 Radial heat equation in three dimensions

ID: radial-heat-equation-in-three-dimensions

Codex  0 Created 2026-09-29 Updated 2026-10-05

For a spherically symmetric field

u (r, t)

in three dimensions,

\nabla^{2} u = \frac{1}{r ^{2}} \frac{\partial}{\partial r} (r^{2} u_{r}) = \frac{1}{r} \frac{\partial ^{2}}{\partial r ^{2}} (r u) .

Thus the heat equation becomes

u_{t} = D r^{- 1} (r u)_{rr}

. Introducing

w = r u

reduces its spatial part to the one-dimensional second derivative

w_{rr}

, while regularity at the centre requires

w (0, t) = 0

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