What is Boundary conditions: Definition and 415 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.

View More On Wikipedia.org
Recent contents Top users
  1. patricio ramos

    Question about initial and boundary conditions with the heat equation

    I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature. When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have...
  2. F

    Boundary conditions for a purely inductive load in an AC circuit

    Hi all, Kirchhoff's equation for this simple circuit is equivalent to \dot I=\frac{V}{L} Where V=V_0 \sin(\omega t). Integrating both sides should give I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c where c is an arbitrary constant (current). Here, most of the derivations I've found simply drop...
  3. Leonardo Machado

    A Boundary conditions for the Heat Equation

    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 $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...
  4. tworitdash

    A Dirichlet and Neumann boundary conditions in cylindrical waveguides

    The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is: F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z} There is no mention of how to solve for the constant A_{mn} . Then, from a paper...
  5. T

    I Equal or larger/smaller versus larger/smaller in boundary conditions

    Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...
  6. maistral

    Boundary conditions for convective heat and mass transfer + wall Temperature

    I am operating via finite differences. Say for example, I have this pipe that contains a fluid. I have the boundary condition at x = x1: k is the effective thermal conductivity of the fluid, T is the temperature of the fluid at any point x, hw is the wall heat transfer coefficient, and Tw is...
  7. H

    A Crank-Nicholson method and Robin boundary conditions

    I have the following PDE I wish to solve: \frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}} With the following boundary conditions: \frac{\partial u}{\partial x}(t,1)+u(t,1)=f(t),\quad u(t,0)=0 Now, I wish to do this via the Crank-Nicholson method and I would naively...
  8. Terrycho

    Partial Differential Equation: a question about boundary conditions

    Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...
  9. maistral

    A Deriving the derivative boundary conditions from natural formulation

    PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...
  10. U

    I Boundary conditions and discontinuity of EM fields

    Premise: everything that follows is done in the frequency domain. Boundary conditions If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields...
  11. B

    I Finding CDF given boundary conditions (simple stats and calc)

    I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
  12. Othman0111

    How to deal with a circular light boundary conditions

    Hi everyone, in the attached file I tried to find the transmitted and the reflected coefficients. I ran into trouble applying the boundary conditions to the linear components of the electric field. Check the outlined boxes and see if they make sense. Thanks
  13. L

    What are the boundary conditions for this plate/ring system?

    Homework Statement [/B] I have a metal disc adhesively bonded at its edges to a piezoelectric ring. The piezoelectric ring vibrates radially which leads to the plate vibrating transversely. I am looking to work out the resonant frequency of the metal disc which I believe will depend on the...
  14. CharlieCW

    Conductor sphere floating on a dielectric fluid

    Homework Statement A conductor sphere of radius R without charge is floating half-submerged in a liquid with dielectric constant ##\epsilon_{liquid}=\epsilon## and density ##\rho_l##. The upper air can be considered to have a dielectric constant ##\epsilon_{air}=1##. Now an infinitesimal...
  15. B

    Green's Function Boundary Conditions

    Homework Statement I am trying to fill in the gaps of a calculation (computing the deflection potential ##\psi##) in this paper: http://adsabs.harvard.edu/abs/1994A%26A...284..285K We have the Poisson equation: ##\frac{1}{x}\frac{\partial}{\partial x} \left( x \frac{\partial \psi}{\partial...
  16. G

    Poisson's equation boundary conditions (electrostatics)

    Hi everyone! I have to solve a problem using Poisson's equation. There are two parallel infinite conductor planes in vacuum. The distance between them is d and they are both kept at a potential V=0. Between them there is a uniform volume density charge \rho_0>0 infinite along the directions...
  17. B

    I Confused about the boundary conditions on a conductor

    In the textbook (attached image) it says that the boundary condition is V=0 at r=R. This creates a correlation that ##B_l=-A_l R^{2l+1}## but the potential at any boundary is continuous so when we take this account, we get. ##B_l=A_l R^{2l+1}## These two clearly contradict each other. I'd...
  18. Phys pilot

