Partial Differential Equations
Introduction
Partial Differential Equations (PDEs) are a type of differential equation, the general form of which is an equation that contains some function and its derivatives. In mathematics, PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.
Definition
A partial differential equation is an equation which involves 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; for the fluid, they happen in an infinite-dimensional configuration space. This distinction often leads to different mathematical treatments, such as differential equations for the rigid body and partial differential equations for the fluid.
Types of Partial Differential Equations
There are several types of partial differential equations, classified in a variety of ways, including by degree, linearity and the properties of the equations in mathematical physics.
First Order
First order partial differential equations are classified as linear and nonlinear PDEs. Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, one has found the general solution to the equation.
Second Order
Second order partial differential equations are classified as either elliptic, hyperbolic, or parabolic. Integrating factor methods are often the most straightforward way to solve second order equations.
Solutions of Partial Differential Equations
Solving PDEs can be hard, even impossible in terms of expressing the solution in a closed-form, i.e., in terms of functions whose names are known. Many of the standard methods for solving PDEs are applicable only for linear homogeneous equations, and for the most part, cannot be used for other cases. Thus, one often resorts to numerical methods for solving PDEs.
Applications of Partial Differential Equations
Partial differential equations have a wide range of applications in various physical and engineering disciplines. These include wave propagation, heat conduction, fluid flow, elasticity, electrostatics, quantum mechanics, and many others.