Numerical Methods for Partial Differential Equations

ISBN-13: 9783540761259 / Angielski / Miękka / 1999 / 290 str.

G. A. Evans; P. D. Yardley; J. M. Blackledge

Opis

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.

Spis treści

1. Background Mathematics.- 1.1 Introduction.- 1.2 Vector and Matrix Norms.- 1.3 Gerschgorin’s Theorems.- 1.4 Iterative Solution of Linear Algebraic Equations.- 1.5 Further Results on Eigenvalues and Eigenvectors.- 1.6 Classification of Second Order Partial Differential Equations.- 2. Finite Differences and Parabolic Equations.- 2.1 Finite Difference Approximations to Derivatives.- 2.2 Parabolic Equations.- 2.3 Local Truncation Error.- 2.4 Consistency.- 2.5 Convergence.- 2.6 Stability.- 2.7 The Crank-Nicolson Implicit Method.- 2.8 Parabolic Equations in Cylindrical and Spherical Polar Coordinates.- 3. Hyperbolic Equations and Characteristics.- 3.1 First Order Quasi-linear Equations.- 3.2 Lax-Wendroff and Wendroff Methods.- 3.3 Second Order Quasi-linear Hyperbolic Equations.- 3.4 Reetangular Nets and Finite Difference Methods for Second Order Hyperbolic Equations.- 4. Elliptic Equations.- 4.1 Laplace’s Equation.- 4.2 Curved Boundaries.- 4.3 Solution of Sparse Systems of Linear Equations.- 5. Finite Element Method for Ordinary Differential Equations.- 5.1 Introduction.- 5.2 The Collocation Method.- 5.3 The Least Squares Method.- 5.4 The Galerkin Method.- 5.5 Symmetrie Variational Forrnulation.- 5.6 Finite Element Method.- 5.7 Some Worked Examples.- 6. Finite Elements for Partial Differential Equations.- 6.1 Introduction.- 6.2 Variational Methods.- 6.3 Some Specific Elements.- 6.4 Assembly of the Elements.- 6.5 Worked Example.- 6.6 A General Variational Principle.- 6.7 Assembly and Solution.- 6.8 Solution of the Worked Example.- 6.9 Further Interpolation Functions.- 6.10 Quadrature Methods and Storage Considerations.- 6.11 Boundary Element Method.- A. Solutions to Exercises.- References and Further Reading.