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.

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  1. Spinnor

    Time independent Klein–Gordon equation with boundary conditions.

    Say we consider the time independent Klein–Gordon equation, see: http://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation Lets impose the following boundary conditions, the function is zero at infinity and on some small ball of radius R centered on some origin the function is some...
  2. H

    Alternative boundary conditions - Thomas-algorithm

    Alternative boundary conditions -- Thomas-algorithm Hello, I have to solve a diffusion equation: MatrixL * Csim(:,i+1) = MatrixR * Csim(:,i) + BoundaryConditions where Csim = concentration, j = location, i = time. Boundary conditions are of type Dirichlet (Csim = 5 at j = 1, Csim = 0...
  3. R

    Fourier transform for loaded string with periodic boundary conditions.

    Homework Statement So we have a string of N particles connected by springs like so: *...*...*...*...* A corresponding Hamiltonian that looks like: H= 1/2* \Sigma P_j^2 + (x_j - x_(j+1) )^2 Where x is transverse position of the particle as measured from the equilibrium position, and...
  4. N

    Steady State 2-D Heat Equation with Mixed Boundary Conditions

    Homework Statement I am trying to solve the Laplacian Equation with mixed boundary conditions on a rectangular square that is 1m x 1m. Homework Equations \nabla2T=0 .....T=500K ....________ ....|@@@@| T=500K...|@@@@|...T=500K ....|@@@@| ....|______.| ....Convection ....dT...
  5. Y

    What is the magnetic boundary conditions between air and copper?

    I understand \vec J_{free} only exist on boundary surface of perfect conductors. Copper is close enough and have surface current. Also copper is paramagnetic material which implies \mu_{cu} = \mu_0 or very very close. In order to find the exact angle of the of the magnetic field inside the...
  6. R

    Boundary Conditions for Pressurized Cylinder in FEM

    boundary conditions for pressurized cylinder in fem?
  7. X

    Solving for constants given boundary conditions

    Homework Statement http://img843.imageshack.us/img843/3515/11193469.png Homework Equations The Attempt at a Solution [PLAIN][PLAIN]http://img801.imageshack.us/img801/4829/scan0001i.jpg An upload of my attempt to solve the problem. Not sure to interpret the results. A = B...
  8. D

    Solving Wave Equations with Boundary Conditions

    Homework Statement http://img685.imageshack.us/img685/5585/63862334.png Homework Equations -The Attempt at a Solution y_1(0,t)=y_2(0,t) \longrightarrow 1+\frac{B}{A}e^{2i \omega t} = \frac{C}{A} y_1_x(0,t)=y_2_x(0,t) \longrightarrow 1+\frac{B}{A}e^{2i \omega t} =\frac{k_2}{k_1}...
  9. N

    Sturm-Liouville Like Equation with Boundary Conditions on Second Derivative

    Hello, I am facing a diffusion equation.. \frac{du(x,t)}{dt} = D \frac{d^2u}{dx^2} .. with slightly exotic boundary conditions: u(0,t) = 0 \frac{d^2u(a,t)}{dx^2}+ \alpha \frac{du(a,t)}{dx} = 0 I expected the solution to be relatively easy to find, since separation of variables quickly...
  10. O

    Differential Equation with Boundary Conditions II

    Homework Statement d^2T/dx^2 + S^2*T+B=0 Boundary Conditions: dT/dx=0 @ x=0 T=T_2 @ x=L Homework Equations The Attempt at a Solution I think you either have to make some type of substitution or find the roots and do it that way. P.S. This is assignment is a review of diff...
  11. O

    Differential equuation with boundary conditions

    Homework Statement d^2T/dx^2+S/K=0 Boundary Conditions T=Tsub1 @ x=0 and T=Tsub2 @ x=L Homework Equations The Attempt at a Solution d^2T/dx^2 = -(S/K) <--- intergrate to get dT=-(S/K)dx+ C1 <--- intergrate to get T=(-S/K)x+c1+c2 apply both boundary conditions to get...
  12. D

    How to deal with this Neumann boundary conditions?

    hi all, I am trying to solve this PDE by separation of variables, it goes like this: \frac{\partial u}{\partial t} = \alpha\frac{\partial ^2 u}{\partial z^2} for 0\leq z\leq infty the initial condition I have is: t=0; u = uo. the boundary condtions: z=0; \frac{\partial...
  13. W

    Magnetostatic field: solution to Poisson's equation and Boundary Conditions

    How to derive boundary conditions for interfaces between ferromagnetic material and air? Please see the attached figure. Any hints will be greatly appreciated!
  14. H

    Is the topological insulators a result of boundary conditions with SO coupling ?

