What is Boundary conditions: Definition + 416 Threads

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.

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1. I Dirichlet problem boundary conditions

The Dirichlet problem asks for the solution of Poisson or Laplace equation in an open region ##S## of ##\mathbb R^n## with a condition on the boundary ##\partial_S##. In particular the solution function ##f()## is required to be two-times differentiable in the interior region ##S## and...
2. A Rigid body motion (RBM) transformation

Above are pictures of a problem in mechanics. An elastic body occupying domain ##\Omega## is supported below with a fixed support and from above with a rigid body. The following calculations aim to express the movements of the points located at the interface ##\Gamma_F## between the rigid and...
3. I Lagrangian density of a linearly elastic string under gravity

This was inspired by this:Dropping an extended Slinky -- Why does the bottom of the Slinky not fall?. There is that famous demonstration of dropping a slinky, and the bottom of the slinky does not move until the center of mass reaches the bottom. I was trying to figure out how hard are the...
4. A Why is "method of image current" valid in magnetostatics?

Hi wise folks, I am working through Jackson problems, and have just encountered problem 5.17: It is pretty straightforward to show that the given image current distributions will satisfy the boundary conditions (both tangent and normal) at the ##z=0## plane. But my question is actually: "why...
5. I The boundary conditions at infinity

Section 5 (pg. 29) of the Michel Janssen's paper EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920(*) says: 1. From the above I understand that the application of general relativity to an infinite universe was considered problematic. 2. On the other hand, I understand that it is currently...
6. A Boundary conditions for variable length bar

Suppose I'm looking at a bar of length L(t) in 1D and I have the conservation of mass: \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0 In order to make things easier, I make the change of variable x'=x/L(t) so that in this frame of reference, the length remains constant...
7. A Two degrees of freedom in GR after boundary conditions specified?

I read there are 2 degrees of freedom in GR after boundary conditions specified. Does that mean 2 equations are enough for EFE equivalent? Those two seem like the amplitude and a phase.
8. Fortran Need help with Jacobi relaxation method for Dirichlet boundary conditions

program r_jacobi implicit none !!!!Variables!!! real*8 V, V_1, V_2, Lx, Ly integer n ,i , j, k, nx, ny real*8, allocatable :: arrx(:), arry(:), phi(:,:,:) real*8 x, xi, xf, y, yi, yf, dx, dy real*8 d, q, bx, by V=1 V_1=V V_2=-V Lx = 2 Ly = 1 nx = 200 ny = nx/2...
9. A The boundary conditions in reference to Laplace's equation

We have inhomogenous dirichlet boundary conditions (well understood)....the laplace equation is a steady state equation and we can clearly see that in 2D..it will be defined by 4 boundary conditions and NO initial condition...having said that; kindly have a look at the continuation below... I...
10. Checking assumptions in boundary conditions of double well system

The idea here (as I'm told) is to use the boundary conditions to get a transcendental equation, and then that transcendental equation can be solved numerically. So I'm making a few assumptions in this problem: 1. The potential ##V(x)## is even, so the wavefunction ##\psi(x)## is either even or...
11. A Solve the heat equation having Dirichlet boundary conditions

I am going through these notes...they are pretty easy to follow. I would like more insight on the initial condition. In this problem, (attachment below), i guess the choice of initial condition is convenient as its easier to plug in the values of ##n=2## and ##b=3## (highlighted on the...
12. I Query on Electromagnetic Theory (Dielectric Boundary Conditions)

The given question from Electromagnetic Theory (which is based on Dielectric Boundary Conditions) is as follows: Interface b/w two dielectric medium has a surface charge density (suppose xyz C / (m ^ 2) ). Using boundary condition find field in 1 (relative permittivity =xyz) if field in 2...
13. MATLAB No End" Boundary Conditions in Wave on a String

How did they impose boundary conditions here if "no end" is selected? Here: https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html I would like to do the same thing without changing the wave equation of the string.
14. MATLAB Boundary conditions in the resolution of a PDE with the FFT method

How to impose boundary conditions when solving a PDE with fft? For example here: If I copy this code I get periodic boundary conditions. Thank you
15. Engineering Hand tool boundary conditions - Forces determination

Hello, I' m trying to make a linear static analysis (Finite Element Analysis) on the following hand tool. I want to determine the boundary conditions. In order to do that I have decided to use a force couple to represent the forces that a bolt exerts on the jaws of this spanner. Despite using...
16. Boundary Conditions for an infinite rectangular pipe

Does setting up the problem symmetrically on this axis and the boundary conditions applied make sense? I don't believe I will have a problem solving for the potential inside, but i just want to make sure I have my B.C and axis correct before proceeding. EDIT: Or should this be a 2-D lapace...
17. Evaluating the boundary conditions for a rectangular pipe

I have attached an image of the pipe in the attachmnts. The pipe is parallel to z-axis form (-∞,∞) and sides of length a. So my boundary conditions for this problem are as follows 1.) V=0 at y=0 2.)V=0 at y=a 4.)∂v/∂x=0 @ x=0 3.)V0 @ x=a I am a little confused on the fourth boundary...

