Advanced Engineering Mathematics with Modeling Applications

by Kelly; S. Graham

• ISBN13: 9780849395338
• ISBN10: 084939533X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-12-05
• Publisher: CRC Press

Summary

Engineers require a solid knowledge of the relationship between engineering applications and their underlying mathematical theory. However, most books do not present sufficient theory or they do not fully explain its importance and relevance in understanding those applications. Advanced Engineering Mathematics with Modeling Applications employs a balanced approach to address this need, providing a solid comprehension of mathematical theory that will enhance understanding of applications - and vice versa. With a focus on modeling, this book illustrates why mathematical methods work, when they apply, and what their limitations are.

Table of Contents

Preface	p. xi
Author	p. xiii
Foundations of mathematical modeling	p. 1
Engineering analysis	p. 1
Conservation laws and mathematical modeling	p. 6
Problem formulation	p. 10
Nondimensionalization	p. 27
Scaling	p. 45
Problems	p. 56
Linear algebra	p. 65
Introduction	p. 65
Three-dimensional space	p. 66
Vector spaces	p. 73
Linear independence	p. 77
Basis and dimension	p. 80
Inner products	p. 86
Norms	p. 93
Gram-Schmidt orthonormalization	p. 98
Orthogonal expansions	p. 105
Linear operators	p. 109
Adjoint operators	p. 121
Positive definite operators	p. 129
Energy inner products	p. 132
Problems	p. 139
Ordinary differential equations	p. 143
Linear differential equations	p. 143
General theory for second-order differential equations	p. 146
Differential equations with constant coefficients	p. 149
Differential equations with variable coefficients	p. 167
Singular points of second-order equations	p. 173
Bessel functions	p. 180
Differential equations whose solutions are expressible in terms of Bessel functions	p. 193
Legendre functions	p. 198
Problems	p. 201
Variational methods	p. 209
Introduction	p. 209
The general variational problem	p. 210
Variational solutions of operator equations	p. 224
Method of least squares	p. 224
Rayleigh-Ritz method	p. 226
Finite-element method	p. 242
Galerkin's method	p. 270
Problems	p. 273
Eigenvalue problems	p. 277
Eigenvalue and eigenvector problems	p. 277
Eigenvalues of adjoint operators	p. 278
Eigenvalues of positive definite operators	p. 279
Eigenvalue problems for operators in finite-dimensional vector spaces	p. 280
Second-order differential operators	p. 287
Eigenvector expansions	p. 303
Completeness theorem	p. 303
Trigonometric fourier series	p. 309
Completeness of eigenvectors for self-adjoint and non-self-adjoint operators	p. 312
Solution of nonhomogeneous equations using eigenvector expansions	p. 314
Fourth-order differential operators	p. 318
Differential operators with eigenvalues in boundary conditions	p. 329
Eigenvalue problems involving Bessel functions	p. 336
Eigenvalue problems in other infinite-dimensional vector spaces	p. 349
Solvability conditions	p. 354
Asymptotic approximations to solutions of eigenvalue problems	p. 358
Rayleigh's quotient	p. 370
Rayleigh-Ritz method	p. 376
Green's functions	p. 394
Problems	p. 408
Partial differential equations	p. 419
Homogeneous partial differential equations	p. 419
Second-order steady-state problems, Laplace's equation	p. 421
Time-dependent problems: Initial value problems	p. 441
Nonhomogeneous partial differential equations	p. 455
Problems in cylindrical coordinates	p. 471
Problems in spherical coordinates	p. 491
Problems	p. 508
Index	p. 517
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