2nd Edition

Advanced Mathematical Methods in Science and Engineering

By S.I. Hayek Copyright 2010
    872 Pages 96 B/W Illustrations
    by Chapman & Hall

    Classroom-tested, Advanced Mathematical Methods in Science and Engineering, Second Edition presents methods of applied mathematics that are particularly suited to address physical problems in science and engineering. Numerous examples illustrate the various methods of solution and answers to the end-of-chapter problems are included at the back of the book.

    After introducing integration and solution methods of ordinary differential equations (ODEs), the book presents Bessel and Legendre functions as well as the derivation and methods of solution of linear boundary value problems for physical systems in one spatial dimension governed by ODEs. It also covers complex variables, calculus, and integrals; linear partial differential equations (PDEs) in classical physics and engineering; the derivation of integral transforms; Green’s functions for ODEs and PDEs; asymptotic methods for evaluating integrals; and the asymptotic solution of ODEs. New to this edition, the final chapter offers an extensive treatment of numerical methods for solving non-linear equations, finite difference differentiation and integration, initial value and boundary value ODEs, and PDEs in mathematical physics. Chapters that cover boundary value problems and PDEs contain derivations of the governing differential equations in many fields of applied physics and engineering, such as wave mechanics, acoustics, heat flow in solids, diffusion of liquids and gases, and fluid flow.

    An update of a bestseller, this second edition continues to give students the strong foundation needed to apply mathematical techniques to the physical phenomena encountered in scientific and engineering applications.

    Ordinary Differential Equations
    DEFINITIONS
    LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER
    LINEAR INDEPENDENCE AND THE WRONSKIAN
    LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF ORDER N WITH CONSTANT COEFFICIENTS
    EULER’S EQUATION
    PARTICULAR SOLUTIONS BY METHOD OF UNDETERMINED COEFFICIENTS
    PARTICULAR SOLUTIONS BY THE METHOD OF VARIATIONS OF PARAMETERS
    ABEL’S FORMULA FOR THE WRONSKIAN
    INITIAL VALUE PROBLEMS

    Series Solutions of Ordinary Differential Equations
    INTRODUCTION
    POWER SERIES SOLUTIONS
    CLASSIFICATION OF SINGULARITIES
    FROBENIUS SOLUTION

    Special Functions
    BESSEL FUNCTIONS
    BESSEL FUNCTION OF ORDER ZERO
    BESSEL FUNCTION OF AN INTEGER ORDER N
    RECURRENCE RELATIONS FOR BESSEL FUNCTIONS
    BESSEL FUNCTIONS OF HALF ORDERS
    SPHERICAL BESSEL FUNCTIONS
    HANKEL FUNCTIONS
    MODIFIED BESSEL FUNCTIONS
    GENERALIZED EQUATIONS LEADING TO SOLUTIONS IN TERMS OF BESSEL FUNCTIONS
    BESSEL COEFFICIENTS
    INTEGRAL REPRESENTATION OF BESSEL FUNCTIONS
    ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR SMALL ARGUMENTS
    ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR LARGE ARGUMENTS
    INTEGRALS OF BESSEL FUNCTIONS
    ZEROES OF BESSEL FUNCTIONS
    LEGENDRE FUNCTIONS
    LEGENDRE COEFFICIENTS
    RECURRENCE FORMULAE FOR LEGENDRE POLYNOMIALS
    INTEGRAL REPRESENTATION FOR LEGENDRE POLYNOMIALS
    INTEGRALS OF LEGENDRE POLYNOMIALS
    EXPANSIONS OF FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS
    LEGENDRE FUNCTION OF THE SECOND KIND QN(X)
    ASSOCIATED LEGENDRE FUNCTIONS
    GENERATING FUNCTION FOR ASSOCIATED LEGENDRE FUNCTIONS
    RECURRENCE FORMULAE FOR Pnm
    INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS
    ASSOCIATED LEGENDRE FUNCTION OF THE SECOND KIND Qnm

    Boundary Value Problems and Eigenvalue Problems
    INTRODUCTION
    VIBRATION, WAVE PROPAGATION OR WHIRLING OF STRETCHED STRINGS
    LONGITUDINAL VIBRATION AND WAVE PROPAGATION IN ELASTIC BARS
    VIBRATION, WAVE PROPAGATION AND WHIRLING OF BEAMS
    WAVES IN ACOUSTIC HORNS
    STABILITY OF COMPRESSED COLUMNS
    IDEAL TRANSMISSION LINES (TELEGRAPH EQUATION)
    TORSIONAL VIBRATION OF CIRCULAR BARS
    ORTHOGONALITY AND ORTHOGONAL SETS OF FUNCTIONS
    GENERALIZED FOURIER SERIES
    ADJOINT SYSTEMS
    BOUNDARY VALUE PROBLEMS
    EIGENVALUE PROBLEMS
    PROPERTIES OF EIGENFUNCTIONS OF SELF-ADJOINT SYSTEMS
    STURM-LIOUVILLE SYSTEM
    STURM-LIOUVILLE SYSTEM FOR FOURTH-ORDER EQUATIONS
    SOLUTION OF NON-HOMOGENEOUS EIGENVALUE PROBLEMS
    FOURIER SINE SERIES
    FOURIER COSINE SERIES
    COMPLETE FOURIER SERIES
    FOURIER-BESSEL SERIES
    FOURIER–LEGENDRE SERIES

