What is Boundary condition: Definition and 136 Discussions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.
for the boundary conditions for this problem I understand that Electric field and Electric potential will be continuous on the boundaries.
I also know that I can set the reference point for Electric potential, wherever it is convenient. This gives me the fifth boundary condition. I am confused...
The original differential equation is:
My solution is below, where C and D are constants. I have verified that it satisfies the original DE.
When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
Hey, I have a really short question about electrostatics.
The boundary conditions are :
\mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} ,
\mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}.
My question is what is...
Hi everyone,
I'm trying to understand the rationale behind the boundary condition for the problem "Finite bending of an incompressible elastic block". (See here from page 180).Here we have as Cauchy Stress tensor (see eq. (5.82)):
##T = - \pi I + \mu (\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r...
I really have no idea as to how to attack the problem in the first place. I am here to ask for some generous help on how to start. The figure is shown below for reference.
The Hawking-Hartle no boundary condition is well known. The authors considered a many worlds/histories model considering a sum over all compact euclidean metrics.
But are there any models or theories that consider a sum over all possible metrics or boundaries?
And finally, if all possible...
Hi everyone,
I am trying to solve the 1 dimensional diffusion equation over an interval of 0 < x < L subject to the boundary conditions that C = kt at x = 0 and C = 0 at x = L. k is a constant. The diffusion equation is
\frac{dC}{dt}=D\frac{d^2C}{dx^2}
I am using the Laplace transform method...
Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some...
Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...
Hi everyone!
I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example,
$$
u_t=k u_{xx},
$$
$$
u(t,-1)=\alpha,
$$
$$
u(t,1)=\beta,
$$
$$
u(0,x)=f(x),
$$
$$...
Hi! I want to use Euler's equations to model a 2 dimensional, incompressible, non-viscous fluid under steady flow (essentially the simplest case I can think of). I'm trying to use the finite difference method and convert the differential equations into matrices to be solved using MATLAB. I set...
Hello guys.
I am studying the heat equation in polar coordinates
$$
u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta})
$$
via separation of variables.
$$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$
which gives the ODEs
$$T''+k \lambda^2 T=0$$
$$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...
Summary: Questions about the Multiverse hypothesis and the 'No boundary' conditions approach in cosmology
I have some questions about James Hartle and Stephen Hawking's 'No-boundary' proposal:
- In their approach multiple histories would exist. These histories could yield universes with...
the image is given here along with some numerical information:
Now I know that the formula for the electric field in a capacitor is given as:
$$E = \frac{V}{d}$$
which I can use to obtain the three following fomulas:
$$E_1 = \frac{V_1}{d}$$
$$E_2 = \frac{V_2}{d}$$
$$E_3 = \frac{V_3}{d}$$
where...
I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic.
I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which...
Hi, I'd like to be clarified regarding the general natural/Neumann boundary condition for a PDE.
1. The natural boundary condition is generally defined as:
(1)
and can be expressed as, according to this resource:
(2)
But apparently, according to...
I am trying to understand Aubry-Andre model. It has the following form
$$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$
This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with...
Hi everyone,
I'm working through the boundary conditions and I could not figure out what to do with the last boundary condition (when z=L)
I know that the values for K are:
How so?
1. Homework Statement
A hollow right angle cylinder of radius a and length l. The sides and bottom are...
I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
Hi! I have a question related to boundary condition in a one dimensional beam subject to compression and traction efforts.
In my class notes I have the following: If we consider a 1D beam of length L which is fixed at x=0 and subject to an effort F at x=0 we have the following boundary...
Hello,
If I have a homgeneous linear differential equation like this one (or any other eq):
$$y''(x)-y'(x)=0$$
And they give me these Dirichlet boundary conditions:
$$y(0)=y(1)=0$$
Can I transform them into a mixed boundary conditions?:
$$y(0)=y'(1)=0$$
I tried solving the equation, derivating...
Homework Statement
Two magnetic materials are separated by a planar boundary. The first magnetic material has a relative permeability μr2=2; the second material has a relative permeability μr2=3. A magnetic field of magnitude B1= 4 T exists within the first material. The boundary is...
Homework Statement
I am confused on how it's using the surrounding temperature minus the surface temperature as its the other way around in the Newton's law of cooling, Doing that would change the sign of convection right? I don't see the reason to do that, since if left side is hotter, then...
Homework Statement
[/B]Homework Equations
shear bending diagram
The Attempt at a Solution
May I ask for supports like these, if shear force V= 0 or not when x=L?
From the shear diagram, it is a shape increase from -0.5P to 0 there.
As the shear go back to zero is owing to the reaction force...
Homework Statement
Suppose we have the standard rectangular potential barrier in 1D, with
$$
V =
\left\{
\!
\begin{aligned}
0 & \,\text{ if } x<0, x>d\\
V_0 & \,\text{ if } x>0,x<d\\
\end{aligned}
\right.
$$
The standard approach to solve for tunneling through the barrier is to match the...
I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter.
The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...
Homework Statement
I am having an issue, not with the maths but with the boundary conditions for this question.
A bar 10 cm long with insulated sides, is initially at ##100 ^\circ##. Starting at ##t=0##
Find the temperature distribution in the bar at time t.
The heat flow equation is...
Hi PF!
Given a fluid/fluid interface ##F(t,x,y,z) = 0## the kinematic boundary condition states ##DF/Dt = 0##. Given ##y=F## the text states ##v = \partial_tF+u\partial_xF+w\partial_zF##. How is this possible? I thought ##DF/Dt = 0 = \partial_tF+u\partial_xF+v\partial_yF+w\partial_zF## and...
Homework Statement
I'm trying to keep the post brief and will post more info if needed. But I am trying to understand how the value of two "A" constants were found. This is from Griffiths Electrodynamics.
In this part of the problem, I am given a boundary condition that is a function of theta...
Homework Statement
So I have an equation V = Ae(kx)+Be(-kx)
And boundary conditons V= V0 when x=0 and V= 0 when x=b
2. Homework Equations
I have solved ones where v=0 at x=0 where it nicely simplifies as the exponentials =1 and the Coeffecients A=-B which leads to a sinh function and I...
Homework Statement
I have to calculate the stationary field inside a room.
Homework EquationsThe Attempt at a Solution
I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Eeknx+Fe-knx)cos(kny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary...
The question is basically find the boundary conditions when ##l=0##, for energies minor than 0.
Homework Equations
$$V(r)=\begin{cases}
& 0\text{ $r<a_0$}\\
&V_0\text{ $a_0<r<a_1$}\\
& 0\text{ $r>a_1$}\\
\end{cases}
$$
$$...
1. Homework Statement
A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##.
How is the general form for the eigenfunctions?
What is the boundary conditions for this eigenfunctions?
Find the degeneracy level for the energy, when it is ##E<V_0##
2...
Hello gents,
Q:/ what is the reason for letting the inner boundary condition = (-1) when solving the radial flow of infinite form of diffusivity equation, and i would like to know what will happened if i didn't equate it with (-1).
as in the attached pic:
https://i.imgur.com/AVesmHM.png
The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time.
xsize = 10; % Model size, m
xnum = 10; % Number of nodes
xstp =...
Homework Statement
[/B]
Inside a sperical dielectric mass there is a electric dipole on the center of the sphere. The sphere has radius a. This dieletric sphere is inside and on the center of a conductive spherical shell of radius b. The problem asks to find the potentials and then the...
hi guys, i am new user on ansys
i have task from my lecturer. it is about static test for seat bus. i try simulate the seat bus on ansys statical structural. you can see my model on attachment (untitle1.png).
at that picture, there are 2 bars will push the seat back. the stiffness beahvior of...
So I have this problem, taken from Kraus's heat transfer book.
So deriving the computational molecule, the conditions for (3.251a), (3.251b) is a bit of a no brainer. The issue I am having is about the boundaries for (3.251c) and (3.251d). This is actually the first time I have seen this...
Hello everybody. I'm about to take a final exam and I've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation:
∂u/∂x = ∂u/∂t
Plus i have a condition...
Hey everyone
Just a picture of my configuration.
The assumption here is $$\epsilon_a,\epsilon_b,\epsilon_c$$ are different from one another. Really the interest of this problem is to find the scalar potential $$\phi$$, such that $$\nabla^2 \phi = 0$$.
So now my question, about jump...
I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function.
I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...
Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at r=R ?
Hi everyone,
This is my first time posting here
I am looking to get some help with Abaqus,
I wish to compare two models and find the residual stress which causes a original model to deform to the other. The deflection between the two models can be calculated by other software.
I plan to do...
Dear all,
I made a cad of floating structure (the frame only), figure attached below. It should be located in the water and moored, so it can't go anywhere. The CFD simulation was done. So I have fluid force on the structure. Now I want to do mechanical analysis of the structure by apply the...
Hello everyone,
The boundary condition :
P=0, z=ζ
is very common when studying irrotational flows. When cast with the Bernoulli equation, it gives rise to the famous dynamic boundary conditionn, which is much more convenient :
∂tφ+½(∇φ)2+gζ=0, z=ζ
But what happens if the motion is rotational ...
If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have
\psi(x) = A\sin(kx) + B\cos(kx)
with boundary conditions \psi(x) = \psi(x+L)
In the fixed boundary case, we had \psi(0) = 0 which meant B=0 and...