Analytical and Computational Methods of Advanced Engineering Mathematics

ISBN-13: 9781461268475 / Angielski / Miękka / 2012 / 733 str.

Grant B. Gustafson; Calvin H. Wilcox; Grant B. Gustafson

Opis

This book focuses on the topics which provide the foundation for practicing engineering mathematics: ordinary differential equations, vector calculus, linear algebra and partial differential equations. Destined to become the definitive work in the field, the book uses a practical engineering approach based upon solving equations and incorporates computational techniques throughout.

Recenzje

G.B. Gustafson and C. H. Wilcox

Analytical and Computational Methods of Advanced Engineering Mathematics

"This is a modern version of the textbook on applied mathematics for undergraduate engineering students. It provides a quick survey to ordinary differential equations, linear algebra, the Laplace transform, vector analysis, Fourier series and other eigenfunction expansions, and Fourier transforms, with applications to the wave equation and heat equation . . . has an excellent set of exercises, and the appendix provides answers."-ZENTRALBLATT MATH

Spis treści

1 Numerical Analysis.- 1.1 The Nature of Numerical Analysis.- 1.2 Polynomial Interpolation.- 1.3 Numerical Integration and Differentiation.- 1.4 Solution of Equations.- 1.5 Inverse Functions.- 1.6 Implicit Functions.- 1.7 Numerical Summation of Infinite Series.- 2 Ordinary Differential Equations of First Order.- 2.1 The Nature of Differential Equations.- 2.2 Separable Equations.- 2.3 Linear First-Order Equations.- 2.4 Exact Equations.- 2.5 Applications to Some Second-Order Equations.- 2.6 The Initial Value Problem.- 2.7 Numerical Methods for the Initial Value Problem.- 3 Ordinary Differential Equations of Higher Order.- 3.1 Examples from Engineering and Physics.- 3.2 Linear Second-Order Equations — Structure of Solutions.- 3.3 Linear Second-Order Equations with Constant Coefficients.- 3.4 Linear Second-Order Equations with Analytic Coefficients.- 3.5 Numerical Methods for Second-Order Equations.- 3.6 Linear Equations of Order n > 2.- 4 The Laplace Transform.- 4.1 The Nature of the Laplace Transform.- 4.2 The Laplace Transforms of Some Elementary Functions.- 4.3 Operational Rules for the Laplace Transform.- 4.4 Applications to Differential Equations.- 4.5 Applications to Systems of Differential Equations.- 5 Linear Algebra.- 5.1 Systems of Linear Equations.- 5.2 The Gauss Elimination Method.- 5.3 Vector Spaces.- 5.4 Matrices and Matrix Algebra.- 5.5 The Fundamental Theorem of Linear Algebra.- 5.6 Determinants and Cramer’s Rule.- 5.7 Eigenvalues and Eigenvectors.- 6 Vector Analysis.- 6.1 Vector Algebra.- 6.2 Vector Calculus of Curves in Space.- 6.3 Vector Calculus of Surfaces in Space.- 6.4 Calculus of Scalar and Vector Fields.- 6.5 Integral Theorems of Vector Calculus.- 6.6 X-Ray Diffraction and Crystal Structure.- 7 Partial Differential Equations of Mathematical Physics.- 7.1 Vibrating Strings: D’Alembert’s Wave Equation.- 7.2 Heat Diffusion in Rods: Fourier’s Heat Equation.- 7.3 Heat Diffusion in Plates.- 7.4 Steady-State Heat Diffusion in Plates: The Laplace Equation.- 7.5 Vibrations of Drums.- 7.6 Heat Diffusion in Solids.- 7.7 Steady-State Heat Diffusion in Solids.- 8 Fourier Analysis and Sturm-Liouville Theory.- 1 Fourier Series.- 8.1 Dirichlet Boundary Conditions and Fourier Sine Series.- 8.2 Orthogonality and Fourier Coefficients.- 8.3 Convergence of Fourier Sine Series.- 8.4 Neumann Boundary Conditions and Fourier Cosine Series.- 8.5 Periodic Boundary Conditions and the Complete Fourier Series.- 8.6 Proofs of the Convergence Theorems (Optional).- II Fourier Integrals.- 8.7 Heat Diffusion in an Infinite Rod.- 8.8 Orthogonality Calculation.- 8.9 The Fourier Integral.- 8.10 Fourier Sine and Cosine Integrals.- III Sturm-Liouville Theory.- 8.11 Heat Diffusion in Nonhomogeneous Rods.- 8.12 Sturm-Liouville Problems: Basic Theory.- 8.13 Construction of Eigenvalues and Eigenfunctions.- 8.14 Singular Sturm-Liouville Problems.- 9 Boundary Value Problems of Mathematical Physics.- 9.1 Heat Diffusion in One Dimension.- 9.2 Vibration of Strings and Traveling Waves.- 9.3 Steady-State Diffusion of Heat in Plates.- 9.4 Transient Diffusion of Heat in Plates.- 9.5 Vibrations of Drums.- 9.6 Steady-State Diffusion of Heat in Solids.- 9.7 The Laplace Transform Method.- Appendix: Answers and Hints to Selected Exercises.- References.