Skip to main content
MathidiaMathematics Academy
Partial differential equations
Lesson 65 of 75
Review

Laplace equation

Objective. Use harmonic functions and boundary conditions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The Laplace equation ∇²u = 0 describes steady-state fields with no sources, such as equilibrium temperature or electrostatic potential, and its solutions (harmonic functions) are entirely determined by boundary values. On a rectangle, separation of variables gives solutions like sinh and sin combinations that satisfy prescribed boundary temperatures, with no time dependence since the state has already settled. Example: on a square with three sides at zero and one side at u=sin(πx/L), the interior solution is u(x,y) = sin(πx/L)sinh(πy/L)/sinh(πH/L), matching the boundary exactly as y approaches it. Harmonic functions obey the maximum principle: their extreme values always occur on the boundary, never inside.

Worked example

A harmonic function u satisfies Laplace's equation on a square, with three sides held at u=0. By the maximum principle, where must the maximum value of u occur?

Diagram of a square
  1. 1Recall the maximum principle for harmonic functions: extrema occur on the boundary, not the interior
  2. 2Three sides are fixed at u=0, so these cannot hold the maximum unless u≡0
  3. 3The remaining boundary side carries the nonzero data
  4. 4Hence the maximum of u occurs on the fourth (non-zero) boundary side

Answer: On the fourth boundary side (the one not held at zero)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students look for time-dependent behaviour or decaying modes as in the heat equation, but Laplace's equation has no time variable at all; its solution is a single steady spatial pattern fixed by boundary data.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard