Master Partial Differential Equations with advanced Fourier analysis techniques. This essential guide leverages the power of Fourier transforms and heuristic methods to provide significant insight into solving standard PDE models. Discover explicit formulas for PDEs governed by canonical operators, including the Laplacian on Euclidean space, the Hermite operator, and the sub-Laplacian on the Heisenberg group. Designed as a crucial bridge, it prepares undergraduate science and engineering students for advanced graduate-level mathematics and research careers in analysis.