    Transform Dirichlet condition into mixed boundary condition

    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...
  19. N

    Need help understanding "Electrostatic Boundary Conditions"

    We are using griffith's 4 edition in my electromagnetic course atm. and there's something I just don't understand about boundary conditions. It says that if we have a surface charge, and we put a pillbox on it, in such a way that half of it extends under the surface charge, and the other half...
  20. B

    Simple Harmonic Oscillator with Boundary Conditions

    How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...
  21. H

    MATLAB Crank-Nicholson solution of 1D heat equation

    I wish to numerically compute solutions of the 1D heat equation using the Crank-Nicholson scheme: The equation is: \partial_{t}u=\partial^{2}_{x}u I use the discretisation: u_{i+1,j}-u_{i,j}=s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1})+s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1}) Where s=\delta...
  22. Zubair Ahmad

    I Boundary conditions for displacement vector D

    Griffith's writes in chapter 7 electrodynamics that D1.a - D2.a = sigma. a. But minus sine comes when we evaluate the dot product first. How does the minus sign occur without evaluating the dot product?
  23. T

    Boundary conditions errantly applied to pressure across flui

    I am trying to decipher if an error occurred in a calculation given in this paper. It is understandable that if two compressible fluids of different uniform densities have a common interface (e.g. Figure 1), then to be in equilibrium and supported against gravity, there must be a pressure...
  24. BookWei

    I Solving Boundary Conditions in 4D Spacetime Volume

    Hi, my classmate asks me an interesting question: For a finite 4D volume in spacetime, its boundary is a 3D close surface. If the 4D volume is a 4D rectangular, the boundary consists of eight 3D surfaces. The boundary condition is specified on these eight 3D surface. Please explain the physical...
  25. Dor

    I What should be continuous at the interface of two materials?

    At the interface between: 1) conductor/conductor 2) conductor/semiconductor (or dielectric) 3) semiconductor/semiconductor (or dielectric/dielectric) What quantity should be continuous? Is it the electrochemical potential, only the chemical potential or is it the electric potential? Since they...
  26. stephen8686

    Introductory QM boundary conditions

    Homework Statement A particle is represented by the following wave function: ψ(x)=0 x<-L/2 =C(2x/L+1) -L/2<x<0 =C(-2x/L+1) 0<x<+L/2 =0 x>+L/2 use the normalization condition to find C Homework Equations ψ(x) must be...
  27. cromata

    I Boundary conditions of a forced oscillator (string)

    -If we have string of length L that has fixed ends, then we can easily find frequencies with which this string can oscillate: We just need to solve wave equation: ∂2y/∂x2=1/c2*∂2∂t2 (c is determined by strings properties (linear density and tension), with Dirichlet boundary conditions...
  28. H

    A Applying boundary conditions on an almost spherical body

    I am solving the Laplace equation in 3D: \nabla^{2}V=0 I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are: V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
  29. B

    Schrodinger equation and boundary conditions

    Hi at all, I'm tring to solve Schrodinger equation in spherically symmetry with these bondary conditions: ##\lim_{r \rightarrow 0} u(r)\ltimes r^{l+1}## ##\lim_{r \rightarrow 0} u'(r)\ltimes (l+1)r^{l}## For eigenvalues, the text I'm following says that I have to consider that the...
  30. Hypercube

    Electrostatic Boundary Conditions

    Hello PF community, I am currently self-studying electrodynamics from Griffiths textbook, and I'm at a point where the book discusses electrostatic boundary conditions. If someone can please check if my reasoning is right. So, as I am approaching an infinite, uniformly charged plane (let the...
  31. Mzzed

    I Boundary Conditions for System of PDEs

    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...
  32. M

    Green's Function with Neumann Boundary Conditions

    Homework Statement [/B] Determine the Green's functions for the two-point boundary value problem u''(x) = f(x) on 0 < x < 1 with a Neumann boundary condition at x = 0 and a Dirichlet condition at x = 1, i.e, find the function G(x; x) solving u''(x) = delta(x - xbar) (the Dirac delta...
  33. Dor

    Steady state boundary conditions between metal/dielectric?

