Fourier series and boundary value problems


Fourier series and boundary value problems

James Ward Brown, Ruel V. Churchill

(Churchill-Brown series)

McGraw-Hill, 2012

8th ed., International ed

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Includes bibliographical references (p. 377-380) and index



Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations. The book is a thorough revision of the seventh edition and much care is taken to give the student fewer distractions when determining solutions of eigenvalue problems, and other topics have been presented in their own sections like Gibbs' Phenomenon and the Poisson integral formula.


Preface1 Fourier SeriesPiecewise Continuous FunctionsFourier Cosine SeriesExamplesFourier Sine SeriesExamplesFourier SeriesExamplesAdaptations to Other Intervals2 Convergence of Fourier SeriesOne-Sided DerivativesA Property of Fourier CoefficientsTwo LemmasA Fourier TheoremA Related Fourier TheoremExamplesConvergence on Other IntervalsA LemmaAbsolute and Uniform Convergence of Fourier SeriesThe Gibbs PhenomenonDifferentiation of Fourier SeriesIntegration of Fourier Series3 Partial Differential Equations of PhysicsLinear Boundary Value ProblemsOne-Dimensional Heat EquationRelated EquationsLaplacian in Cylindrical and Spherical CoordinatesDerivationsBoundary ConditionsDuhamel's PrincipleA Vibrating StringVibrations of Bars and MembranesGeneral Solution of the Wave EquationTypes of Equations and Boundary Conditions4 The Fourier MethodLinear OperatorsPrinciple of SuperpositionExamplesEigenvalues and EigenfunctionsA Temperature ProblemA Vibrating String ProblemHistorical Development5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated Temperature Problems Temperatures in a SphereA Slab with Internally Generated HeatSteady Temperatures in Rectangular CoordinatesSteady Temperatures in Cylindrical CoordinatesA String with Prescribed Initial ConditionsResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Fourier Integrals and ApplicationsThe Fourier Integral FormulaDirichlet's IntegralTwo LemmasA Fourier Integral TheoremThe Cosine and Sine IntegralsSome Eigenvalue Problems on Undounded IntervalsMore on Superposition of SolutionsSteady Temperatures in a Semi-Infinite StripTemperatures in a Semi-Infinite SolidTemperatures in an Unlimited Medium7 Orthonormal SetsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and Parseval's EquationApplications to Fourier Series8 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of Eigenfunctions adn Real EigenvaluesReal-Valued EigenfunctionsNonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsA Temperature Problem in Rectangular CoordinatesSteady TemperaturesOther CoordinatesA Modification of the MethodAnother ModificationA Vertically Hung Elastic Bar9 Bessel Functions and ApplicationsThe Gamma FunctionBessel Functions Jn(x)Solutions When v = 0,1,2,...Recurrence RelationsBessel's Integral FormSome Consequences of the Integral FormsThe Zeros of Jn(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of the TheoremsTwo LemmasFourier-Bessel SeriesExamplesTemperatures in a Long CylinderA Temperature Problem in Shrunken FittingsInternally Generated HeatTemperatures in a Long Cylindrical WedgeVibration of a Circular Membrane10 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsRodrigues' FormulaLaplace's Integral FormSome Consequences of the Integral FormOrthogonality of Legendre PolynomialsNormalized Legendre PolynomialsLegendre SeriesThe Eigenfunctions Pn(cos )Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere11 Verification of Solutions and UniquenessAbel's Test for Uniform ConvergenceVerification of Solution of Temperature ProblemUniqueness of Solutions of the Heat EquationVerification of Solution of Vibrating String ProblemUniqueness of Solutions of the Wave EquationAppendixesBibliographySome Fourier Series ExpansionsSolutions of Some Regular Sturm-Liouville Problems Some Fourier-Bessel Series ExpansionsIndex

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