Advanced Mathematical Methods in Science and Engineering: 2nd Edition (Hardback) book cover

Advanced Mathematical Methods in Science and Engineering

2nd Edition

By S.I. Hayek

Chapman and Hall/CRC

866 pages | 96 B/W Illus.

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pub: 2010-06-22
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Description

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.

Reviews

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

Table of Contents

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.

About the Author

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

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics