Numerical Methods for Engineers and Scientists, Second Edition,

by Hoffman; Joe D.

• ISBN13: 9780824704438
• ISBN10: 0824704436
• eBook ISBN(s): 9781351990943
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-05-31
• Publisher: CRC Press
• More Book Details

Summary

This text places emphasis on solving differential equations using the finite difference approach. Among the other features are problem-solving methods, clear identification of the pitfalls of using selected algorithms, and more on practical solutions than theoretical abstraction.

Author Biography

Joe D. Hoffman is Professor of Mechanical Engineering and Director of the Maurice J. Zucrow Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana.

Table of Contents

Preface	p. v
Introduction	p. 1
Objectives and Approach	p. 1
Organization of the Book	p. 2
Examples	p. 2
Programs	p. 3
Problems	p. 3
Significant Digits, Precision, Accuracy, Errors, and Number Representation	p. 4
Software Packages and Libraries	p. 6
The Taylor Series and the Taylor Polynomial	p. 7
Basic Tools of Numerical Analysis	p. 11
Systems of Linear Algebraic Equations	p. 11
Eigenproblems	p. 13
Roots of Nonlinear Equations	p. 14
Polynomial Approximation and Interpolation	p. 14
Numercial Differentiation and Difference Formulas	p. 15
Numerical Integration	p. 16
Summary	p. 16
Systems of Linear Algebraic Equations	p. 17
Introduction	p. 18
Properties of Matrices and Determinants	p. 21
Direct Elimination Methods	p. 30
LU Factorization	p. 45
Tridiagonal Systems of Equations	p. 49
Pitfalls of Elimination Methods	p. 52
Iterative Methods	p. 59
Programs	p. 67
Summary	p. 76
Exercise Problems	p. 77
Eigenproblems	p. 81
Introduction	p. 81
Mathematical Characteristics of Eigenproblems	p. 85
The Power Method	p. 89
The Direct Method	p. 101
The QR Method	p. 104
Eigenvectors	p. 110
Other Methods	p. 111
Programs	p. 112
Summary	p. 118
Exercise Problems	p. 119
Nonlinear Equations	p. 127
Introduction	p. 127
General Features of Root Finding	p. 130
Closed Domain (Bracketing) Methods	p. 135
Open Domain Methods	p. 140
Polynomials	p. 155
Pitfalls of Root Finding Methods and Other Methods of Root Finding	p. 167
Systems of Nonlinear Equations	p. 169
Programs	p. 173
Summary	p. 179
Exercise Problems	p. 181
Polynomial Approximation and Interpolation	p. 187
Introduction	p. 188
Properties of Polynomials	p. 190
Direct Fit Polynomials	p. 197
Lagrange Polynomials	p. 198
Divided Difference Tables and Divided Difference Polynomials	p. 204
Difference Tables and Difference Polynomials	p. 208
Inverse Interpolation	p. 217
Multivariate Approximation	p. 218
Cubic Splines	p. 221
Least Squares Approximation	p. 225
Programs	p. 235
Summary	p. 242
Exercise Problems	p. 243
Numerical Differentiation and Difference Formulas	p. 251
Introduction	p. 251
Unequally Spaced Data	p. 254
Equally Spaced Data	p. 257
Taylor Series Approach	p. 264
Difference Formulas	p. 270
Error Estimation and Extrapolation	p. 270
Programs	p. 273
Summary	p. 279
Exercise Problems	p. 279
Numerical Integration	p. 285
Introduction	p. 285
Direct Fit Polynomials	p. 288
Newton-Cotes Formulas	p. 290
Extrapolation and Romberg Integration	p. 297
Adaptive Integration	p. 299
Gaussian Quadrature	p. 302
Multiple Integrals	p. 306
Programs	p. 311
Summary	p. 315
Exercise Problems	p. 316
Ordinary Differential Equations	p. 323
Introduction	p. 323
General Features of Ordinary Differential Equations	p. 323
Classification of Ordinary Differential Equations	p. 325
Classification of Physical Problems	p. 326
Initial-Value Ordinary Differential Equations	p. 327
Boundary-Value Ordinary Differential Equations	p. 330
Summary	p. 332
One-Dimensional Initial-Value Ordinary Differential Equations	p. 335
Introduction	p. 336