    Functions of a Complex Variable
    COMPLEX NUMBERS
    ANALYTIC FUNCTIONS
    ELEMENTARY FUNCTIONS
    INTEGRATION IN THE COMPLEX PLANE
    CAUCHY’S INTEGRAL THEOREM
    CAUCHY’S INTEGRAL FORMULA
    INFINITE SERIES
    TAYLOR’S EXPANSION THEOREM
    LAURENT’S SERIES
    CLASSIFICATION OF SINGULARITIES
    RESIDUES AND RESIDUE THEOREM
    INTEGRALS OF PERIODIC FUNCTIONS
    IMPROPER REAL INTEGRALS
    IMPROPER REAL INTEGRAL INVOLVING CIRCULAR FUNCTIONS
    IMPROPER REAL INTEGRALS OF FUNCTIONS HAVING SINGULARITIES ON THE REAL AXIS
    THEOREMS ON LIMITING CONTOURS
    INTEGRALS OF EVEN FUNCTIONS INVOLVING LOG X
    INTEGRALS OF FUNCTIONS INVOLVING Xa
    INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS
    INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INVOLVING LOG X
    INVERSE LAPLACE TRANSFORMS

    Partial Differential Equations of Mathematical Physics
    INTRODUCTION
    THE DIFFUSION EQUATION
    THE VIBRATION EQUATION
    THE WAVE EQUATION
    HELMHOLTZ EQUATION
    POISSON AND LAPLACE EQUATIONS
    CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS
    UNIQUENESS OF SOLUTIONS
    THE LAPLACE EQUATION
    THE POISSON EQUATION
    THE HELMHOLTZ EQUATION
    THE DIFFUSION EQUATION
    THE VIBRATION EQUATION
    THE WAVE EQUATION

    Integral Transforms
    FOURIER INTEGRAL THEOREM
    FOURIER COSINE TRANSFORM
    FOURIER SINE TRANSFORM
    COMPLEX FOURIER TRANSFORM
    MULTIPLE FOURIER TRANSFORM
    HANKEL TRANSFORM OF ORDER ZERO
    HANKEL TRANSFORM OF ORDER ν
    GENERAL REMARKS ABOUT TRANSFORMS DERIVED FROM THE FOURIER INTEGRAL THEOREM
    GENERALIZED FOURIER TRANSFORM
    TWO-SIDED LAPLACE TRANSFORM
    ONE-SIDED GENERALIZED FOURIER TRANSFORM
    LAPLACE TRANSFORM
    MELLIN TRANSFORM
    OPERATIONAL CALCULUS WITH LAPLACE TRANSFORMS
    SOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
    OPERATIONAL CALCULUS WITH FOURIER COSINE TRANSFORM
    OPERATIONAL CALCULUS WITH FOURIER SINE TRANSFORM
    OPERATIONAL CALCULUS WITH COMPLEX FOURIER TRANSFORM
    OPERATIONAL CALCULUS WITH MULTIPLE FOURIER TRANSFORM
    OPERATIONAL CALCULUS WITH HANKEL TRANSFORM

