Dasgupta B. Applied Mathematical Methods

New Delhi: Pearson, 2012. — 524 p.Applied Mathematical Methods refreshes the reader's undergraduate mathematics background and enhances it to the point of enabling him/her to take up research challenges in a mature manner. Organized in the style of classroom discourses, the book can be used equally well in a regular course and for self-study.This book covers the material vital for research in today's world and can be covered in a regular semester course. It is the consolidation of the efforts of teaching the compulsory first semester postgraduate applied mathematics course at the Department of Mechanical Engineering at IIT Kanpur in two successive years.Key Features:
Coverage of the right material for applied areas
Just enough detail for completion within a semester course
Structured lecture-size chapters for convenient teaching and learning
A style of dialogue to keep the reader tuned in partnership rather than high-handed prescriptions
Careful framing of exercises with enough guideline to encourage the reader in problem solving as well as theory development
A detailed appendix with complete answers (including plots of results), solution outlines, interpretations, comments and further extensions
A rich bibliography and appropriate citations all over the text to indicate further study material on all topicsList of Figures
List of TablesAcknowledgements
Preliminary Background
Theme of the Book
Course Contents
Sources for More Detailed Study
Directions for Using the Book
Expected Background
Preliminary Test
Prerequisite Problem Sets*
Problem set 1
Problem set 2
Problem set 3
Problem set 4
Problem set 5
Problem set 6
Matrices and Linear Transformations
Matrices
Geometry and Algebra
Linear Transformations
Matrix Terminology
Exercises
Operational Fundamentals of Linear Algebra
Range and Null Space: Rank and Nullity
Basis
Change of Basis
Elementary Transformations
Exercises
Systems of Linear Equations
Nature of Solutions
Basic Idea of Solution Methodology
Homogeneous Systems
Pivoting
Partitioning and Block Operations
Exercises
Gauss Elimination Family of Methods
Gauss-Jordan Elimination
Gaussian Elimination with Back-Substitution
LU Decomposition
Exercises
Special Systems and Special Methods
Quadratic Forms, Symmetry and Positive Definiteness
Cholesky Decomposition
Sparse Systems*
Exercises
Numerical Aspects in Linear Systems
Norms and Condition Numbers
Ill-conditioning and Sensitivity
Rectangular Systems
Singularity-Robust Solutions
Iterative Methods
Exercises
Eigenvalues and Eigenvectors
Eigenvalue Problem
Generalized Eigenvalue Problem
Some Basic Theoretical Results
Eigenvalues of transpose
Diagonal and block diagonal matrices
Triangular and block triangular matrices
Shift theorem
Deflation
Eigenspace
Similarity transformation
Power Method
Exercises
Diagonalization and Similarity Transformations
Diagonalizability
Canonical Forms
Symmetric Matrices
Similarity Transformations
Exercises
Jacobi and Givens Rotation Methods
Plane Rotations
Jacobi Rotation Method
Givens Rotation Method
Exercises
Householder Transformation and Tridiagonal Matrices
Householder Reflection Transformation
Householder Method
Eigenvalues of Symmetric Tridiagonal Matrices
Exercises
QR Decomposition Method
QR Decomposition
QR Iterations
Conceptual Basis of QR Method*
QR Algorithm with Shift*
Exercises
Eigenvalue Problem of General Matrices
Introductory Remarks
Reduction to Hessenberg Form*
QR Algorithm on Hessenberg Matrices*
Inverse Iteration
Recommendation
Exercises
Singular Value Decomposition
SVD Theorem and Construction
Properties of SVD
Pseudoinverse and Solution of Linear Systems
Optimality of Pseudoinverse Solution
SVD Algorithm
Exercises
Vector Spaces: Fundamental Concepts*
Group
Field
Vector Space
Linear Transformation
Isomorphism
Inner Product Space
Function Space
Vector space of continuous functions
Linear dependence and independence
Inner product, norm and orthogonality
Linear transformations
Exercises
Topics in Multivariate Calculus
Derivatives in Multi-Dimensional Spaces
Taylor’s Series
Chain Rule and Change of Variables
