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. M

    Boundary Conditions and Hilbert Space

    Hey all, Last year, I took my university's undergraduate QM sequence. We mainly used Griffiths' book, but we also used a little of Shankar's. Anyway, I decided to go through Shankar's book this year, in a more formal treatment of QM. After the first chapter, I already have some questions that...
  2. D

    Solving Light Refraction Boundary Conditions: Find T^-1

    Can someone please help me with this question: Light with frequency \omega in media 1 ,with refractive index n_{1} , is incident (normal) to an interface of media 2, with refractive index n_{2}, and then is incident on a second interface with refractive index n_{3}. Using boundary conditions...
  3. W

    Diff EQ with the boundary conditions

    I took an ODE course last year, but I seem to have forgotten some stuff. I need to solve this equation: \frac{d^2u}{dt^2} + {\omega}^2u = f_osin({\mu}t) with the boundry conditions: u(0) = 0, du/dt(0) = 0 When I tried to solve the homogenenous equation first, I got...
  4. S

    Appropriate boundary conditions ?

    "Appropriate boundary conditions"...? I am stumped by a question in my Electromagnetism asignment that asks, after determining the potential (V) and electric field (E) of a hollow conductive sphere containing a point charge system, to "show that E and V satisfy the appropriate boundary...
  5. A

    Electrostatic boundary conditions

    Im having trouble following how this is derived: The normal component of the electric field is discontinuous by an amount sigma/epsilon_0 at any boundary (when you cross a continuous surface charge). They talk about taking a little box so that the surface integral E dot da = 1/epsilon_0 * sigma...
  6. M

    Boundary Conditions for Waves in Joined Strings with a Knot of Mass

    I'm given the fact that two strings under tension T are joined by a knot of mass m... I'm supposed to find the appropriate boundary conditions. I know that the tensions are the same in both ropes and that the boundary will be continuous. I know the "trick" in this problem is knowing the...
  7. M

    Show that the boundary conditions X(b)=wX(a) +zX'(a)

    I need a little help getting started here, Show that the boundry conditions X(b)=wX(a) +zX'(a) and X'(b) = yX(a) + dX'(a) on the interval a<=x<=b are symmetric if and only if wd-zy=1 i know that the a set of boundries are symmetric if f '(x)g(x) - f(x)g'(x) = 0 evaluated at x=a and...
  8. Clausius2

    Solving z(r) Equation with Boundary Conditions

    I am looking for the general solutions of this equation in z(r) If someone remembers well, this equation arises in surface tension physics. z(r)=\frac{1}{r}\frac{d}{dr}\left[\frac{z_r r}{(1+z_r^2)^{1/2}}\right] subject to the boundary conditions z_r(0)=z_{ro} and z(\infty)=0 I only...
  9. S

    Reconciling the Use of Periodic Boundary Conditions in Solid State Physics

    I'm taking solid state, and again and again we use the periodic boundary conditions, that the wavefunction should be unchanged by displacements of the length of the sample, L (assume 1D for simplicity). The argument was that the surface is so far away that it shouldn't have an effect on the...
  10. B

    Boundary conditions for charged cylinder

    Hello, Charge density \sigma(\phi) = k \sin 5\phi (where k is a constant is glued over the surface of an infinite cylinder of radius R with axis along the z-direction. Find the potential inside and outside the cylinder. Two things I'm having trouble with: 1. Is the potential of an...
  11. P

    Imposing boundary conditions on a string

    I need to know how different boundary conditions on the DE representing a string under a force can be physically implemented. For example, if you need y(0)= 0, just tie the string to y=0 at that end. If you need y'(0)=0, attatch it so that it can freely slide up and down a pole at x=0. But...
  12. N

    What's the difference between initial conditions and boundary conditions?

    As the thread title says: What's the difference between initial conditions and boundary conditions? Thanks in advance for helpful replies. :)
  13. O

    Fresnel Equations Boundary Conditions

    I'm having trouble with the meaning of the boundary condition in the derivation of fresnels quations, namely that the component of E tangental to the surface is continuous across the boundary. My trouble is, what physically does this corresspond to. Is it something to do with the divergence...
  14. L

    Green function and the boundary conditions

    Hello there, I am glad that I found this forum. Because I have a little bit trouble with theoretical physics. The problem is the Green function in theoretical electrodynamic. I try to understand the difference between the Dirichlet Condition and the Neumann Condition. I understand...
  15. Norman

    Solving Electric Field Boundary Conditions Across a Dipole Layer

    This is a homework question so please do not just tell me the answer, but please point me in the right direction. A dipole layer, D(y,z), exists on the plane x=0. Find the boundary conditions (discontinuities, if any) for [phi](x,y,z), E_x(x,y,z), E_y(x,y,z), and E_z(x,y,z) across the...
Back
Top