Laplace Equation

Fourier Analysis · Axiom Academy

LESSON Laplace Equation and Steady-State Solutions Understanding equilibrium solutions in heat conduction and electrostatics through the Laplace equation The Laplace equation is obtained when the heat equation reaches equilibrium (time-independent solution): In two dimensions, this expands to: This is a second-order elliptic PDE that describes how a quantity is distributed in space when all time-dependent effects have vanished. The function u represents the steady-state value at each point in the domain. 2. Physical Meaning: Steady-State Heat and Electrostatics Heat Conduction: When a metal plate is heated along its edges and given sufficient time, the temperature distribution stabilizes. At each interior point, the temperature equals the average of nearby temperatures. Electrostatics: The electric potential in a charge-free region satisfies the Laplace equation. This describes the potential field between conductors held at fixed voltages. 3. Solving on a Rectangular Domain Using Fourier Series Consider the Laplace equation on a rectangle [0,L] × [0,H] with boundary conditions. We use separation of variables: This leads to two ODEs. The general solution is a Fourier series: The coefficients are determined by the boundary conditions. For example, with u=0 on three sides and u=f(x) on y=H: The Laplace equation satisfies a powerful property that has no analogue in ordinary differential equations: This principle has profound implications:

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