PDE Boundary Value Problems — Mathematics Knowledge System

Overview

Boundary value problems (BVPs) for PDEs ask: find a function u satisfying a PDE on a domain Ω together with conditions on the boundary ∂Ω (Dirichlet: prescribe u; Neumann: prescribe ∂u/∂n; Robin: prescribe a linear combination). The three classical linear second-order PDEs — the heat equation, wave equation, and Laplace equation — represent parabolic, hyperbolic, and elliptic types respectively. The method of separation of variables reduces each to a Sturm–Liouville eigenvalue problem, and the solution is an eigenfunction expansion (Fourier series or its generalization).

Intuition

The key insight of separation of variables: assume u(x,t) = X(x)T(t). The PDE splits into two ODEs — one for X (a Sturm–Liouville problem with BCs giving discrete eigenvalues), one for T (a simple ODE solved by exponentials or trig functions). The general solution is a superposition (series) of these separated solutions. The initial condition determines the coefficients via orthogonality — exactly the Fourier series idea. For Laplace's equation, the eigenfunctions in one direction determine growth/decay in the other direction.

Formal Definition

Definition

The three classical PDEs and their standard BVPs:

u_t = k\,u_{xx},\quad x \in (0,L),\ t > 0 \quad \text{(heat equation)}

Heat equation

u_{tt} = c^2 u_{xx},\quad x \in (0,L),\ t > 0 \quad \text{(wave equation)}

Wave equation

u_{xx} + u_{yy} = 0,\quad (x,y) \in \Omega \quad \text{(Laplace equation)}

Laplace equation

\text{Dirichlet: } u|_{\partial\Omega} = g,\quad \text{Neumann: } \frac{\partial u}{\partial n}\bigg|_{\partial\Omega} = h

Boundary condition types

Worked Examples

Separate: u = X(x)T(t). Then T′/T = X″/X = −λ. BCs give X(0)=X(π)=0.
$X'' + \lambda X = 0,\quad X(0) = X(\pi) = 0$
Eigenvalues λₙ=n², eigenfunctions Xₙ=sin(nx). Each Tₙ satisfies T′=−n²T, giving Tₙ=e^{−n²t}.
$X_n = \sin(nx),\quad T_n = e^{-n^2 t}$
General solution: u(x,t) = ∑ cₙ sin(nx) e^{−n²t}. Apply IC: u(x,0) = ∑ cₙ sin(nx) = sin(x)+sin(3x)/2.
$u(x,0) = \sum c_n \sin(nx) = \sin x + \tfrac{1}{2}\sin 3x$
By orthogonality: c₁=1, c₃=1/2, all other cₙ=0.
$c_1 = 1,\ c_3 = 1/2$

Answer: u(x,t) = sin(x)e^{−t} + (1/2)sin(3x)e^{−9t}.

Practice Problems

Difficulty 7/10

Solve the wave equation u_{tt} = u_{xx} on [0,π] with u(0,t)=u(π,t)=0, u(x,0)=sin(2x), u_t(x,0)=0.

Difficulty 8/10

State the maximum principle for the Laplace equation and explain its physical meaning.

Difficulty 8/10

What are the three types of linear second-order PDE (elliptic, parabolic, hyperbolic) and give one example of each?

Common Mistakes

Common Mistake

Separation of variables always works for any PDE.

Separation of variables works for linear PDEs with separable geometry and coefficients; nonlinear PDEs or irregular domains generally require other methods.

Common Mistake

All BVPs for PDEs have unique solutions.

Existence and uniqueness depend on the type of PDE, boundary conditions, and domain. Neumann BCs for Laplace's equation have solutions only if a compatibility condition is met.

Quiz

Separation of variables for u_t = k u_{xx} leads to which eigenvalue problem?

An algebraic equationX″ + λX = 0 with BCsT′ = kT∇²u = 0

A Dirichlet boundary condition specifies:

The normal derivative of u on the boundaryThe value of u on the boundaryA linear combination of u and ∂u/∂nThat u=0 everywhere

Summary

The three classical PDE types are elliptic (Laplace), parabolic (heat), and hyperbolic (wave).
Boundary conditions: Dirichlet (u prescribed), Neumann (∂u/∂n prescribed), Robin (combination).
Separation of variables u=X(x)T(t) reduces a PDE to SL eigenvalue problems.
The solution is an eigenfunction series with coefficients from orthogonality (Fourier coefficients).
The maximum principle for harmonic functions ensures uniqueness of the Dirichlet problem.

References

WebsiteWikipedia — Boundary value problem