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Separation and Laplace's equation

Separation of variables assumes the solution can be written as a product $u(x,t)=X(x)T(t)$ , turning a PDE into two simpler ordinary differential equations. Laplace's equation describes a stationary (time-independent) state, such as a stationary heat distribution.

u(x,t) = X(x)\,T(t)

the separation ansatz

\nabla^2u = 0

Laplace's equation (stationary state)

Symbols

X(x),\ T(t)

factors that each solve an ordinary equation

Example

Stationary heat conduction in a rod with $u(0)=25$ and $u(10)=93$ : the solution is linear, $u(x)=25+6.8x$ , so $u(0)=25$ .

Stationary means

u_t=0

, so the heat equation becomes

u_{xx}=0

, giving a linear solution in 1D.

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Part of Laplace, Fourier and PDEs: Partial differential equations.