Partial Differential Equations

Mathematical equations that involve partial derivatives and are often used to describe physical phenomena.

What is a partial differential equation?

"A partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function."

How are solutions to partial differential equations typically obtained?

"it is usually impossible to write down explicit formulas for solutions of partial differential equations."

What is the role of numerical approximation in solving partial differential equations?

"There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers."

What are some qualitative features that researchers seek to identify in solutions of partial differential equations?

"The usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability."

What specific open question is mentioned in the paragraph?

"Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000."

In which scientific fields are partial differential equations commonly used?

"they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics."

Are partial differential equations only applicable in scientific fields?

"They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations."

What is an example of a notable application of partial differential equations in mathematics?

"they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology."

Is there a unified theory for partial differential equations?

"it is usually acknowledged that there is no 'general theory' of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields."

What is the relationship between ordinary differential equations and partial differential equations?

"Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable."

What are some extensions or variations of the concept of partial differential equations?

"Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the 'PDE' notion."

What are some classical topics or areas of active research in the field of partial differential equations?

"classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations." Note: Since the paragraph is relatively short, not all questions will have specific quotes addressing them.