Questions: Laplace's Equation and Boundary Value Problems

Question 1

Why can't Laplace's equation ∇²u = 0 be solved by specifying an initial condition u(x,y,0) and marching forward in time?

• A. Because Laplace's equation requires Fourier series, which are defined only on bounded domains
• B. Because Laplace's equation has no time variable — it describes a steady state, and the solution at any interior point depends on the entire boundary simultaneously
• C. Because Laplace's equation is nonlinear, making time-marching unstable
• D. Because time-marching only fails near the boundaries; interior points can be solved sequentially

Answer: (B)

Laplace's equation is elliptic: it has no time derivative and describes equilibrium, not evolution. The solution at any interior point depends on conditions on the entire boundary, not on an initial state. This is structurally different from parabolic PDEs (heat equation) or hyperbolic PDEs (wave equation), which have time derivatives and can be stepped forward from initial data. For elliptic PDEs, boundary conditions on the entire closed boundary must be specified and the whole domain solved simultaneously.

Question 2

A metal plate has its edges held at fixed temperatures. After a very long time, the temperature distribution no longer changes. Which statement correctly describes the governing equation at steady state?

• A. The heat equation ∂u/∂t = α∇²u still applies, but with ∂u/∂t ≈ 0 for practical purposes
• B. The temperature satisfies ∇²u = 0 exactly — the steady-state distribution is a harmonic function
• C. The wave equation governs the steady state because temperature disturbances propagate outward
• D. No differential equation applies at steady state; the temperature is simply whatever the boundary conditions dictate

Answer: (B)

At steady state, ∂u/∂t = 0 exactly. Substituting into the heat equation gives α∇²u = 0, which (since α ≠ 0) reduces to Laplace's equation ∇²u = 0. The steady-state temperature is precisely a harmonic function. This is how Laplace's equation arises physically: it is what remains of the heat equation after all transients die away and time evolution has stopped.

Question 3

A solution to Laplace's equation on a bounded domain can achieve its maximum value at an interior point of the domain.

• T. True
• F. False

Answer: False

The maximum principle states that harmonic functions achieve their maximum and minimum values on the boundary of the domain, not in the interior — unless the function is constant throughout. Physically, the temperature at any interior point of a steady-state plate is a weighted average of nearby temperatures and cannot exceed the maximum boundary temperature. This principle also explains why Dirichlet boundary conditions uniquely determine the solution: the boundary controls the interior completely.

Question 4

Laplace's equation can be understood as the time-independent limit of the heat equation, representing the distribution that the heat equation approaches as t → ∞.

• T. True
• F. False

Answer: True

The heat equation ∂u/∂t = α∇²u describes evolution from an initial distribution. As t → ∞, the distribution converges to a steady state where ∂u/∂t = 0, yielding ∇²u = 0 — Laplace's equation. This connection is general: Laplace's equation characterizes the equilibrium of any diffusive process. The same relationship holds for the Poisson equation ∇²u = f, which describes steady states with sources.

Question 5

Explain why Laplace's equation requires boundary conditions on the entire closed boundary, whereas the wave equation requires initial conditions at a single time. What structural difference accounts for this?

Model answer: The wave equation is hyperbolic: it has two time derivatives, and disturbances propagate at finite speed from initial data. Specifying u and ∂u/∂t at t=0 determines the future because information flows forward. Laplace's equation is elliptic: it has no time variable, and the value at any interior point depends on conditions in all directions simultaneously. The solution is not built from one boundary inward — it is determined by a global balance. Specifying conditions on only part of the boundary leaves the problem underdetermined; the entire boundary must be constrained.

This structural difference — hyperbolic vs. elliptic — determines computational strategy as well. A hyperbolic problem can be solved step by step in time; an elliptic problem requires solving a large coupled system simultaneously. Numerically, Laplace's equation leads to sparse linear systems (finite difference or finite element methods), while wave equations can use explicit time-stepping. The physics matches the math: equilibrium is a global property, while propagation is a local, sequential one.