    There are few thing I'm not sure of and be happy for clarifications. In general: at steady state, what are the electric-field,potential, and current boundary conditions between a conductor and a dielectric medium? more specific: a) When dealing with a perfect conductor there exist a surface...
  34. K

    Boundary conditions of a bending plate

    Homework Statement I'm trying to find the boundary conditions for the following problem: A plate with length 2L is placed on supports at x = L/2 and x = - L/2. The plate is deforming elastically under its own weight (maximum displacement bowing up at x = 0). Both ends of the plate are free...
  35. A

    A Fourier Transform for 3rd kind of boundary conditions?

    I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is ## \Gamma \small[ f(x) \small] = \bar{f}(a) =...
  36. J

    A What is the method for calculating the dampening of thermal oscillations?

    Hello, I am attempting to solve the 1 d heat equation using separation of variables. 1d heat equation: ##\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}## I used the standard separation of variables to get a solution. Without including boundary conditions right now...
  37. Marcin H

    Boundary Conditions & finding E

    Homework Statement Find E1, E3, and ps2 Homework Equations boundary conditions The Attempt at a Solution (these are class notes)[/B] I understand how to find E1, but I am a bit confused about the reasoning behind finding E3... Why do we leave the 2x(hat) for E3...? I though that only...
  38. F

    Stationary temperature field inside a house

    Homework Statement House: a room (see figure) has perfectly isolated walls, except the two windows where a convective heat exchange takes place (with the same transfer coefficient). Outside temperature in front of a sun-faced wall-sized panoramic window is T1, while at the back it is...
  39. A

    I How to simulate a lattice with boundary conditions?

    Hello, My question is very simple but I do not have a lot of experience with simulation. I want to write some code to simulate a lattice with boundary conditions and then I will perform calculations with the Hubbard model to find different kinds of properties of interest. I would like to know...
  40. D

    Boundary conditions for a step potential

    Homework Statement A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##. What is the boundary conditions for this eigenfunctions? Find the degeneracy level for the energy, when it is ##E<V_0## Homework Equations Radial equation \begin{equation}...
  41. D

    Boundary conditions with l=0

    The question is basically find the boundary conditions when ##l=0##, for energies minor than 0. Homework Equations $$V(r)=\begin{cases} & 0\text{ $r<a_0$}\\ &V_0\text{ $a_0<r<a_1$}\\ & 0\text{ $r>a_1$}\\ \end{cases} $$ $$...
  42. D

    Boundary conditions for eigenfunctions in a potential step

    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...
  43. M

    Newtonian energy integral and suitable boundary conditions

    I have a (somewhat) strange energy equation which has the following form: KE = A + B W + C \exp(-D W), where A,B,D are known constant, C is an unknown constant to be determined and kinetic and potential energy are given by KE and W respectively with W\equiv W(r) i.e. is a function of...
  44. Ibix

    I Deriving Schwarzschild Interior Boundary Conditions

    I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere. I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the...
  45. R

    Boundary conditions in dielectric problems

    Q) A conducting sphere of radius R floats half submerged in a liquid dielectric medium of permittivity e1. The region above the liquid is a gas of permittivity e2. The total free charge on the sphere is Q. Find a radial inverse-square electric field satisfying all boundary conditions and...
  46. A

    A Boundary conditions on the Euclidean Schwarzschild black hole

    This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} +...
  47. C

    I Laplace equation boundary conditions

    Hi, I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...
  48. 1

    Boundary conditions of a plane wave on a conductor

    Homework Statement Consider a plane monochromatic wave incident on a flat conducting surface. The incidence angle is ##θ##. The wave is polarized perpendicular to the plane of incidence. Find the radiation pressure (time-averaged force per unit area) exerted on the surface. Homework Equations...
  49. henry wang

    Why can we use periodic boundary conditions?

    (Mentor note: moved here from noon homework thread hence no template) I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC). I wanted to justify the use of PBC, so I came up with this: atoms deep inside the crystal sees an...
  50. abilolado

    A Free boundary conditions on vibrating rectangular membranes

    I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges. What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a...
Back
Top