Differentiation of implicit functions
Multiple integrals
Exact differentials
Differentiation under the integral sign
Numerical Differentiation
An Introduction to Tensors*
Exercises
Vector Analysis: Curves and Surfaces
Recapitulation of Basic Notions
Curves in Space
Surfaces*
Exercises
Scalar and Vector Fields
Differential Operations on Field Functions
The del or nabla (∇) operator
Gradient
Divergence
Curl
Composite operations
Second order differential operators
Integral Operations on Field Functions
Green’s theorem in the plane
Gauss’s divergence theorem
Green’s identities (theorem)
Stokes’s theorem
Line integral
Surface and volume integrals
Closure
Exercises
Polynomial Equations
Basic Principles
Analytical Solution
Cubic equations (Cardano)
Quartic equations (Ferrari)
General Polynomial Equations
Two Simultaneous Equations
Elimination Methods*
Sylvester’s dialytic method
Bezout’s method
Advanced Techniques*
Exercises
Solution of Nonlinear Equations and Systems
Methods for Nonlinear Equations
Bracketing and bisection
Fixed point iteration
Newton-Raphson method
Secant method and method of false position
Quadratic interpolation method
Van Wijngaarden-Dekker Brent method
Systems of Nonlinear Equations
Newton’s method for systems of equations
Broyden’s secant method
Closure
Exercises
Optimization: Introduction
The Methodology of Optimization
Single-Variable Optimization
Optimality criteria
Iterative methods
Conceptual Background of Multivariate Optimization
Optimality criteria
Convexity
Trust region and line search strategies
Global and local convergence
Exercises
Multivariate Optimization
Direct Methods
Cyclic coordinate search
Rosenbrock’s method
Hooke-Jeeves pattern search
Box’s complex method
Nelder and Mead’s simplex search
Remarks
Steepest Descent (Cauchy) Method
Newton’s Method
Modified Newton’s method
Hybrid (Levenberg-Marquardt) Method
Least Square Problems
Exercises
Methods of Nonlinear Optimization*
Conjugate Direction Methods
Conjugate gradient method
Extension to general (non-quadratic) functions
Powell’s conjugate direction method
Quasi-Newton Methods
Closure
Exercises
Constrained Optimization
Constraints
Optimality Criteria
Sensitivity
Duality*
Structure of Methods: An Overview*
Penalty methods
Primal methods
Dual methods
Lagrange methods
Exercises
Linear and Quadratic Programming Problems*
Linear Programming
The simplex method
General perspective
Quadratic Programming
Active set method
Linear complementary problem
A trust region method
Duality
Exercises
Interpolation and Approximation
Polynomial Interpolation
Lagrange interpolation
Newton interpolation
Limitations of single-polynomial interpolation
Hermite interpolation
Piecewise Polynomial Interpolation
Piecewise cubic interpolation
Spline interpolation
Interpolation of Multivariate Functions
Piecewise bilinear interpolation
Piecewise bicubic interpolation
A Note on Approximation of Functions
Modelling of Curves and Surfaces*
Exercises
Basic Methods of Numerical Integration
Newton-Cotes Integration Formulae
Mid-point rule
Trapezoidal rule
Simpson’s rules
Richardson Extrapolation and Romberg Integration
Further Issues
Exercises
Advanced Topics in Numerical Integration*
Gaussian Quadrature
Gauss-Legendre quadrature
Weight functions in Gaussian quadrature
Multiple Integrals
Double integral on rectangular domain
Monte Carlo integration
Exercises
Numerical Solution of Ordinary Differential Equations
Single-Step Methods
Euler’s method
Improved Euler’s method or Heun’s method
Runge-Kutta methods
Practical Implementation of Single-Step Methods
Runge-Kutta method with adaptive step size
Extrapolation based methods
Systems of ODE’s
Multi-Step Methods*
Exercises
ODE Solutions: Advanced Issues
Stability Analysis
Implicit Methods
Stiff Differential Equations
Boundary Value Problems
Shooting method
Finite difference (relaxation) method
Finite element method
Exercises
Existence and Uniqueness Theory
Well-Posedness of Initial Value Problems
Existence of a solution
Uniqueness of a solution
Continuous dependence on initial condition
Uniqueness Theorems
Extension to ODE Systems
Closure
Exercises
First Order Ordinary Differential Equations
