PDEs Summary

Differential Equations · Axiom Academy

SUMMARY Partial Differential Equations Let's review how PDEs model multidimensional phenomena and how separation of variables transforms them into solvable ODEs. Dimensionality matters: PDEs extend calculus to functions of multiple variables, enabling models of spatial-temporal phenomena Classification guides solutions: Elliptic, parabolic, and hyperbolic equations have fundamentally different behaviors and require different solution approaches Separation reduces complexity: Product solutions transform PDEs into ODEs, making them tractable through familiar techniques Fourier series are essential: Matching initial conditions requires representing arbitrary functions as infinite series of eigenfunctions Boundary conditions determine eigenfunctions: The spatial domain and boundary conditions dictate the separation constants and basis functions Superposition builds solutions: Linear PDEs allow general solutions to be constructed from infinite sums of simple modes Core Concept From ODEs to PDEs PDEs involve partial derivatives with respect to multiple independent variables — space and time, or several spatial dimensions — unlike an ODE's single-variable functions. Motivation: models phenomena that vary in both space and time, like heat diffusing along a rod Solutions: are functions of multiple variables u(x,t) , not single-variable y(t) Conditions needed: both initial conditions (at t=0 ) and boundary conditions (at the endpoints of the spatial domain)

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