Partial Differential Equation (PDE)

A Partial Differential Equation (PDE) is a differential equation that imposes relations between the various partial derivatives of a multivariable function.

• Context:
  • It can range from linear partial differential equation to nonlinear partial differential equation.
  • It can range from homogeneous partial differential equation to nonhomogeneous partial differential equation.
  • It can be models of various physical and geometrical problems, arising when the unknown functions (the solutions) depend on two or more variables, usually on time [math]\displaystyle{ t }[/math] and one or several space variables.
  • It can be associated to a Partial Derivative.
  • It can be solved by PDE Solver (solving a PDE task).
• Example(s):
  • one dimensional wave equation ([math]\displaystyle{ \frac{\partial^2 u}{\partial t^2}=c^2 \frac{\partial^2 u}{\partial x^2} }[/math]).
  • a Schrödinger Equation.
  • a Stochastic PDE.
  • a Black-Scholes Equation.
  • …
• Counter-Example(s):
  • an Ordinary Differential Equation.
• See: Differential Equation, Finite Element Algorithm, System of Equations, Fluid Flow, Continuous Distribution, Electrodynamics, Linear Map, Heat Equation, Wave Equation, Laplace's Equation.

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References

2020

• (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Partial_differential_equation Retrieved:2020-9-11.
  • In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.

    The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number, to be solved for, in an algebraic equation like x² − 3x + 2 0. However, 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. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations.

    Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, 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; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.

    Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, 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. More 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.

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partial_differential_equation#Introduction Retrieved:2014-12-3.
  • Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, electrodynamics, and heat transfer.

    A partial differential equation (PDE) for the function [math]\displaystyle{ u(x_1, \cdots, x_n) }[/math] is an equation of the form

     : [math]\displaystyle{ F \left (x_1, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \ldots, \frac{\partial u}{\partial x_n}, \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots, \frac{\partial^2 u}{\partial x_1 \partial x_n}, \ldots \right) = 0. }[/math]

    If F is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

    A relatively simple PDE is

     : [math]\displaystyle{ \frac{\partial u}{\partial x}(x,y) = 0.~ }[/math]

    This relation implies that the function u(x,y) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is

     : [math]\displaystyle{ u(x,y) = f(y), }[/math]

    where f is an arbitrary function of y. The analogous ordinary differential equation is

     : [math]\displaystyle{ \frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0, }[/math]

    which has the solution

     : [math]\displaystyle{ u(x) = c, }[/math]

    where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function f(y) can be determined if u is specified on the line x = 0.

1950

• (Hopf, 1950) ⇒ Eberhard Hopf. (1950). “The partial differential equation ut + uux = xx.” In: Communications on Pure and Applied Mathematics, 3(3).

1944

• (Bateman, 1944) ⇒ Harry Bateman. (1944). “Partial Differential Equations." Read Books. ISBN:1406743747

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