# What is Boundary value problem: Definition and 78 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. ### Boundary value problem- Random Walker

I want to solve this using difference equation. So I set up the general equation to be Pi = 0.5 Pi+1 + 0.5 i-1 I changed it to euler's form pi = z 0.5z2-z+0.5 = 0 z = 1 since z is a repeated real root I set up general formula Pn = A(1)n+B(1)n then P0 = A = 1 PN = A+BN = 0 -> A= -BN...
2. ### Green's function for a boundary value problem

Homework Statement I try to integral as picture 1. The result that is found by me, it doesn't satisfy Green's function for boundary value problem. Homework EquationsThe Attempt at a Solution show in picture 2 & picture 3.

11. ### I Applications needing fast numerical conformal maps

What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask. As part of an engineering project I've implemented...
12. ### MHB Boundary Value Problem: Does it Have a Solution?

Hello! (Wave) I want to check if the following boundary value problem has a solution $\left\{\begin{matrix} -u_{xx}-4u=\sin {2x}, x \in (0,\pi)\\ u(0)=u(\pi)=0 \end{matrix}\right.$ I have thought the following: We consider the corresponding homogeneous equation $-u_{xx}-4u=0$. The...
13. ### A Solving a Boundary Value Problem: Proving u(x) < 0

I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0 where f(x) and g(x) are positive functions. I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this. I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...
14. ### A Boundary Value Problem Requiring Quarterwave Symmetry

I can't seem to find an explicit or analytical solution to a boundary value problem and thought I might ask those more knowledgeable on the subject than me. If t is an independent variable and m(t) and n(t) are two dependent variables with the following 8 constraints: a) m' =0 @T=0 and...
15. ### Lipschitz perturbations and Hammerstein integral equations

Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...
16. ### Finding a solution to Laplace's equation

So here I have Laplace's equation with non-homogeneous, mixed boundary conditions in both x and y. 1. Homework Statement Solve Laplace's equation $$\label{eq:Laplace}\nabla^2\phi(x,y)=0$$ for the following boundary conditions: \phi(0, y)=2; \phi(1, y)=0; \phi(x...
17. ### MHB Solving a Boundary Value Problem: Non-Uniform vs. Uniform Partitioning

Hello! (Wave)Consider the boundary value problem $\left\{\begin{matrix} - \epsilon u''+u'=1 &, x \in [0,1] \\ u(0)=u(1)=0 & \end{matrix}\right.$ where $\epsilon$ is a positive given constant. I have to express a finite difference method for its numerical solution. How can we know whether it...

41. ### Boundary Value Problem; Eigenvalues and Eigenfunctions

Homework Statement Find the eigenvalues and eigenfunction for the BVP: y'''+\lambda^2y'=0 y(0)=0, y'(0)=0, y'(L)=0 Homework Equations m^3+\lambdam=0, auxiliary equation The Attempt at a Solution 3 cases \lambda=0, \lambda<0, \lambda>0 this first 2 give y=0 always, as the only...
42. ### Non-homogeneous Boundary value Problem

Hello, I am trying to solve a vibration problem analytically but I don't understand how to implement the non-homogeneous boundary conditions. The problem is defined as below: y_{t}_{t}(x,t) = a^{2}y_{x}_{x}(x,t) With Boundary conditions: y(0,t) = 0 [ fixed...
43. ### Heat Equation: Boundary Value Problem

http://img821.imageshack.us/img821/7901/heatp.png Uploaded with ImageShack.us I'm having difficulty with the boundary conditions on this problem. I don't need a solution or a step by step. I've just never solved a boundary condition like this. Its the u(pi,t) = cos(t) that is giving me...
44. ### Solving a Boundary Value Problem: y + y = 0 ; 0<x<2π, y(0)=0 , y(2π)=1

Homework Statement Determine all the solutions, if any, to the given boundary value problem by first finding a general solution to the differential equation: y" + y = 0 ; 0<x<2π y(0)=0 , y(2π)=1 The attempt at a solution So the general solution is given by: y = c1sin(x) +...
45. ### Eigenvalues for a 4th order boundary value problem

Homework Statement y^{(4)}+\lambda y=0 y(0)=y'(0)=0 y(L)=y'(L)=0 Homework Equations The hint says... let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution Listening to the hint, I got r=\pm\mu With multiplicity 2 of each. So that means.. y=c_1 e^{\mu t}+c_2te^{\mu...
46. ### Could anyone help me out for this Boundary Value Problem?

Can't seem to work this out, any solutions would be greatly appreciated! Thanks in advance! Solve the boundary-value problem Uxx + Uyy + U = 0 , 0<x<1,0<y<1 U(0,y) = 0 , Ux(a,y)= f(y) U(x,0) = 0 , Uy(x,1)= sin(3*pi*x)
47. ### Two point boundary value problem

Homework Statement Solve the given boundary value problem or else show that it has no solutions: y'' + 4y = cos x, y'(0) = 0, y'(pi) = 0. Homework Equations N/A The Attempt at a Solution So I made it all the way through the problem I think, but I am not getting the correct answer...
48. ### Boundary value problem for heat conduction (HELP)

Hi, I am trying to work this problem out but I don't know how to solve the boundary value. here is the problem statement thanks in advance
49. ### Laplace Boundary Value Problem

Homework Statement A cantilever beam has uniform load w over a length of L as described by the eq. EI y'''' = -w y(0) = y'(0) = 0 y''(L) = y'''(L) = 0 EI are constants find y(x) Homework Equations L[y^4] = S^4*Y(s) - S^3*Y(0) - S^2*Y'(0) - s*Y''(0) - Y'''(0) The...
50. ### Green's Function ODE Boundary Value Problem

Homework Statement Use a Green's function to solve: u" + 2u' + u = e-x with u(0) = 0 and u(1) = 1 on 0\leqx\leq1 Homework Equations This from the lecture notes in my course: The Attempt at a Solution Solving for the homogeneous equation first: u" + 2u' + u = 0...