19. I Is the scalar magnetic Potential the sum of #V_{in}# and ##V_{out}##

Hi, I'm wondering if I have an expression for the scalar magnetic potential (V_in) and (V_out) inside and outside a magnetic cylinder and the potential is continue everywhere, which mean ##V^1 - V^2 = 0## at the boundary. Does it means that ##V^1 - V^2 = V_{in} - V_{out} = 0## ?
20. Error when creating boundary conditions (OpenFOAM)

Let's give some context. I created a Mesh via snappyHexMesh and next I am setting the boundary conditions (BCs). First I copied the patches folder "createPatchDict" to my system's folder via the command "cp \$FOAM_ETC/caseDicts/mesh/manipulation/patches/createPatchDict ./system" Taking into...
21. Boundary conditions ##\vec{B}## and ##\vec{H}##

When asking for boundary conditions I'm wondering if this is enough in this situation to give ##\vec{\nabla} \cdot \vec{B} = 0 , B_{2\perp} - B_{1 \perp} = 0## ##\vec{\nabla} \cdot \vec{H} = - \vec{\nabla} \cdot \vec{M}, H_{2\perp} - H_{1 \perp} = - (M_{2\perp} - M_{1 \perp})## ##\vec{\nabla}...
22. Boundary conditions (E and D) for a dielectric sphere

Since there is no free charge ##\int_S \vec{D} \cdot d\vec{a} = 0## and ##\rho_f = 0## ##\sigma_f = 0## ##\vec{nabla} \cdot \vec{P} = 0## since P is a constant ##\rho_b = - \vec{nabla} \cdot \vec{P} = 0## For a simple surface we can find the boundary conditions for ##\vec{E}## using a Gauss'...
23. I Non-Homogeneous Robin Boundary conditions and Interpretations of Signs

I have been solving the constant coefficient 1D advection-diffusion equation ##\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} = D\frac{\partial^2 c}{\partial x^2}## on ##0<x<L,t>0## with a variety of robin BC's. Namely $$vc + D\frac{\partial c}{\partial x} = J^f ~~at~~ x=L$$...
24. A Solving Laplace's equation in polar coordinates for specific boundary conditions

Hello everybody, Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions...
25. Confused About Boundary Conditions for ##y## and ##x##

I am going through this notes, i can follow quite well...my only issue is on the highlighted part...i thought that we had two boundary conditions for ##y## ( of which one of them is non homogenous) and two boundary conditions for ##x##( of which both are homogenous)...kindly clarify on this part...
26. Boundary conditions in Ansys

Hi, I have one question related to the boundary conditions I should apply in a Static Structural simulation for the following support. The support is subjected to the following loading conditions shown below...
27. I Unraveling Laplace's Equation: Exploring Valid Domains and Boundary Conditions

Hi! This thread might well be similar to: https://www.physicsforums.com/threads/thread-about-jacksons-classical-electrodynamics-3rd-edition.910410/ I'm self-studying Vanderlinde and having a great time. However, I think that I am conflating and confusing many different things. Let me just ask...
28. Boundary conditions for 2 ropes fixed to a massless ring with a damper

Hi, I'm not quite sure if I'm correct. I need to find the boundary conditions for 2 ropes ##T_1 \mu_1, T_2 \mu_2## fixed at ##x=0## to a massless ring with a massless damper of force ##F_d - -bv_y## Here what I think, since the ring and the damper is massless ##\sum F_y = 0##. Thus, ##-T_1...
29. Periodic boundary conditions -> Shouldn't supports hinder all motion?

Hello everyone, I am currently trying to understand periodic boundary conditions for the mechanical investigation of mechanical properties of a RVE. I found a good video explaining the theory behind it: But something is unclear to me: At the above linked time step, the individual conical...
30. Unusual boundary conditions in FEA software

For a 2D problem with unknown displacements u(x,y) and v(x,y), is it allowed to give such a set of BCs u(0,y)=1 and vy(0,y)=0, the former being a displacement BC, the latter being a force BC (vy is the y strain)? How is this implemented in FEA software?
31. Set up boundary conditions for a simple elasticity problem

[Mentor Note -- Thread moved to the ME forum to get better views] Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by ##B=[0,b] \times [0,b] \times [0,h]##. The professor gave me the following task: Of course there can be many...
32. A Ising model open chain and periodic boundary conditions

One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?
33. Is d'Alembert's Formula Correct for Neumann Boundary Conditions in PDEs?

Hi all, I was hoping someone could check whether I computed part (4) correctly, where i find the solution u(t,x) using dAlembert's formula: $$\boxed{\tilde{u}(t,x)=\frac{1}{2}\Big[\tilde{g}(x+t)+\tilde{g}(x-t)\Big]+\frac{1}{2}\int^{x+t}_{x-t}\tilde{h}(y)dy}$$ Does the graph of the solution look...
34. A Implementing FDM Boundary Conditions at a Red Point

in Finite Difference Method (FDM), the boundary conditions can be implemented by applying the continuity of parallel component of magnetic field intensity. when it comes to the interface of two areas, it is done at ease, but consider this case at the red point: in FDM we exactly require on...
35. Solve PDE Uxx+Uyy=-2 with Boundary Conditions

ial this is the question.

37. Python Tackling Boundary Conditions in Python (Griffins Example)

How to run a numerical simulation of Laplace equation if one of the boundary condition is like this: $$V(x,y) = 0 \text{ when } x \to \infty$$ I am trying to use Python to plot the solution of this Example 3.5. in Griffins EM
38. Boundary conditions of linear materials

The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$\vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...
39. I Boundary conditions for TM and TE waves

Theta in the incident angle Phi is the refraction angle '' denotes everything that propagates to the other medium, that is, everything related to refraction ' denotes the reflection in the original medium I am rather confused, would appreciate any help. I see the second equation of TE is...

47. I Boundary Conditions For Modelling of a Fluid Using Euler's Equations

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...
48. Heat Equation with Periodic Boundary Conditions

I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
49. I The density of states independent of Boundary Conditions

Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...
50. RVE Periodic Boundary Conditions in ANSYS Workbench Modal Analysis

Hi I have a project regarding micromechanics of composites. I'm starting my analysis on the Fiber Matrix RVE. Right now I'm trying to find the natural frequency of the unit cell. The Unit cell has some unique geometry which I will keep on changing to see how natural frequency changes. I have...