Laplace on a Rectangle

PDEs · Axiom Academy

Step-by-step guide to solving Laplace's equation with separation of variables 1. Problem Setup and Boundary Conditions We solve Laplace's equation on a rectangle [0, a] × [0, b]: The key to separation of variables is choosing homogeneous boundary conditions on opposite sides. A typical setup: Homogeneous x-boundaries: u(0, y) = 0 and u(a, y) = 0 Homogeneous bottom: u(x, 0) = 0 Non-homogeneous top: u(x, b) = f(x) We assume the solution has the form u(x, y) = X(x)Y(y), where X depends only on x and Y only on y. Substituting into Laplace's equation: Dividing by X(x)Y(y) separates the variables: Since the left side depends only on x and the right only on y, both must equal the same constant, which we call −λ (the separation constant). Setting each side equal to −λ gives us two ordinary differential equations: The choice of sign (−λ vs +λ) is deliberate: we want the X equation to have oscillatory solutions (sin, cos) to satisfy the homogeneous boundary conditions at x = 0 and x = a. 4. Applying Homogeneous Boundary Conditions The general solution to X'' + λX = 0 is X(x) = A cos(√λ x) + B sin(√λ x). Applying boundary conditions: u(0, y) = 0 ⟹ X(0) = 0 ⟹ A = 0 u(a, y) = 0 ⟹ X(a) = B sin(√λ a) = 0 For non-trivial solutions (B ≠ 0), we need sin(√λ a) = 0, giving: This yields the eigenvalues and eigenfunctions: With λ_n = (nπ/a)², the Y equation becomes: This has the general solution: Applying the homogeneous condition u(x, 0) = 0 gives Y(0) = 0, so C_n = 0. This leaves:

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