    Hi, these days I have been trying to understand the essentials of the so-called topological insulators (TBI), such as Bi2Te3, which seem very hot in current research. As i understand, these materials should possesses at the same time gapped bulk bands but gapless surface bands, and spin-orbit...
  15. C

    Maxwell equations with time-dependent boundary conditions

    Hi folks, I was wondering how to code a Maxwell solver for a problem with time-dependent boundary conditions. This is not my homework, but I love programming and would like to implement what I learned in my physics undergrad course to get a better understanding. More precisely, if I have an...
  16. H

    Stiff spring boundary conditions?

    Let me see if I can line out my question a little better to hopefully get some sort of input. I am trying to understand where a type of boundary condition approx. called stiff spring BCs. I have, among a couple of other examples, an example comsol "dialysis" model that uses it. I have been...
  17. N

    Wave equation boundary conditions at infinity

    Are there general boundary conditions for the wave equation PDE at infinity? If there is, could someone suggest a book/monograph that deals with these boundary conditions? More specifically, if we have the following wave equation: \[ \nabla ^2 p = A\frac{{\partial ^2 p}}{{\partial t^2...
  18. T

    Boundary conditions, Sturm-Liouville, & Gauss Divergence

    Homework Statement I'm getting through a paper and have a few things I can't wrap my head around. 1. In defining the boundary conditions for a membrane (a function of vector 'r'), the author claims that for a small displacement (u) and a boundary movement (f), the boundary condition can be...
  19. G

    Boundary conditions in String Theory

    I have this, probably quite simple, problem. In the RNS superstring, when varying the action, we obtain in general a term \int d\tau [X'_{\mu}\delta X^{\mu}|_{\sigma=\pi}-X'_{\mu}\delta X^{\mu}|_{\sigma=0} + (\psi_+ \delta \psi_+ - \psi_- \delta \psi_-)|_{\sigma=\pi}-(\psi_+ \delta \psi_+ -...
  20. P

    Boundary Conditions for 1D heat flow in Wire with source

    I'm trying to understand how to set up the problem of a 1D wire that is insulated at one end and has a heat source at the other. I know the heat law, from my textbook: du/dt = B d2u/dx2 + q(x,t) 0 < x < L, t > 0 Where q(x,t) is the source of heat. The problem is, I want the heat...
  21. M

    Wave equation with initial and boundary conditions.

    Hallo Every one, Homework Statement y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct) Boundary Condition: y(0,t)=y(2pi,t)=(1/c)sin(ct) fot t>0 Initial Condition : y(x,0)=sin(x),( partial y / Partial t ) (x,0) = cos(x) for 0<x<2pi show that y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct)...
  22. B

    Cauchy Boundary Conditions on a Wave

    Homework Statement So using the D'Alembert solution, I know the solution of the wave equation is of the form: y(x,t) = f(x-ct) + g(x+ct) I'm told that at t=0 the displacement of an infinitely long string is defined as y(x,t) = sin (pi x/a) in the range -a<= x <= a and y =0...
  23. A

    Neumann boundary conditions on S^1/Z_2

    Hello everybody, I've been puzzling over something (quite simple I assume). Take S^1. Now consider the action of a Z_2 which takes x to -x, where x is a natural coordinate on the cylinder ( -1< x <1). Now we mod out by this action. The new space is an orbifold: smooth except at x=0. It...
  24. P

    What would be my boundary conditions? Heat Equation

    1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celcius. Find the...
  25. P

    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    Homework Statement The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation: \frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0 Separate the variables and find T everywhere on a square flat plate of sides S with...
  26. T

