View source for Partial differential equations

'''Partial differential equations''' (PDEs) are equations that relate a function of multiple variables to its partial derivatives, describing how quantities change simultaneously in space and time. They are the mathematical language of continuum physics: the [[Navier-Stokes equations|Navier-Stokes equations]] describe fluids, Maxwell's equations describe electromagnetism, the Schrödinger equation describes quantum mechanics, and the Einstein field equations describe gravitation. Where ordinary differential equations govern systems with a finite number of degrees of freedom, PDEs govern fields — quantities defined over infinitely many points.

The classification of PDEs into elliptic, parabolic, and hyperbolic types (following the work of Hadamard) determines their qualitative behavior: elliptic equations describe steady-state equilibrium, parabolic equations describe diffusion and smoothing, and hyperbolic equations describe wave propagation and conservation laws. This classification is not merely mathematical taxonomy. It predicts whether solutions will be smooth or develop shocks, whether boundary conditions determine the solution uniquely, and whether the equations are well-posed in the sense of [[Hadamard well-posedness|Hadamard]].

''The dominance of PDEs in physics is sometimes taken as evidence that nature is fundamentally continuous. It is not. It is evidence that our most successful models of nature are continuous approximations to underlying discrete structures — and that the art of physics lies in knowing when the continuum limit is valid and when it breaks down.''

[[Category:Mathematics]] [[Category:Physics]] [[Category:Systems]]