Linear Differential Operators

Häftad, Engelska, 1996

AvCornelius Lanczos

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Don't let the title fool you! If you are interested in numerical analysis, applied mathematics, or the solution procedures for differential equations, you will find this book useful. Because of Lanczos's unique style of describing mathematical facts in nonmathematical language, Linear Differential Operators also will be helpful to nonmathematicians interested in applying the methods and techniques described.Originally published in 1961, this Classics edition continues to be appealing because it describes a large number of techniques still useful today. Although the primary focus is on the analytical theory, concrete cases are cited to forge the link between theory and practice. Considerable manipulative skill in the practice of differential equations is to be developed by solving the 350 problems in the text. The problems are intended as stimulating corollaries linking theory with application and providing the reader with the foundation for tackling more difficult problems.Lanczos begins with three introductory chapters that explore some of the technical tools needed later in the book, and then goes on to discuss interpolation, harmonic analysis, matrix calculus, the concept of the function space, boundary value problems, and the numerical solution of trajectory problems, among other things. The emphasis is constantly on one question: ""What are the basic and characteristic properties of linear differential operators?""In the author's words, this book is written for those ""to whom a problem in ordinary or partial differential equations is not a problem of logical acrobatism, but a problem in the exploration of the physical universe. To get an explicit solution of a given boundary value problem is in this age of large electronic computers no longer a basic question. But of what value is the numerical answer if the scientist does not understand the peculiar analytical properties and idiosyncrasies of the given operator? The author hopes that this book will help in this task by telling something about the manifold aspects of a fascinating field.

Produktinformation

Utgivningsdatum1996-12-31
Mått152 x 229 x undefined mm
Vikt789 g
FormatHäftad
SpråkEngelska
SerieClassics in Applied Mathematics
Antal sidor581
FörlagSociety for Industrial & Applied Mathematics,U.S.
ISBN9780898713701

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PrefaceBibliographyChapter 1: Interpolation. IntroductionThe Taylor expansionThe finite Taylor series with the remainder termInterpolation by polynomialsThe remainder of Lagrangian interpolation formulaEquidistant interpolationLocal and global interpolationInterpolation by central differencesInterpolation around the midpoint of the rangeThe Laguerre polynomialsBinomial expansionsThe decisive integral transformBinomial expansions of the hypergeometric typeRecurrence relationsThe Laplace transformThe Stirling expansionOperations with the Stirling functionsAn integral transform of the Fourier typeRecurrence relations associated with the Stirling seriesInterpolation of the Fourier transformThe general integral transform associated with the Stirling series Interpolation of the Bessel functionsChapter 2: Harmonic Analysis. IntroductionThe Fourier series for differentiable functionsThe remainder of the finite Fourier expansionFunctions of higher differentiabilityAn alternative method of estimationThe Gibbs oscillations of the finite Fourier seriesThe method of the Green's functionNon-differentiable functions. Dirac's delta functionSmoothing of the Gibbs oscillations by Fejér's methodThe remainder of the arithmetic mean methodDifferentiation of the Fourier seriesThe method of the sigma factorsLocal smoothing by integrationSmoothing of the Gibbs oscillations by the sigma methodExpansion of the delta functionThe triangular pulseExtension of the class of expandable functionsAsymptotic relations for the sigma factorsThe method of trigonometric interpolationError bounds for the trigonometric interpolation methodRelation between equidistant trigonometric and polynomial interpolationsThe Fourier series in the curve fittingChapter 3: Matrix Calculus. IntroductionRectangular matricesThe basic rules of matrix calculusPrincipal axis transformation of a symmetric matrixDecomposition of a symmetric matrixSelf-adjoint systemsArbitrary n x m systemsSolvability of the general n x m systemThe fundamental decomposition theoremThe natural inverse of a matrixGeneral analysis of linear systemsError analysis of linear systemsClassification of linear systemsSolution of incomplete systemsOver-determined systemsThe method of orthogonalisationThe use of over-determined systemsThe method of successive orthogonalisationThe bilinear identityMinimum property of the smallest eigenvalueChapter 4: The Function Space. IntroductionThe viewpoint of pure and applied mathematicsThe language of geometryMetrical spaces of infinitely many dimensionsThe function as a vectorThe differential operator as a matrixThe length of a vectorThe scalar product of two vectorsThe closeness of the algebraic approximationThe adjoint operatorThe bilinear identityThe extended Green's identityThe adjoint boundary conditionsIncomplete systemsOver-determined systemsCompatibility under inhomogeneous boundary conditionsGreen's identity in the realm of partial differential operatorsThe fundamental field operations of vector analysisSolution of incomplete systemsChapter 5: The Green's Function. IntroductionThe role of the adjoint equationThe role of Green's identityThe delta function d(x, x) The existence of the Green's functionInhomogeneous boundary conditionsThe Green's vectorSelf-adjoint systemsThe calculus of variationsThe canonical equations of HamiltonThe Hamiltonisation of partial operatorsThe reciprocity theoremSelf-adjoint problems. Symmetry of the Green's functionReciprocity of the Green's vectorThe superposition principle of linear operatorsThe Green's function in the realm of ordinary differential operatorsThe change of boundary conditionsThe remainder of the Taylor seriesThe remainder of the Lagrangian interpolation formulaLagrangian interpolation with double points

'This scholarly volume describes Lanczos' life (1893-1974) and presents a clear development of the many fields he opened. ... a collection of seven photographs of Lanczos from 1910 to 1972 ... is followed by a fascinating twenty-eight page annotated story, 'Cornelius Lanczos: A Biographical Essay', by Barbara Gellai. The fortuitous inclusion of this excellent biography makes the Proceedings a more complete and desirable volume!' Mathematics of Computation