Advanced Mathematical Methods in Science and Engineering 2nd Edition

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 EquationsDEFINITIONS 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 EquationsINTRODUCTION POWER SERIES SOLUTIONS CLASSIFICATION OF SINGULARITIES FROBENIUS SOLUTION Special FunctionsBESSEL FUNCTIONS BESSEL FUNCTION OF ORDER ZERO BESSEL FUNCTION OF AN INTEGER ORDER NRECURRENCE RELATIONS FOR BESSEL FUNCTIONS BESSEL FUNCTIONS OF HALF ORDERS SPHERICAL BESSEL FUNCTIONSHANKEL 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 ProblemsINTRODUCTION 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 HORNSSTABILITY 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 VariableCOMPLEX NUMBERSANALYTIC FUNCTIONSELEMENTARY FUNCTIONSINTEGRATION IN THE COMPLEX PLANECAUCHY’S INTEGRAL THEOREM CAUCHY’S INTEGRAL FORMULA INFINITE SERIES TAYLOR’S EXPANSION THEOREM LAURENT’S SERIES CLASSIFICATION OF SINGULARITIESRESIDUES 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 CONTOURSINTEGRALS 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 PhysicsINTRODUCTION THE DIFFUSION EQUATIONTHE VIBRATION EQUATIONTHE WAVE EQUATIONHELMHOLTZ EQUATIONPOISSON AND LAPLACE EQUATIONSCLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS UNIQUENESS OF SOLUTIONSTHE LAPLACE EQUATION THE POISSON EQUATION THE HELMHOLTZ EQUATIONTHE DIFFUSION EQUATION THE VIBRATION EQUATION THE WAVE EQUATION Integral TransformsFOURIER INTEGRAL THEOREM FOURIER COSINE TRANSFORM FOURIER SINE TRANSFORM COMPLEX FOURIER TRANSFORM MULTIPLE FOURIER TRANSFORM HANKEL TRANSFORM OF ORDER ZERO HANKEL TRANSFORM OF ORDER ?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 TRANSFORMSSOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS OPERATIONAL CALCULUS WITH FOURIER COSINE TRANSFORMOPERATIONAL CALCULUS WITH FOURIER SINE TRANSFORM OPERATIONAL CALCULUS WITH COMPLEX FOURIER TRANSFORM OPERATIONAL CALCULUS WITH MULTIPLE FOURIER TRANSFORM OPERATIONAL CALCULUS WITH HANKEL TRANSFORM Green’s FunctionsINTRODUCTIONGREEN’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 LAPLACIANFUNDAMENTAL SOLUTION FOR THE BI-LAPLACIAN FUNDAMENTAL SOLUTION FOR THE HELMHOLTZ OPERATORFUNDAMENTAL SOLUTION FOR THE OPERATOR, - ?2 + µ2CAUSAL FUNDAMENTAL SOLUTION FOR THE DIFFUSION OPERATORCAUSAL FUNDAMENTAL SOLUTION FOR THE WAVE OPERATORFUNDAMENTAL SOLUTIONS FOR THE BI-LAPLACIAN HELMHOLTZ OPERATOR GREEN’S FUNCTION FOR THE LAPLACIAN OPERATOR FOR BOUNDED MEDIACONSTRUCTION OF THE AUXILIARY FUNCTION-METHOD OF IMAGES GREEN’S FUNCTION FOR THE LAPLACIAN FOR HALF-SPACEGREEN’S FUNCTION FOR THE LAPLACIAN BY EIGENFUNCTION EXPANSION FOR BOUNDED MEDIA GREEN’S FUNCTION FOR A CIRCULAR AREA FOR THE LAPLACIANGREEN’S FUNCTION FOR SPHERICAL GEOMETRY FOR THE LAPLACIANGREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR BOUNDED MEDIA GREEN’S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR HALF-SPACEGREEN’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 MethodsINTRODUCTION METHOD OF INTEGRATION BY PARTSLAPLACE’S INTEGRAL STEEPEST DESCENT METHOD DEBYE’S FIRST ORDER APPROXIMATION ASYMPTOTIC SERIES APPROXIMATIONMETHOD 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 ONETHE 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 MethodsINTRODUCTION ROOTS OF NON-LINEAR EQUATIONSROOTS OF A SYSTEM OF NON-LINEAR EQUATIONFINITE DIFFERENCESNUMERICAL DIFFERENTIATIONNUMERICAL INTEGRATIONORDINARY DIFFERENTIAL EQUATIONS: INITIAL VALUE PROBLEMSORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS ORDINARY DIFFERENTIAL EQUATIONS: EIGENVALUE PROBLEMS PARTIAL DIFFERENTIAL EQUATIONS Appendix A: Infinite SeriesAppendix B: Special FunctionsAppendix C: Orthogonal Coordinate SystemsAppendix D: Dirac Delta FunctionsAppendix E: Plots of Special FunctionsAppendix F: Vector AnalysisAppendix G: Matrix Algebra References Answers Index Problems appear at the end of each chapter.

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