    PDE-Heat Equation with weird boundary conditions help

    Homework Statement Consider the Heat Equation： du/dt=k(d2u/dx2), where d is a partial and d2 is the second partial. The B.C.'s are u_x(0,t)=u(0,t) and u_x(L,t)=u(L,t), where u_x is the partial of u with respect to x. The I.C is u(x,0)=f(x) Now, consider the Boundary Value Problem...
  27. G

    Waves under Boundary Conditions

    For a string with one endpoint attached to a wall and the other to an oscillator (so that it is under boundary conditions), what is the character of waves that are not at a harmonic frequency?
  28. L

    Heat Transfer Boundary Conditions

    Homework Statement A high temperature, gas cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (Ka=57 W/m*K) is encased in graphite (Kb= 3 W/m*K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium...
  29. T

    Linear 1st order PDE (boundary conditions)

    Homework Statement Solve the equation u_{x}+2xy^{2}u_{y}=0 with u(x,0)=\phi(x) Homework Equations Implicit function theorem \frac{dy}{dx}=-\frac{\partial u/\partial x}{\partial u/\partial y}The Attempt at a Solution -\frac{u_x}{u_y}=\frac{dy}{dx}=2xy^2 Separating variables...
  30. W

    Boundary conditions on D-Branes

    Hi there, I recently read that the equations of motion for an classical open string naturally give rise to two boundary conditions, namely Dirichlet and Neumann boundary conditions. (i) Could someone explain to me what do these boundary conditions physically mean, in particular for open...
  31. C

    Solving du/dt = (d^2)u/d(x^2) with Boundary Conditions

    Homework Statement du/dt = (d^2)u/d(x^2), t>0, 0<x<1 u(0,t) = 0 = u(1,t) , t>0 u(x,0) = P(x), 0<x<1 P(x) = {0 , if abs(x-1/2) >epsilon/2 {u/epsilon, if abs(x-1/2) <= epsilon/2 i need to find u(1/2,1/pi^2) Homework Equations i have u(x,t) = SUM{ 2/(n*pi)...
  32. B

    Dielectric-Dielectric Boundary Conditions Problem

    Homework Statement The problem can be found http://whites.sdsmt.edu/classes/ee382/homework/382Homework4.pdf" . It is the first one. Note: The subscript x = 0 is supposed to be y = 0 (the teacher typed it in wrong). Homework Equations \vec{\boldsymbol{D}}_{2t} =...
  33. P

    Solving 1d Helmholtz with boundary conditions

    Hello all, This is to do with forced longitudinal vibration of a rod (bar). It's basically a problem to do with the linearised plane wave equation (1d). The rod is fixed firmly at one end, and excited at the other by a harmonic force. The wave equation (with constant rho/E instead of...
  34. J

    How Do Boundary Conditions Affect Differential Equations?

    Homework Statement d20/de2+1=0 and the boundry condition is -d0/de(evaluated at e=+/- 1)=+/-H0(evaluated at +/-1). The final result yields 0(e)=(1/2)(1-e2)+1/H. What i don't understand is how to use this boundary condition and where the 1/H comes from. The Attempt at a Solution...
  35. M

    Boundary Conditions for infinite grounded cylinder (Laplace Equation)

    Homework Statement Find the potential outside of a long grounded conducting cylindrical rod of radius R placed perpendicular to a uniform electric field E0. Homework Equations V(s,\phi) = a_{0}+b_0{}ln(s) + \sum(A_n{}cos(n\phi)+B_n{}sin(n\phi))*(C_n{}s^n{}+D_n{}s^{-n}) The sum being...
  36. R

    Generic question on boundary conditions

    A partial differential equation requires boundary conditions. Consider a 2-dimensional problem, where the variables are 'x' and 'y'. The boundary is the line x=0 and you are given all sorts of information about the function on that line. If you are given just the values of the function on the...
  37. K

    1-D Wave equation with mixed boundary conditions

    Homework Statement Solve, u_{t} = u_{xx}c^{2} given the following boundary and initial conditions u_{x}(0,t) = 0, u(L,t) = 0 u(x,0) = f(x) , u_{t}(x,0) = g(x)Homework Equations u(x,t) = F(x)G(t) The Attempt at a Solution I solved it, I am just not sure if it is right. u(x,t) =...
  38. X

    Boundary conditions for a 4th order beam deflection equation

    What would the boundary conditions be for a fourth order differential equation describing the deflection (elastic curve) of a propped cantilever beam with a uniform distributed load applied? i.e. a beam with a built in support on the left and a simple support on the right. I need 4 obviously but...
  39. R

    Do Boundary Conditions Only Require Information at a Single Point?

