Synopses & Reviews
The subject of partial differential equations holds an exciting place in mathematics. Inevitably, the subject falls into several areas of mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The book presents a clear introduction of the methods and underlying theory used in the numerical solution of partial differential equations. After revising the mathematical preliminaries, the book covers the finite difference method of parabolic or heat equations, hyperbolic or wave equations and elliptic or Laplace equations. Throughout, the emphasis is on the practical solution rather than the theoretical background, without sacrificing rigour.
Synopsis
The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. James Clerk Maxwell, for example, put electricity and magnetism into a unified theory by establishing Maxwell's equations for electromagnetic theory, which gave solutions for prob- lems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechanical processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier- Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forecasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics.
Synopsis
Partial differential equations fall into several areas of mathematics: many of the greatest advances in modern science have been based on discovering the underlying PDE for the process in question. In this book, the emphasis is on the practical solution of problems rather than the theoretical background. An introductory chapter recaps the mathematical preliminaries required and exercises with solutions are provided. With its companion volume, Analytic Methods for Partial Differential Equations, it provides a complete introduction to the subject.
Synopsis
The subject of PDE's holds an exciting place in mathematics. At one extreme, the interest lies in the existence and uniqueness of solutions, and in theoretical treatment of the material. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, which can be used in design and construction. These books aim to actually solve equations rather than discuss the theoretical properties of their solutions, and will serve as a comprehensive, highly accessible introduction to the subject.
Description
Includes bibliographical references (p. 287-288) and index.
Table of Contents
1. Background Mathematics.- 2. Finite Differences and Parabolic Equations.- 3. Hyperbolic Equations and Characteristics.- 4. Elliptic Equations.- 5. Finite Element Method for Ordinary Differential Equations.- 6. Finite Elements.- Appendix.- References.- Index.