    Green’s Functions
    INTRODUCTION
    GREEN’S FUNCTION FOR ORDINARY DIFFERENTIAL BOUNDARY VALUE PROBLEM
    GREEN’S FUNCTION FOR AN ADJOINT SYSTEM
    SYMMETRY OF THE GREEN’S FUNCTIONS AND RECIPROCITY
    GREEN’S FUNCTION FOR EQUATIONS WITH CONSTANT COEFFICIENTS
    GREEN’S FUNCTIONS FOR HIGHER ORDERED SOURCES
    GREEN’S FUNCTION FOR EIGENVALUE PROBLEMS
    GREEN’S FUNCTION FOR SEMI-INFINITE ONE DIMENSIONAL MEDIA
    GREEN’S FUNCTION FOR INFINITE ONE-DIMENSIONAL MEDIA
    GREEN’S FUNCTION FOR PARTIAL DIFFERENTIAL EQUATIONS
    GREEN’S IDENTITIES FOR THE LAPLACIAN OPERATOR
    GREEN’S IDENTITY FOR THE HELMHOLTZ OPERATOR
    GREEN’S IDENTITY FOR BI-LAPLACIAN OPERATOR
    GREEN’S IDENTITY FOR THE DIFFUSION OPERATOR
    GREEN’S IDENTITY FOR THE WAVE OPERATOR
    GREEN’S FUNCTION FOR UNBOUNDED MEDIA—FUNDAMENTAL SOLUTION
    FUNDAMENTAL SOLUTION FOR THE LAPLACIAN
    FUNDAMENTAL SOLUTION FOR THE BI-LAPLACIAN
    FUNDAMENTAL SOLUTION FOR THE HELMHOLTZ OPERATOR
    FUNDAMENTAL SOLUTION FOR THE OPERATOR, - ∇2 + μ2
    CAUSAL FUNDAMENTAL SOLUTION FOR THE DIFFUSION OPERATOR
    CAUSAL FUNDAMENTAL SOLUTION FOR THE WAVE OPERATOR
    FUNDAMENTAL SOLUTIONS FOR THE BI-LAPLACIAN HELMHOLTZ OPERATOR
    GREEN’S FUNCTION FOR THE LAPLACIAN OPERATOR FOR BOUNDED MEDIA
    CONSTRUCTION OF THE AUXILIARY FUNCTION-METHOD OF IMAGES
    GREEN’S FUNCTION FOR THE LAPLACIAN FOR HALF-SPACE
    GREEN’S FUNCTION FOR THE LAPLACIAN BY EIGENFUNCTION EXPANSION FOR BOUNDED MEDIA
    GREEN’S FUNCTION FOR A CIRCULAR AREA FOR THE LAPLACIAN
    GREEN’S FUNCTION FOR SPHERICAL GEOMETRY FOR THE LAPLACIAN
    GREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR BOUNDED MEDIA
    GREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR HALF-SPACE
    GREEN’S FUNCTION FOR A HELMHOLTZ OPERATOR IN QUARTER-SPACE
    CAUSAL GREEN’S FUNCTION FOR THE WAVE OPERATOR IN BOUNDED MEDIA
    CAUSAL GREEN’S FUNCTION FOR THE DIFFUSION OPERATOR FOR BOUNDED MEDIA
    METHOD OF SUMMATION OF SERIES SOLUTIONS IN TWO DIMENSIONAL MEDIA

    Asymptotic Methods
    INTRODUCTION
    METHOD OF INTEGRATION BY PARTS
    LAPLACE’S INTEGRAL
    STEEPEST DESCENT METHOD
    DEBYE’S FIRST ORDER APPROXIMATION
    ASYMPTOTIC SERIES APPROXIMATION
    METHOD OF STATIONARY PHASE
    STEEPEST DESCENT METHOD IN TWO DIMENSIONS
    MODIFIED SADDLE POINT METHOD: SUBTRACTION OF A SIMPLE POLE
    MODIFIED SADDLE POINT METHOD: SUBTRACTION OF POLE OF ORDER N
    SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS FOR LARGE ARGUMENTS
    CLASSIFICATION OF POINTS AT INFINITY
    SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH REGULAR SINGULAR POINTS
    ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK ONE
    THE PHASE INTEGRAL AND WKBJ METHOD FOR AN IRREGULAR SINGULAR POINT OF RANK ONE
    ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK HIGHER THAN ONE
    ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH LARGE PARAMETERS

    Numerical Methods
    INTRODUCTION
    ROOTS OF NON-LINEAR EQUATIONS
    ROOTS OF A SYSTEM OF NON-LINEAR EQUATION
    FINITE DIFFERENCES
    NUMERICAL DIFFERENTIATION
    NUMERICAL INTEGRATION
    ORDINARY DIFFERENTIAL EQUATIONS: INITIAL VALUE PROBLEMS
    ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS
    ORDINARY DIFFERENTIAL EQUATIONS: EIGENVALUE PROBLEMS
    PARTIAL DIFFERENTIAL EQUATIONS

    Appendix A: Infinite Series
    Appendix B: Special Functions
    Appendix C: Orthogonal Coordinate Systems
    Appendix D: Dirac Delta Functions
    Appendix E: Plots of Special Functions
    Appendix F: Vector Analysis
    Appendix G: Matrix Algebra

    References

    Answers

    Index

    Problems appear at the end of each chapter.

    Biography

    S.I. Hayek is a Distinguished Professor of Engineering Mechanics at Pennsylvania State University.

    S.I. Hayek’s Advanced Mathematical Methods in Science and Engineering covers a wide range of applied mathematics centered around differential equations. … Hayek’s book contains a great deal of useful information …
    MAA Reviews, October 2010