General Features of Initial-Value ODEs	p. 340
The Taylor Series Method	p. 343
The Finite Difference Method	p. 346
The First-Order Euler Methods	p. 352
Consistency, Order, Stability, and Convergence	p. 359
Single-Point Methods	p. 364
Extrapolation Methods	p. 378
Multipoint Methods	p. 381
Summary of Methods and Results	p. 391
Nonlinear Implicit Finite Difference Equations	p. 393
Higher-Order Ordinary Differential Equations	p. 397
Systems of First-Order Ordinary Differential Equations	p. 398
Stiff Ordinary Differential Equations	p. 401
Programs	p. 408
Summary	p. 414
Exercise Problems	p. 416
One-Dimensional Boundary-Value Ordinary Differential Equations	p. 435
Introduction	p. 436
General Features of Boundary-Value ODEs	p. 439
The Shooting (Initial-Value) Method	p. 441
The Equilibrium (Boundary-Value) Method	p. 450
Derivative (and Other) Boundary Conditions	p. 458
Higher-Order Equilibrium Methods	p. 466
The Equilibrium Method for Nonlinear Boundary-Value Problems	p. 471
The Equilibrium Method on Nonuniform Grids	p. 477
Eigenproblems	p. 480
Programs	p. 483
Summary	p. 488
Exercise Problems	p. 490
Partial Differential Equations	p. 501
Introduction	p. 501
General Features of Partial Differential Equations	p. 502
Classification of Partial Differential Equations	p. 504
Classification of Physical Problems	p. 511
Elliptic Partial Differential Equations	p. 516
Parabolic Partial Differential Equations	p. 519
Hyperbolic Partial Differential Equations	p. 520
The Convection-Diffusion Equation	p. 523
Initial Values and Boundary Conditions	p. 524
Well-Posed Problems	p. 525
Summary	p. 526
Elliptic Partial Differential Equations	p. 527
Introduction	p. 527
General Features of Elliptic PDEs	p. 531
The Finite Difference Method	p. 532
Finite Difference Solution of the Laplace Equation	p. 536
Consistency, Order, and Convergence	p. 543
Iterative Methods of Solution	p. 546
Derivative Boundary Conditions	p. 550
Finite Difference Solution of the Poisson Equation	p. 552
Higher-Order Methods	p. 557
Nonrectangular Domains	p. 562
Nonlinear Equations and Three-Dimensional Problems	p. 570
The Control Volume Method	p. 571
Programs	p. 575
Summary	p. 580
Exercise Problems	p. 582
Parabolic Partial Differential Equations	p. 587
Introduction	p. 587
General Features of Parabolic PDEs	p. 591
The Finite Difference Method	p. 593
The Forward-Time Centered-Space (FTCS) Method	p. 599
Consistency, Order, Stability, and Convergence	p. 605
The Richardson and DuFort-Frankel Methods	p. 611
Implicit Methods	p. 613
Derivative Boundary Conditions	p. 623
Nonlinear Equations and Multidimensional Problems	p. 625
The Convection-Diffusion Equation	p. 629
Asymptotic Steady State Solution to Propagation Problems	p. 637
Programs	p. 639
Summary	p. 645
Exercise Problems	p. 646
Hyperbolic Partial Differential Equations	p. 651
Introduction	p. 651
General Features of Hyperbolic PDEs	p. 655
The Finite Difference Method	p. 657
The Forward-Time Centered-Space (FTCS) Method and the Lax Method	p. 659
Lax-Wendroff Type Methods	p. 655
Upwind Methods	p. 673
The Backward-Time Centered-Space (BTCS) Method	p. 677
Nonlinear Equations and Multidimensional Problems	p. 682
The Wave Equation	p. 683
Programs	p. 691
Summary	p. 701
Exercise Problems	p. 702
The Finite Element Method	p. 711
Introduction	p. 711
The Rayleigh-Ritz, Collocation, and Galerkin Methods	p. 713
The Finite Element Method for Boundary Value Problems	p. 724
The Finite Element Method for the Laplace (Poisson) Equation	p. 739
The Finite Element Method for the Diffusion Equation	p. 752
Programs	p. 759
Summary	p. 769
Exercise Problems	p. 770
References	p. 775
Answers to Selected Problems	p. 779
Index	p. 795
Table of Contents provided by Syndetics. All Rights Reserved.