Numerical Methods for Engineers and Scientists Using MATLAB®

Ramin S. Esfandiari

• Verlag: CRC Press
• Jahr: 2017
• ISBN:
  9781498777452
• Auflage: 2
• Thema: Mathematics
• Thema: Applied Mathematics

This book provides a pragmatic, methodical and easy-to-follow presentation of numerical methods and their effective implementation using MATLAB, which is introduced at the outset. The author introduces techniques for solving equations of a single variable and systems of equations, followed by curve fitting and interpolation of data. The book also provides detailed coverage of numerical differentiation and integration, as well as numerical solutions of initial-value and boundary-value problems. The author then presents the numerical solution of the matrix eigenvalue problem, which entails approximation of a few or all eigenvalues of a matrix. The last chapter is devoted to numerical solutions of partial differential equations that arise in engineering and science. Each method is accompanied by at least one fully worked-out example showing essential details involved in preliminary hand calculations, as well as computations in MATLAB.

1

Background and Introduction

This chapter is divided into two parts. In Part 1, a review of some essential mathematical concepts as related to differential equations and matrix analysis is presented. In Part 2, fundamentals of numerical methods, such as sources of computational errors, as well as iterations and rates of convergence are introduced. The materials presented here will be fully integrated throughout the book.

Part 1: Background

1.1 Differential Equations

Differential equations are divided into two main groups: ordinary differential equations (ODEs) and partial differential equations (PDEs). An equation involving an unknown function and one or more of its derivatives is called a differential equation. When there is only one independent variable, the equation is called an ODE. If the unknown function is a function of several independent variables, the equation is a PDE. For example,

is an ODE involving the unknown function x(t), its first derivative with respect to t, as well as a given function e−t. Similarly,

is an ODE relating x(t) and its first and second derivatives with respect to t, as well as the f...