Where ordinary differentialequations have solutions that are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think that the parameters are function data (informally put, this means that the set of solutions is much larger).
Partial differentialequations are ubiquitous in science, as they describe phenomena such as fluid flow, gravitational fields, and electromagnetic fields.
In the WKB approximation it is the Hamilton-Jacobi equation.
A stochastic differentialequation (SDE) is a differentialequation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
The theory of differentialequations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates.
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