Poisson's Equation

PDEs · Axiom Academy

Understanding one of the most fundamental equations in mathematical physics Poisson's equation relates the Laplacian of a function to a source term. In its simplest form: Here, ∇² is the Laplacian operator , which measures how much a function differs from its average value at nearby points. The function u represents a potential (like electric potential or gravitational potential), and f(x) is the source term that creates this potential. In three dimensions, the Laplacian expands to: 2. Laplace's Equation: The Homogeneous Case When there are no sources in the region of interest (f = 0), Poisson's equation reduces to Laplace's equation : Laplace's equation describes equilibrium states and steady-state solutions. Functions satisfying this equation are called harmonic functions and possess remarkable properties: They satisfy the mean value property : the value at any point equals the average over any surrounding sphere They have no local maxima or minima in the interior (maximum principle) They are infinitely differentiable wherever they're defined 3. Physical Interpretation: Sources and Sinks The source term f(x) determines the behavior of the solution: Positive f(x) > 0: A source that creates outward flow (like a positive charge or heat source) Negative f(x) A sink that draws inward flow (like a negative charge or heat drain) Zero f(x) = 0: Equilibrium region (reduces to Laplace's equation)

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