    If you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then haven't you determined the entire function? You can use a Taylor expansion about that point to build the entire function because you have the...
  40. B

    Find Flux Density On One Side of Dielectric Boundary Given Boundary Conditions

    Homework Statement A dielectric interface is defined as 4x + 3y = 10 m. The region including the origin is free space, where D1 = 2ax - 4ay + 6.5az nC/m2. In the other region, εr2 = 2.5. Find D2 given the previous conditions. Homework Equations an12 = ± grad(f)/|grad(f)| D2n = D1n =...
  41. D

    Boundary conditions for P

    Homework Statement Use a Gaussian surface and an Amperian loop to derive the electrostatic boundary conditions for the polarisation field P at an interface between electric media 1 and 2 of relative permittivities e1 and e2. (Hint: determine results for D and E first) Homework Equations...
  42. U

    Pde with boundary conditions

    u_t=u_{xx}+2u_x 0<=x<=L, t>=0, u(x,0)=f(x), u_x(0,t)=u_x(L,t)=0 How to do this?
  43. P

    Boundary conditions for two dimensional problems in Quantum mechanics

    I am stuck at the problems of Boundary conditions for two dimensional problem in QM. iIf we have a two-dimensional domain, along the boundary, we can define two directions, one is tangential, the other is normal, assuming that there is no current flowing in and out along the normal direction...
  44. J

    Finding the Boundary Conditions of a Potential Well

    Homework Statement Show that the conditions for a bound state, Eqn1 and Eqn2, can be obtained by requiring the vanquishing of the denominators in Eqn3 at k=i\kappa. Can you give the argument for why this is not an accident? Homework Equations Eqn1: \alpha=q*tan(qa) Eqn2...
  45. P

    Heat Equation + 2 Robin Boundary Conditions

    Homework Statement Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a given initial temperature u(x,0) = f(x). The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is u = 0. The radiation is taken...
  46. L

    Green's Function Solution to ODE. Boundary Conditions Problem.

    Use Green's Functions to solve: \frac{d^{2}y}{dx^{2}} + y = cosec x Subject to the boundary conditions: y\left(0\right) = y\left(\frac{\pi}{2}\right) = 0 Attempt: \frac{d^{2}G\left(x,z\right)}{dx^{2}} + G\left(x,z\right) = \delta\left(x-z\right) For x\neq z the RHS is zero...
  47. N

    Thin conducting plate boundary conditions

    Homework Statement A thin conductor plate is in free space. Its conductivity is finite and thickness is approaching zero. Relate the tangential electric field in either side of the conductor. Repeat for tangential magnetic field. How are electric and magnetic fields related. Homework...
  48. M

    Boundary conditions for wave fixed at one end

    For a string fixed at x=0 and free at x=l I know \frac{dy}{dx}(l,t)=0 (d's are meant to be partials) but what is the other boundry associated with the end of the string? Is the second derivative also equal to 0?
  49. A

    Second Order Numerical Integration w/ Neumann Boundary Conditions

    I hope this is the right place to post this question. I'm trying to figure out how to run a numeric integration for a nonlinear second order ODE with Neumann B.C. I've started programming up Runge Kutta 4, but clearly without a boundary condition on the function, but only on its derivative...
  50. M

    Solving an equation with boundary conditions

    Hey all, I finally figured out how to solve the integral: \int{dp} = \int{6U\eta(\frac{h-\overline{h}}{h^{3}})}{dx} + C using maple and have it export to MATLAB where: h=R+h0-\sqrt{R+x}\sqrt{R-x} \overline{h}=R+h0-\sqrt{R+\overline{x}}\sqrt{R-\overline{x}} how do i find the...
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