Abstract
Scientists, engineers and applied mathematicians have always seen the value of obtaining precise solutions to the problems that they work on. Because of the deterministic nature of physical laws, the solution of many of these problems is governed by partial differential equations (PDEs). In this book we focus on the most frequently occurring partial differential equations (PDEs) that engineers or scientists might have to solve numerically in the course of their work. The emphasis throughout this book will not be on mathematical rigor but rather on intuitive understanding that is later supported by several detailed examples that help one develop a practical knowledge of the solution techniques. In this chapter we introduce some of the partial differential equations, understand how they arise and show that they fall into certain familiar categories. The PDEs that interest us can be divided into some simple classes that go under interesting names such as: hyperbolic, parabolic and elliptic. We explain these new names with some intuitive definitions and familiar examples. Several examples of useful PDEs that play an important role in science and engineering are discussed in the later sections of this chapter.
Outline of the Chapter
- 1.1) Introduction
- 1.2) Derivation of the Fluid Equations
- 1.2.1) The Role of Collisions
- 1.2.2) The Boltzmann Equation
- 1.2.3) Moments of the Boltzmann Equation
- 1.2.4) The Continuity Equation
- 1.2.5) The Momentum Equation
- 1.2.6) The Energy Equation
- 1.3) The Euler Equations
- 1.4) The Navier-Stokes Equations
- 1.5) Classifying and Understanding PDEs
- 1.6) Incompressible Flow Equations
- 1.7) The Shallow Water Equations
- 1.8) Maxwell’s Equations
- 1.9) The Magnetohydrodynamic Equations
- 1.10) Flux Limited Diffusion Radiation Hydrodynamics
- 1.11) The Radiative Transfer Equation
- 1.12) The Equations of Linear Elasticity
- 1.13) Relativistic Hydrodynamics and Magnetohydrodynamics