Laplace Equation Examples

PDEs · Axiom Academy

EXAMPLE Laplace Equation Examples Worked solutions using separation of variables and the Poisson integral formula Example 1: Rectangle with One Non-Zero Boundary Excellent work! You've completed these Laplace equation examples. Here's what we learned: Separation of Variables: For rectangular domains, assuming u(x,y) = X(x)Y(y) transforms the PDE into two ODEs. Boundary conditions determine which variable carries the eigenvalue. Homogeneous vs. Non-Homogeneous BCs: Apply homogeneous boundary conditions first to find eigenfunctions, then use the non-homogeneous condition to find coefficients via Fourier series. Poisson Integral Formula: For circular domains, the solution has an explicit integral representation. The formula naturally satisfies Laplace's equation and matches boundary data. Verifying Harmonic Functions: To verify a function is harmonic, compute both second partial derivatives and check that their sum is zero. Remember to apply the chain rule carefully for composite functions. Physical Interpretation: Solutions to Laplace's equation represent steady-state temperature distributions, electrostatic potentials, or other equilibrium phenomena with no sources or sinks in the interior. These techniques form the foundation for solving elliptic PDEs in various geometries. Practice identifying which method to use based on the domain shape and boundary conditions!

This is the written version of the interactive lesson above. See the full PDEs course.