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Methods of Applied Mathematics for Engineers and Scientists

Book Description

Based on course notes from over twenty years of teaching engineering and physical sciences at Michigan Technological University, Tomas Co's engineering mathematics textbook is rich with examples, applications and exercises. Professor Co uses analytical approaches to solve smaller problems to provide mathematical insight and understanding, and numerical methods for large and complex problems. The book emphasises applying matrices with strong attention to matrix structure and computational issues such as sparsity and efficiency. Chapters on vector calculus and integral theorems are used to build coordinate-free physical models with special emphasis on orthogonal co-ordinates. Chapters on ODEs and PDEs cover both analytical and numerical approaches. Topics on analytical solutions include similarity transform methods, direct formulas for series solutions, bifurcation analysis, Lagrange–Charpit formulas, shocks/rarefaction and others. Topics on numerical methods include stability analysis, DAEs, high-order finite-difference formulas, Delaunay meshes, and others. MATLAB® implementations of the methods and concepts are fully integrated.

Table of Contents

  1. Cover
  2. Half Title
  3. Title
  4. Copyright
  5. Table of Contents
  6. Preface
  7. I MATRIX THEORY
    1. 1 Matrix Algebra
      1. 1.1 Definitions and Notations
      2. 1.2 Fundamental Matrix Operations
      3. 1.3 Properties of Matrix Operations
      4. 1.4 Block Matrix Operations
      5. 1.5 Matrix Calculus
      6. 1.6 Sparse Matrices
      7. 1.7 Exercises
    2. 2 Solution of Multiple Equations
      1. 2.1 Gauss-Jordan Elimination
      2. 2.2 LU Decomposition
      3. 2.3 Direct Matrix Splitting
      4. 2.4 Iterative Solution Methods
      5. 2.5 Least-Squares Solution
      6. 2.6 QR Decomposition
      7. 2.7 Conjugate Gradient Method
      8. 2.8 GMRES
      9. 2.9 Newton’s Method
      10. 2.10 Enhanced Newton Methods via Line Search
      11. 2.11 Exercises
    3. 3 Matrix Analysis
      1. 3.1 Matrix Operators
      2. 3.2 Eigenvalues and Eigenvectors
      3. 3.3 Properties of Eigenvalues and Eigenvectors
      4. 3.4 Schur Triangularization and Normal Matrices
      5. 3.5 Diagonalization
      6. 3.6 Jordan Canonical Form
      7. 3.7 Functions of Square Matrices
      8. 3.8 Stability of Matrix Operators
      9. 3.9 Singular Value Decomposition
      10. 3.10 Polar Decomposition
      11. 3.11 Matrix Norms
      12. 3.12 Exercises
  8. II VECTORS AND TENSORS
    1. 4 Vector and Tensor Algebra and Calculus
      1. 4.1 Notations and Fundamental Operations
      2. 4.2 Vector Algebra Based on Orthonormal Basis Vectors
      3. 4.3 Tensor Algebra
      4. 4.4 Matrix Representation of Vectors and Tensors
      5. 4.5 Differential Operations for Vector Functions of One Variable
      6. 4.6 Application to Position Vectors
      7. 4.7 Differential Operations for Vector Fields
      8. 4.8 Curvilinear Coordinate System: Cylindrical and Spherical
      9. 4.9 Orthogonal Curvilinear Coordinates
      10. 4.10 Exercises
    2. 5 Vector Integral Theorems
      1. 5.1 Green’s Lemma
      2. 5.2 Divergence Theorem
      3. 5.3 Stokes’ Theorem and Path Independence
      4. 5.4 Applications
      5. 5.5 Leibnitz Derivative Formula
      6. 5.6 Exercises
  9. III ORDINARY DIFFERENTIAL EQUATIONS
    1. 6 Analytical Solutions of Ordinary Differential Equations
      1. 6.1 First-Order Ordinary Differential Equations
      2. 6.2 Separable Forms via Similarity Transformations
      3. 6.3 Exact Differential Equations via Integrating Factors
