PDEs couple a field to how it bends in several directions
A partial differential equation involves a multivariable unknown and one or more of its partial derivatives. Laplace’s equation—sum of second pure partials in x, y, z equals zero—is the classic model of steady potentials. Explicit solutions are rare, so much modern research goes into approximating them by computer.
A partial differential equation involves an unknown function of several variables and some of its partial derivatives. Such equations underpin the modern understanding of sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, general relativity and quantum mechanics through the Schrödinger equation. Explicit solution formulas are often impossible, so a vast research effort goes into approximating solutions numerically by computer, and because the equations vary so widely there is no universal theory, only several specialist subfields.
Ordinary differential equations form a subclass, corresponding to functions of a single variable, while stochastic and nonlocal equations extend the notion. Laplace's equation is among the most important: its solutions, harmonic functions, were studied intensively in the 19th century for classical mechanics, and the equilibrium temperature in a uniform solid is one of them. Harmonic functions take strikingly different forms because no general solution formula exists.
Freedom in PDE solutions usually means a choice of whole functions rather than constants. Any v(x, y) = f(x) + g(y), for arbitrary differentiable f and g, solves the equation whose mixed second partial derivative vanishes. On the unit disc, each continuous function on the boundary circle fixes exactly one solution of Laplace's equation, while the near-identical equation with a minus sign, on the strip R × (−1, 1), needs two freely chosen functions. The domain must therefore count as part of the equation itself.
A minimal-surface equation from differential geometry has a completely explicit solution yet allows the free choice of only three numbers, and it is nonlinear because of its squares and square roots. In a linear homogeneous PDE, by contrast, sums and constant multiples of solutions are again solutions.
Source: Partial differential equation
