Questions: Separation of Variables for Elliptic PDEs

Question 1

After separating Laplace's equation in Cartesian coordinates, you obtain three independent ODEs with solutions like sin(kₓx), cos(kₓx), or e^(kₓx). Why does the complete solution require an infinite series of these product solutions rather than just one?

• A. Because a single product solution V(x,y,z) = X(x)Y(y)Z(z) can only satisfy homogeneous boundary conditions on one face
• B. Because a single product solution generally cannot satisfy arbitrary boundary conditions — superposition of infinitely many is needed to match the full boundary data
• C. Because the separation constants must take on infinitely many values to satisfy conservation of energy
• D. Because Laplace's equation is nonlinear, requiring infinitely many terms to cancel the nonlinear residuals

Answer: (B)

Separation of variables finds a family of particular solutions, each satisfying the PDE but typically not the full boundary conditions. A single product solution might satisfy the equation and boundary conditions on some walls, but boundary data on the remaining surfaces generally requires a whole spectrum of modes. Because Laplace's equation is linear, a superposition (infinite series) of product solutions is also a solution. The coefficients are chosen to match the boundary condition — an expansion analogous to a Fourier series. The genius of the method is that orthogonality of the basis functions makes those coefficients extractable independently.

Question 2

Spherical harmonics appear identically in the description of atomic orbital shapes, gravitational multipole expansions, and electromagnetic radiation patterns. What is the deep reason for this?

• A. Physicists adopted a common mathematical convention across fields in the 19th century
• B. Spherical harmonics are the only complete orthogonal set, so all physical problems eventually reduce to them
• C. Any physical system with spherical symmetry decomposes into the same angular eigenfunctions because they satisfy the same angular part of Laplace's equation
• D. Quantum mechanics and classical field theory share the same governing equation by coincidence

Answer: (C)

When Laplace's equation (or Helmholtz or Schrödinger in the appropriate limit) is separated in spherical coordinates, the angular part produces exactly the same ODE regardless of the physical context. The angular solutions — Legendre polynomials times complex exponentials in φ — are universal because they are eigenfunctions of the angular part of the Laplacian, which depends only on the coordinate geometry, not the physics. Any problem with spherical symmetry decomposes onto this same angular basis. The physics only determines the radial equation, which differs across applications.

Question 3

Assuming V(x,y,z) = X(x)Y(y)Z(z) in the separation of variables method is a restriction to a special class of solutions that automatically satisfies the boundary conditions.

• T. True
• F. False

Answer: False

The product ansatz is a clever guess, not a guarantee. A single product solution V = X(x)Y(y)Z(z) satisfies the PDE but generally does not satisfy non-trivial boundary conditions. The full solution is constructed as an infinite superposition of product solutions — each product satisfying the PDE, the sum chosen to satisfy the boundary conditions. The boundary-matching step uses orthogonality of the basis functions to extract each coefficient independently. The ansatz is justified post hoc by the completeness theorem for the resulting eigenfunctions.

Question 4

The orthogonality of basis functions like sin(nπx/L) makes it possible to extract individual coefficients in the series solution without solving a coupled system of equations.

• T. True
• F. False

Answer: True

Orthogonality means ∫₀ᴸ sin(nπx/L) sin(mπx/L) dx = 0 for n ≠ m. When you multiply the boundary condition by one basis function and integrate, every term in the series except one vanishes — giving you the coefficient directly. This is exactly how Fourier coefficients work, and the same logic applies to Legendre polynomial expansions in spherical coordinates and Bessel function expansions in cylindrical coordinates. Without orthogonality, extracting coefficients would require solving an infinite coupled system — effectively impossible.

Question 5

Explain why Bessel functions appear in cylindrical-coordinate solutions while Legendre polynomials appear in spherical-coordinate solutions to Laplace's equation, even though the same separation-of-variables strategy is used in both cases.

Model answer: The separation-of-variables strategy is the same: assume a product solution, substitute into Laplace's equation, divide by the product, and obtain separate ODEs for each coordinate. The ODEs differ because Laplace's equation takes a different algebraic form in cylindrical versus spherical coordinates — the Laplacian operator involves different geometric factors. In cylindrical coordinates, the radial ODE is Bessel's equation, whose solutions are oscillatory Bessel functions that replace sines and cosines near the axis. In spherical coordinates, the radial equation yields simple power laws (r^ℓ and r^(−ℓ−1)) and the polar-angle equation yields Legendre's equation. The special functions that emerge are determined by the geometry of the coordinate system, not by an independent choice.

The key insight is that separation of variables converts the geometry of the boundary into the structure of the ODEs. Choosing coordinates matched to the boundary geometry (cylindrical for cylindrical boundaries, spherical for spherical ones) is what makes the method tractable — the boundary conditions become simple in those coordinates, and the resulting special functions form a complete orthogonal basis for that geometry.