      4. 6.4 Second-Order Ordinary Differential Equations
      5. 6.5 Multiple Differential Equations
      6. 6.6 Decoupled System Descriptions via Diagonalization
      7. 6.7 Laplace Transform Methods
      8. 6.8 Exercises
    2. 7 Numerical Solution of Initial and Boundary Value Problems
      1. 7.1 Euler Methods
      2. 7.2 Runge Kutta Methods
      3. 7.3 Multistep Methods
      4. 7.4 Difference Equations and Stability
      5. 7.5 Boundary Value Problems
      6. 7.6 Differential Algebraic Equations
      7. 7.7 Exercises
    3. 8 Qualitative Analysis of Ordinary Differential Equations
      1. 8.1 Existence and Uniqueness
      2. 8.2 Autonomous Systems and Equilibrium Points
      3. 8.3 Integral Curves, Phase Space, Flows, and Trajectories
      4. 8.4 Lyapunov and Asymptotic Stability
      5. 8.5 Phase-Plane Analysis of Linear Second-Order Autonomous Systems
      6. 8.6 Linearization Around Equilibrium Points
      7. 8.7 Method of Lyapunov Functions
      8. 8.8 Limit Cycles
      9. 8.9 Bifurcation Analysis
      10. 8.10 Exercises
    4. 9 Series Solutions of Linear Ordinary Differential Equations
      1. 9.1 Power Series Solutions
      2. 9.2 Legendre Equations
      3. 9.3 Bessel Equations
      4. 9.4 Properties and Identities of Bessel Functions and Modified Bessel Functions
      5. 9.5 Exercises
  10. IV PARTIAL DIFFERENTIAL EQUATIONS
    1. 10 First-Order Partial Differential Equations and the Method of Characteristics
      1. 10.1 The Method of Characteristics
      2. 10.2 Alternate Forms and General Solutions
      3. 10.3 The Lagrange-Charpit Method
      4. 10.4 Classification Based on Principal Parts
      5. 10.5 Hyperbolic Systems of Equations
      6. 10.6 Exercises
    2. 11 Linear Partial Differential Equations
      1. 11.1 Linear Partial Differential Operator
      2. 11.2 Reducible Linear Partial Differential Equations
      3. 11.3 Method of Separation of Variables
      4. 11.4 Nonhomogeneous Partial Differential Equations
      5. 11.5 Similarity Transformations
      6. 11.6 Exercises
    3. 12 Integral Transform Methods
      1. 12.1 General Integral Transforms
      2. 12.2 Fourier Transforms
      3. 12.3 Solution of PDEs Using Fourier Transforms
      4. 12.4 Laplace Transforms
      5. 12.5 Solution of PDEs Using Laplace Transforms
      6. 12.6 Method of Images
      7. 12.7 Exercises
    4. 13 Finite Difference Methods
      1. 13.1 Finite Difference Approximations
      2. 13.2 Time-Independent Equations
      3. 13.3 Time-Dependent Equations
      4. 13.4 Stability Analysis
      5. 13.5 Exercises
    5. 14 Method of Finite Elements
      1. 14.1 The Weak Form
      2. 14.2 Triangular Finite Elements
      3. 14.3 Assembly of Finite Elements
      4. 14.4 Mesh Generation
      5. 14.5 Summary of Finite Element Method
      6. 14.6 Axisymmetric Case
      7. 14.7 Time-Dependent Systems
      8. 14.8 Exercises
  11. Bibliography
  12. Index
  13. A Additional Details and Fortification for Chapter 1
    1. A.1 Matrix Classes and Special Matrices
    2. A.2 Motivation for Matrix Operations from Solution of Equations
    3. A.3 Taylor Series Expansion
    4. A.4 Proofs for Lemma and Theorems of Chapter 1
    5. A.5 Positive Definite Matrices
  14. B Additional Details and Fortification for Chapter 2
    1. B.1 Gauss Jordan Elimination Algorithm
    2. B.2 SVD to Determine Gauss-Jordan Matrices Q and W
    3. B.3 Boolean Matrices and Reducible Matrices
    4. B.4 Reduction of Matrix Bandwidth
    5. B.5 Block LU Decomposition
    6. B.6 Matrix Splitting: Diakoptic Method and Schur Complement Method
    7. B.7 Linear Vector Algebra: Fundamental Concepts
