>
I am getting stuck on this partial differential equation.
Ut = Uxx - U + x ; 0<x<1
U(0,t) = 0
U(1,t) = 1
U(x,0) = 0
Here is my work so far :
U = e-tw + x
gives the new eq wt=wxx
to get rid of boundary conditions :
w=x+W
Wt=Wxx...
I'm self-teaching through Tenenbaum & Pollard's "Ordinary Differential Equations", and for some reason I'm completely stuck on one of the problems, Ch.2, lesson 6, problem #6:
Find a 1-parameter family of solutions for [...] the differential equation:
6) yx2dy-y3dx = 2x2dy.
I didn't have...
Homework Statement
Given two two grounded concentric spherical shells with radii a,b (a<b) and a point charge q between them at a<r=R<b find:
1.The surface charge density of the point charge using the delta function, assume the charge is on the z axis
2.By using the separation of variables...
Homework Statement
The wave equation for ψ(t, x) in 3D is
##\frac{\partial ^2 \psi}{\partial t^2}## - Δ ##\psi =0##
Let ϒ(x) satisfy Δϒ = λϒ where λ<0.
The x is in bold presumably to indicate it is in 3D, so represents also y and z?
Show there is a solution of the form ψ(t, x) = sin(ωt)ϒ(x)...
Homework Statement
The wave equation for ψ=ψ(t,x,y) is given by
##\frac{\partial ^2 \phi}{\partial t^2} - \frac{\partial ^2 \phi}{\partial x^2} - \frac{\partial ^2 \phi}{\partial y^2}##
Use separation of variables to separate the equation into 3 ODEs for T, X and Y. Use the separation...
I just wanted to check something. The equation
∂2φ / ∂x2 + ∂2φ / ∂y2 = sin(xy)
Was given as an example of a separable equation. I can't separate it, and I found online that to use separation of variables the equation should be linear, which this isn't? Is there a way of separating this?
Homework Statement
A potential satisfies ##\nabla^2 Φ = 0## in the 2d slab ## -\inf < x < \inf ##, ##-b < y < b ##, with boundary conditions ## Φ(x, +b) = +V_s(x)## on the top and ##Φ(x, b) = -V_s(x)## on the bottom, where[/B]
##V_s (x)= -V_0 for -a<x<0##
##V_s (x)=+V_0 for 0<x<a##
(a) what...
Homework Statement
A square is made up of four plates with a potential of zero on the top and bottom plates at (x,L/2) and (x,-L/2), and a potential of cos(πy/L)+cos(3πy/L). Find the potential and electric fields inside the square.
The Attempt at a Solution
I start with...
Homework Statement
Sorry to be bringing these in quick succession, but this one really has me perplexed. Is it possible that both the solution manual and I have two different but valid answers?
I don't want to just go assuming that I'm right...but I think by subsuming certain parts of the...
Homework Statement
Homework EquationsThe Attempt at a Solution
I managed to do the first part of the question. But I'm not sure how to find u(x,t) with that initial condition.
The solution says; "since ##u(x,0) = \sum_{n=1}^\infty a_{n}\sin{(n\lambda x)}## Then it follows by linearity that...
Homework Statement
[/B]
A shark will in the direction of the most rapidly increasing concentration of blood in water.
Suppose a shark is at a point x_0,y_0 when it first detects blood in the water. Find an equation for the path that the shark will follow by setting up and solving a...
I am using the text by Farlow to study elementary methods of solving PDEs, and there is a point in his illustration of separation of variables where I am not seeing something. I am clear on everything that comes after and before this point, but after having returned to a certain step a few times...
How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...