The Cluster's Separation of Variables

A page about separation of variables — the most widely-used technique for solving partial differential equations.

The technique

Separation of variables assumes the solution can be written as a product of functions, each depending on only one variable: f(x, y, z, t) = X(x) Y(y) Z(z) T(t)

Substituting into the PDE and dividing by the product gives terms that depend on different variables, so each must equal a constant (the separation constant). This reduces the PDE to ODEs.

For the 3D Laplace equation (nabla^2 f = 0) in Cartesian coordinates: X''/X + Y''/Y + Z''/Z = 0 So X'' = -k_x^2 X, Y'' = -k_y^2 Y, Z'' = k_z^2 Z where k_x^2 + k_y^2 = k_z^2.

In the cluster, the edit technique assumes an edit product solution.

The wave equation in spherical coordinates

For the wave equation in spherical coordinates with spherical symmetry (no angular dependence): d^2 R / dr^2 + (2/r) dR/dr + k^2 R = 0 The solutions are spherical Bessel functions: j_l(kr), y_l(kr).

With angular dependence, the angular equation is the Legendre equation, giving P_l(cos theta). The radial equation gives spherical Bessel functions. The full solution: f(r, theta, phi, t) = sum_{l,m} [A_{lm} j_l(kr) + B_{lm} y_l(kr)] Y_{lm}(theta, phi) e^{i omega t}

In the cluster, the edit wave equation in spherical coordinates gives an edit product solution.

The Schrödinger equation in hydrogen

The hydrogen atom wave function separates: R_{nl}(r) Y_{lm}(theta, phi). The radial equation is: d^2 u / dr^2 + [2m / hbar^2 (E + e^2 / (4 pi epsilon_0 r)) - l(l+1) / r^2] u = 0 where u(r) = r R(r). The solutions are associated Laguerre polynomials.

In the cluster, the edit Schrödinger equation separates into edit radial and edit angular parts.

The heat equation on a rectangle

For the 2D heat equation on a rectangle [0, a] x [0, b]: T(x, y, t) = sum_{n,m} A_{nm} sin(n pi x / a) sin(m pi y / b) exp(-lambda_{nm} t) where lambda_{nm} = (n^2 pi^2 / a^2 + m^2 pi^2 / b^2) / alpha, and A_{nm} are determined by the initial condition.

In the cluster, the edit heat equation on a rectangle gives an edit Fourier series.

When separation fails

Separation of variables does not work for:

• Irregular geometries (use numerical methods: finite element, finite difference)
• Non-linear PDEs
• PDEs with variable coefficients that don't separate
• PDEs with non-separable boundary conditions

However, separation of variables works in 11 coordinate systems for Laplace's equation in 3D: Cartesian, cylindrical, spherical, spheroidal, parabolic, elliptic, etc.

In the cluster, the edit separation fails for an edit irregular geometry.

This technique

This page is about separation of variables. f = X(x)Y(y)Z(z)T(t). Laplace in Cartesian: X''/X + Y''/Y + Z''/Z = 0. The technique is real.