Formation of Differential Equations and Their Solutions
Separation of Variables
ODE’s with Rational Slope Functions
Some Special ODE’s
Clairaut’s equation
Second order ODE’s with the function not appearing explicitly
Second order ODE’s with independent variable not appearing explicitly
Exact Differential Equations and Reduction to the Exact Form
First Order Linear (Leibnitz) ODE and Associated Forms
Orthogonal Trajectories
Modelling and Simulation
Exercises
Second Order Linear Homogeneous ODE’sHomogeneous Equations with Constant Coefficients
Euler-Cauchy Equation
Theory of the Homogeneous Equations
Basis for Solutions
Exercises
Second Order Linear Non-Homogeneous ODE’s
Linear ODE’s and Their Solutions
Method of Undetermined Coefficients
Method of Variation of Parameters
Closure
Exercises
Higher Order Linear ODE’s
Theory of Linear ODE’s
Homogeneous Equations with Constant Coefficients
Non-Homogeneous Equations
Euler-Cauchy Equation of Higher Order
Exercises
Laplace TransformsBasic Properties and Results
Application to Differential Equations
Handling Discontinuities
Convolution
Advanced Issues
Exercises
ODE Systems
Fundamental Ideas
Linear Homogeneous Systems with Constant Coefficients
Linear Non-Homogeneous Systems
Method of undetermined coefficients
Method of diagonalization
Method of variation of parameters
Nonlinear Systems
Exercises
Stability of Dynamic Systems
Second Order Linear Systems
Nonlinear Dynamic Systems
Phase plane analysis
Limit cycles
Systems with arbitrary dimension of state space
Lyapunov Stability Analysis
Exercises
Series Solutions and Special Functions
Power Series Method
An elementary example
ODE’s with variable coefficients
Frobenius’ Method
Special Functions Defined as Integrals
Special Functions Arising as Solutions of ODE’s
Legendre’s equation
Bessel’s equation
Exercises
Sturm-Liouville Theory
Preliminary Ideas
Boundary value problems as eigenvalue problems
Second order self-adjoint ODE’s
Sturm-Liouville Problems
Orthogonality of eigenfunctions
Real eigenvalues
Eigenfunction Expansions
Exercises
Fourier Series and Integrals
Basic Theory of Fourier Series
Extensions in Application
Fourier Integrals
Exercises
Fourier Transforms
Definition and Fundamental Properties
Important Results on Fourier Transforms
Discrete Fourier Transform
Exercises
Minimax Approximation*
Approximation with Chebyshev polynomials
Chebyshev polynomials
Minimax property
Chebyshev series
Minimax Polynomial Approximation
Chebyshev’s minimax theorem
Construction of the minimax polynomial
Exercises
Partial Differential EquationsQuasi-linear second order PDE’s
Initial and boundary conditions
Method of separation of variables
Hyperbolic Equations
Solution of the wave equation by separation of variables
D’ Alembert’s solution of the wave equation
Further related problems
Parabolic Equations
Heat conduction in a finite bar
Heat conduction in an infinite wire
Elliptic Equations
Steady-state heat flow in a rectangular plate
Steady-state heat flow with internal heat generation
Two-Dimensional Wave Equation
Exercises
Analytic Functions
Analyticity of Complex Functions
Limits, continuity and differentiability
Cauchy-Riemann conditions
Harmonic functions
Conformal Mapping
Potential Theory
Exercises
Integrals in the Complex Plane
Line Integral
Cauchy’s Integral Theorem
Cauchy’s Integral Formula
Exercises
Singularities of Complex Functions
Series Representations of Complex Functions
Zeros and Singularities
Residues
Evaluation of Real Integrals
Exercises
Variational Calculus*Functionals and their extremization
Examples of variational problems
Euler’s Equation
Functions involving higher order derivatives
Functionals of a vector function
Functionals of functions of several variables
Moving boundaries
Equality constraints and isoperimetric problems
Direct Methods
Finite difference method
Rayleigh-Ritz method
The inverse problem and the Galerkin method
Finite element methods
Exercises
EpilogueAppendices
Answers, Solution Outlines and Comments to Exercises
Basic Formulae and Results
Trigonometric Identities
Algebraic Relations and Series Expansions
Analytic Geometry
Calculus