"A partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function."
Mathematical models used to describe physical phenomena and changes over time involving rates of change in multiple dimensions.
Introduction to Partial Differential Equations: An overview of partial differential equations, their types, and their applications in physics, engineering, and other fields.
Classification of Partial Differential Equations: A classification of partial differential equations based on the order, type, and linearity of the equation and its coefficients.
Separation of Variables: A technique for solving partial differential equations by breaking down the problem into a number of simpler, independent problems.
Method of Characteristics: A method used to solve first-order partial differential equations, which involves finding curves along which the solution is constant.
Fourier Series and Transforms: An important mathematical tool for solving partial differential equations, which involves representing functions as a sum or integral of trigonometric functions.
Numerical Methods: Various numerical methods for solving partial differential equations, including finite difference, finite element, and spectral methods.
Boundary Value Problems: A type of partial differential equation where the solution is determined by conditions specified on the boundary of the domain.
Initial Value Problems: A type of partial differential equation where the solution is determined by specifying the initial conditions at a given time.
Nonlinear Partial Differential Equations: A more complex type of partial differential equation where the coefficients themselves can depend on the solution.
Applications of Partial Differential Equations: Some of the many applications of partial differential equations in physics, engineering, biology, and other fields.
Elliptic PDEs: They describe the steady-state behavior of a system and are characterized by smooth solutions with no singularities.
Parabolic PDEs: They describe the time-dependent behavior of a system. These equations usually have a unique solution, and the solution converges to a steady-state solution as time increases.
Hyperbolic PDEs: They describe the wave-like behavior of a system. Such equations have infinite families of solutions, and the solutions propagate in space and time.
Diffusion Equations: These are parabolic PDEs that describe the diffusion of a quantity over time and space.
Wave Equations: These are hyperbolic PDEs that describe the propagation of waves such as light and sound through a medium.
Laplace Equation: It is an elliptic PDE that describes the steady-state temperature distribution in a metal plate based on the heat diffusion equation.
Heat Equation: It is a parabolic PDE that describes the time-dependent temperature distribution in a system based on the conservation of energy.
Navier-Stokes Equations: The are a set of partial differential equations that describe the motion of fluids, including air and water.
"it is usually impossible to write down explicit formulas for solutions of 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."
"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."
"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."
"they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics."
"They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations."
"they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology."
"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."
"Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable."
"Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the 'PDE' notion."
"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.