What is Boundary value problem: Definition + 78 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. TheFerruccio

    Boundary value problem with substitution

    Homework Statement Find the general solution to the boundary value problem. Homework Equations (xy')' + \lambda x^{-1}y = 0 y(1) = 0 y(e) = 0 use x = e^t The Attempt at a Solution x = e^t so \frac{dx}{dt} = e^t using chain rule: y' = e^{-t}\frac{dy}{dt} Substituting...
  2. B

    Boundary Value Problem + Green's Function

    Boundary Value Problem + Green's Function Consider the BVP y''+4y=e^x y(0)=0 y'(1)=0 Find the Green's function for this problem. I am completely lost can someone help me out?
  3. Z

    Solving Boundary Value Problems: Are Eigenvalues Equal?

    let be the two boundary value problem -D^{2}y(x)+f(x)y(x)= \lambda _{n} y(x) with y(0)=0=y(\infty) and the same problem -D^{2}y(x)+f(x)y(x)= \beta _{n} y(x) with y(-\infty)=0=y(\infty) i assume that in both cases the problem is SOLVABLE , so my question is , are the eigenvalues in...
  4. G

    Boundary value problem for non-conducting surface

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  5. S

    Boundary Value Problem + Green's Function

    Consider the BVP y''+4y=f(x) (0\leqx\leq1) y(0)=0 y'(1)=0 Find the Green's function (two-sided) for this problem. Working: So firstly, I let y(x)=Asin2x+Bcos2x Then using the boundary conditions, Asin(2.0)+Bcos(2.0)=0 => B=0 y'(x)=2Acos(2x)-2Asin(2x) y'(0)=2A=0...
  6. R

    Two-Point Boundary Value Problem

    Homework Statement y''+\lambday=0 y'(0)=0 y'(pi)=0 Homework Equations The Attempt at a Solution What's puzzling me is the case when we check if the eigenvalue is zero. y''=0 y'=C1 y=C1x+C2 Now when I check the first boundary value I get C1=0 now How do I check the second one ? with the...
  7. J

    Two-Point Boundary Value Problem

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  8. B

    Boundary Value Problem for the 1-D Wave

    So here's the problem: I'm asked to find the solutions to the 1-D Wave equation u_{tt} = u_{xx} subject to u(x,0) = g(x), u_t(x,0) = h(x) but also u_t(0,t) = A*u_x(0,t) and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to...
  9. Y

    2nd order Boundary Value Problem.

    I want to solve: y(x)''-(\frac{m\pi}{a})^2y(x)=0 With boundary condition y(0)=y(a)=0. First part is very easy using constant coef. which give: y(x)=c_1 cosh(\frac{m\pi}{a}x) + c_2 sinh(\frac{m\pi}{a}x) y(0)=0 \;\Rightarrow\; c_1=0 \;\Rightarrow\; y(x) = c_2 \; sinh(\frac{m\pi}{a}...
  10. A

    Approximation of boundary value problem using finite differences

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  11. T

    Electric field inside hollow conductor boundary value problem

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  12. S

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  13. E

    Boundary Value Problem for y'' + y = A + sin(2x) with y(0) = y'(\pi/2) = 2

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  14. T

    System of ODE Boundary Value Problem with 2nd Order Backward Difference

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  15. A

    How can I solve a system of BVPs with only boundary conditions given?

    Dear all, I have system(4 equations) of BVPs. Could anybody recommend me, how to solve this system(whatever numericaly or analytical): x'=-y/sqrt(x^2+y^2) + u y'=x/sqrt(x^2+y^2) + v u' = -xy/(x^2+y^2)^3/2 u - [1/sqrt(x^2+y^2) - x^2 /(x^2+y^2)^3/2] v v' = xy/(x^2+y^2)^3/2 v -...
  16. M

    Formally solve the following boundary value problem

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  17. J

    Boundary Value Problem , triangular plate

    This exercise deal with the temperature u(x,y,t) in a homogeneous and thin plate. We assume that the top and bottom of the plate are insulated and the material has diffusivity k. Write the BVP . Problem: The plate is triangular , picture this as a right triangle with this coordinates, (0,0) ...
  18. P

    Electric Boundary Value Problem

    Homework Statement A pair of infinite, parallel planes are equipotential surfaces. The plane at z = 0 has an electric potential of 0 and the plane at z = b also has a potential of zero. The electric field at b is 0 at time t at which there is a constant, positive charge density between the...
  19. U

    Trying to solve a boundary value problem

    Trying to solve the following boundary value problems. y'' + 4y = cos x; y(0) = 0, y(pi) = 0 y'' + 4y = sin x; y(0) = 0, y(pi) = 0 The answer key says that there's no solution to the first part, but there is a solution to the 2nd part. I'm really lost and am not sure how to go...
  20. M

    Boundary Conditions and Trig Identities in Solving Differential Equations

    Homework Statement I have the solution to the differential equation : Phi = A*sin(x) + B*cos(x) and need to apply the boundary conditions Phi(-a/2) = Phi(a/2) = 0. Homework Equations The Attempt at a Solution I am confused. If I plug these in, then I get A*sin(-ka/2) = -...
  21. U

    Problem with a boundary value problem non linear

    i've some problem to solve this kind of differential equation, i 've put the link where is the file of my equation, i search a methods that solve it iterativly. the file is there diablo221 .altervista.org / risultato. nb
  22. L

    Does a Dielectric Sphere in a Uniform Electric Field Exhibit Azimuthal Symmetry?

    Homework Statement A solid sphere is placed in an otherwise uniform electric field. Its upper half is made up from a material with dielectric constant e_1; the other half has dielectric constant e_2. The plane at which the parts of the sphere intersect is parallel to the uniform field at...
  23. radou

    A boundary value problem discussion

    A boundary value problem "discussion" So, let's say we are given a function f : [0, 1] --> R and constants a, b, and we want to find u : [0, 1] --> R such that u''(x) + f = 0 on <0, 1> with u(1) = a and u'(0) = -b. One can easily obtain the exact solution to this problem merely by using...
  24. J

    Deriving Solutions for Schrodinger's Equation with Boundary Conditions

    Schodinger's equation for one-dimensional motion of a particle whose potential energy is zero is \frac{d^2}{dx^2}\psi +(2mE/h^2)^\frac{1}{2}\psi = 0 where \psi is the wave function, m the mass of the particle, E its kinetic energy and h is Planck's constant. Show that \psi = Asin(kx) +...
  25. G

    Nonhomogeneous Boundary Value Problem

    I've got a nonhomogeneous BVP I'm trying to solve. Both my book and my professor tend to focus on the really hard cases and completely skipp over the easier ones like this, so I'm not really sure how to solve it. It's the heat equation in a disk (polar coordinates) with no angle dependence...
  26. D

    Boundary value problem involving eigenvalues

    I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need...
  27. S

    What is the solution to the boundary value problem using Green's Theorem?

    Let D be a domain inside a simple closed curve C in R2. Consider the boundary value problem (\Delta u)(x,y) = 0, \ (x,y) \in D, \\ \frac{\partial u}{\partial n} (x,y) = 0 , \ (x,y) \in C. where n is the outward unit normal on C. Use Green's Theorem to prove taht every solution u...
  28. M

    Boundary Value Problem for Electrostatic Potential in a Channel

    I need some help starting off on this question. Electrostatic potential V (x,y) in the channel - \infty < x < \infty, 0 \leq y \leq a satisfies the Laplace Equation \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2}= 0 the wall y = 0 is earthed so that V (x,0) = 0...
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