    8. B.8 Determination of Linear Independence of Functions
    9. B.9 Gram-Schmidt Orthogonalization
    10. B.10 Proofs for Lemma and Theorems in Chapter 2
    11. B.11 Conjugate Gradient Algorithm
    12. B.12 GMRES Algorithm
    13. B.13 Enhanced-Newton Using Double-Dogleg Method
    14. B.14 Nonlinear Least Squares via Levenberg-Marquardt
  15. C Additional Details and Fortification for Chapter 3
    1. C.1 Proofs of Lemmas and Theorems of Chapter 3
    2. C.2 QR Method for Eigenvalue Calculations
    3. C.3 Calculations for the Jordan Decomposition
    4. C.4 Schur Triangularization and SVD
    5. C.5 Sylvester’s Matrix Theorem
    6. C.6 Danilevskii Method for Characteristic Polynomial
  16. D Additional Details and Fortification for Chapter 4
    1. D.1 Proofs of Identities of Differential Operators
    2. D.2 Derivation of Formulas in Cylindrical Coordinates
    3. D.3 Derivation of Formulas in Spherical Coordinates
  17. E Additional Details and Fortification for Chapter 5
    1. E.1 Line Integrals
    2. E.2 Surface Integrals
    3. E.3 Volume Integrals
    4. E.4 Gauss-Legendre Quadrature
    5. E.5 Proofs of Integral Theorems
  18. F Additional Details and Fortification for Chapter 6
    1. F.1 Supplemental Methods for Solving First-Order ODEs
    2. F.2 Singular Solutions
    3. F.3 Finite Series Solution of dx/dt = Ax + b(t)
    4. F.4 Proof for Lemmas and Theorems in Chapter 6
  19. G Additional Details and Fortification for Chapter 7
    1. G.1 Differential Equation Solvers in MATLAB
    2. G.2 Derivation of Fourth-Order Runge Kutta Method
    3. G.3 Adams-Bashforth Parameters
    4. G.4 Variable Step Sizes for BDF
    5. G.5 Error Control by Varying Step Size
    6. G.6 Proof of Solution of Difference Equation, Theorem 7.1
    7. G.7 Nonlinear Boundary Value Problems
    8. G.8 Ricatti Equation Method
  20. H Additional Details and Fortification for Chapter 8
    1. H.1 Bifurcation Analysis
  21. I Additional Details and Fortification for Chapter 9
    1. I.1 Details on Series Solution of Second-Order Systems
    2. I.2 Method of Order Reduction
    3. I.3 Examples of Solution of Regular Singular Points
    4. I.4 Series Solution of Legendre Equations
    5. I.5 Series Solution of Bessel Equations
    6. I.6 Proofs for Lemmas and Theorems in Chapter 9
  22. J Additional Details and Fortification for Chapter 10
    1. J.1 Shocks and Rarefaction
    2. J.2 Classification of Second-Order Semilinear Equations: n > 2
    3. J.3 Classification of High-Order Semilinear Equations
  23. K Additional Details and Fortification for Chapter 11
    1. K.1 d’Alembert Solutions
    2. K.2 Proofs of Lemmas and Theorems in Chapter 11
  24. L Additional Details and Fortification for Chapter 12
    1. L.1 The Fast Fourier Transform
    2. L.2 Integration of Complex Functions
    3. L.3 Dirichlet Conditions and the Fourier Integral Theorem
    4. L.4 Brief Introduction to Distribution Theory and Delta Distributions
    5. L.5 Tempered Distributions and Fourier Transforms
    6. L.6 Supplemental Lemmas, Theorems, and Proofs
    7. L.7 More Examples of Laplace Transform Solutions
    8. L.8 Proofs of Theorems Used in Distribution Theory
  25. M Additional Details and Fortification for Chapter 13
    1. M.1 Method of Undetermined Coefficients for Finite Difference Approximation of Mixed Partial Derivative
    2. M.2 Finite Difference Formulas for 3D Cases
    3. M.3 Finite Difference Solutions of Linear Hyperbolic Equations
    4. M.4 Alternating Direction Implicit (ADI) Schemes
  26. N Additional Details and Fortification for Chapter 14
    1. N.1 Convex Hull Algorithm
    2. N.2 Stabilization via Streamline-Upwind Petrov-Galerkin (SUPG)