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Applied Numerical Analysis

Matiur Rahman
WIT Press
Publication Date: 
Number of Pages: 
[Reviewed by
Thomas Schulte
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This book springs from the author’s lecture notes for a one-semester course of engineering mathematics taught around the world to undergraduate students in many areas of applied mathematics. The text reads that way and it would be easy to feel brevity is being emphasized at the cost of depth. Sometimes the text reads so much like notes that one half-expects to encounter the chalkboard direction “draw picture here”. However, a slow and careful reading of any section proves that explanations and presentation is complete. Indeed, this thoroughly road-tested overview of numerical analysis is a refined and elegant treatment.

The well-organized textbook covers ordinary and partial differential equations, nonlinear differential equations and stability, calculus of variations, and applications. The breadth of the book is such that the discussion of finite differences will elucidate for the undergraduate the connection to derivatives while the in-depth look at the stability of subsystems via linearization and Liapounov provides suitable grist for the graduate.

The soul of this book is to underscore the importance and relevance of the underlying mathematics for avoiding errors in computer solutions while applying numerical methods and the calculus of variations to physical problems. Toward this end, spotlights of enlightenment occur throughout via more than 100 solved problems with detailed explanations. An additional 120 exercises feature solution hints and answers throughout the six sections of the book. What would really help in this book’s endeavor would be clear algorithms in pseudo-code for ready implementation or, better yet, mating the text with a popular mathematics package, such as Matlab or Maple and syntax-specific examples and exercises.

Accompanying the book is a CD-ROM, but it is limited to the Linux operating system. Among the appendices are tables of Laplace transforms and specialized Laplace inverses.

1 - Introduction

Preliminary background; Formulation of ordinary and partial differential equations; Singular solution and complete primitive; Initial and boundary value problems; Graphical representation; References
2 - Ordinary differential equations

Introduction; Some preliminaries of finite difference; Numerical differentiations; Numerical integrations; Numerical solution of ODE; Second- and higher-order equations; Finite difference methods; Application to practical problems; References
3 - Partial differential equations

Introduction; Finite difference methods; the finite element method; The boundary element method; References
4 - Nonlinear differential equations and stability

Introduction; Solutions of trajectories; The phase plane and the linear system; Stability of almost linear systems; Liapounov's second method; Periodic solutions and limit cycles; Some practical problems; References
5 - The calculus of variations

Introduction; Systems of Euler-Lagrange equations; The extrema of integrals under constraints; Sturm-Liouville problems; Principles of variations; Hamilton's principles; Hamilton's equations; Some practical problems; References
6 - Applications

Introduction; Stability of subsystems; Oceanwaves; Nonlinear wave-wave interactions; Seismic response of dams; Green's function method for waves; References