Partial Differential Equations

Many natural processes can be sufficiently well described on the macroscopic level, without taking into account the individual behavior of molecules, atoms, electrons, or other particles. The averaged quantities such as the deformation, density, velocity, pressure, temperature, concentration, or electromagnetic field are governed by partial differential equations (PDEs). These equations serve as a language for the formulation of many engineering and scientific problems. To give a few examples, PDEs are employed to predict and control the static and dynamic properties of constructions, flow of blood in human veins, flow of air past cars and airplanes, weather, thermal inhibition of tumors, heating and melting of metals, cleaning of air and water in urban facilities, burning of gas in vehicle engines, magnetic resonance imaging and computer tomography in medicine, and elsewhere. Most PDEs used in practice only contain the first and second partial derivatives (we call them second-order PDEs).

Chapter l provides an overview of basic facts and techniques that are essential for both the qualitative analysis and numerical solution of PDEs. After introducing the classification and mentioning some general properties of second-order equations in Section 1.1, we focus on specific properties of elliptic, parabolic, and hyperbolic PDEs in Sections 1.2-1.4. Indeed, there are important PDEs which are not of second order. To mention at least some of them, in Section 1.5 we discuss first-order hyperbolic problems that are frequently used to model transport processes such as, e.g., inviscid fluid flow. Fourth-order problems rooted in the bending of elastic beams and plates are discussed later in Chapter 6.

Partial Differential Equations and the Finite Element Method. By Pavel Solin Copyright © 2006 John Wiley & Sons, Inc.

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