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Linear Differential Operators

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Cornelius Lanczos


1996 / xviii + 554 pages / Softcover / ISBN: 978-0-898713-70-1 / List Price $93.00 / SIAM Member Price $65.10 / Order Code CL18

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' 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."

Preface; Bibliography; Chapter 1: Interpolation. Introduction; The Taylor expansion; The finite Taylor series with the remainder term; Interpolation by polynomials; The remainder of Lagrangian interpolation formula; Equidistant interpolation; Local and global interpolation; Interpolation by central differences; Interpolation around the midpoint of the range; The Laguerre polynomials; Binomial expansions; The decisive integral transform; Binomial expansions of the hypergeometric type; Recurrence relations; The Laplace transform; The Stirling expansion; Operations with the Stirling functions; An integral transform of the Fourier type; Recurrence relations associated with the Stirling series; Interpolation of the Fourier transform; The general integral transform associated with the Stirling series Interpolation of the Bessel functions; Chapter 2: Harmonic Analysis. Introduction; The Fourier series for differentiable functions; The remainder of the finite Fourier expansion; Functions of higher differentiability; An alternative method of estimation; The Gibbs oscillations of the finite Fourier series; The method of the Green's function; Non-differentiable functions; Dirac's delta function; Smoothing of the Gibbs oscillations by FejŽr's method; The remainder of the arithmetic mean method; Differentiation of the Fourier series; The method of the sigma factors; Local smoothing by integration; Smoothing of the Gibbs oscillations by the sigma method; Expansion of the delta function; The triangular pulse; Extension of the class of expandable functions; Asymptotic relations for the sigma factors; The method of trigonometric interpolation; Error bounds for the trigonometric interpolation method; Relation between equidistant trigonometric and polynomial interpolations; The Fourier series in the curve fitting; Chapter 3: Matrix Calculus. Introduction; Rectangular matrices; The basic rules of matrix calculus; Principal axis transformation of a symmetric matrix; Decomposition of a symmetric matrix; Self-adjoint systems; Arbitrary n x m systems; Solvability of the general n x m system; The fundamental decomposition theorem; The natural inverse of a matrix; General analysis of linear systems; Error analysis of linear systems; Classification of linear systems; Solution of incomplete systems; Over-determined systems; The method of orthogonalisation; The use of over-determined systems; The method of successive orthogonalisation; The bilinear identity; Minimum property of the smallest eigenvalue; Chapter 4: The Function Space. Introduction; The viewpoint of pure and applied mathematics; The language of geometry; Metrical spaces of infinitely many dimensions; The function as a vector; The differential operator as a matrix; The length of a vector; The scalar product of two vectors; The closeness of the algebraic approximation; The adjoint operator; The bilinear identity; The extended Green's identity; The adjoint boundary conditions; Incomplete systems; Over-determined systems; Compatibility under inhomogeneous boundary conditions; Green's identity in the realm of partial differential operators; The fundamental field operations of vector analysis; Solution of incomplete systems; Chapter 5: The Green's Function. Introduction; The role of the adjoint equation; The role of Green's identity; The delta function _______; The existence of the Green's function; Inhomogeneous boundary conditions; The Green's vector; Self-adjoint systems; The calculus of variations; The canonical equations of Hamilton; The Hamiltonisation of partial operators; The reciprocity theorem; Self-adjoint problems; Symmetry of the Green's function; Reciprocity of the Green's vector; The superposition principle of linear operators; The Green's function in the realm of ordinary differential operators; The change of boundary conditions; The remainder of the Taylor series; The remainder of the Lagrangian interpolation formula; Lagrangian interpolation with double points; Construction of the Green's vector; The constrained Green's function; Legendre's differential equation; Inhomogeneous boundary conditions; The method of over-determination; Orthogonal expansions; The bilinear expansion; Hermitian problems; The completion of linear operators; Chapter 6. Communication Problems. Introduction; The step function and related functions; The step function response and higher order responses; The input-output relation of a galvanometer; The fidelity problem of the galvanometer response; Fidelity damping; The error of the galvanometer recording; The input-output relation of linear communication devices; Frequency analysis; The Laplace transform; The memory time; Steady state analysis of music and speech; Transient analysis of noise phenomena; Chapter 7: Sturm-Liouville Problems. Introduction; Differential equations of fundamental significance; The weighted Green's identity; Second order operators in self-adjoint form; Transformation of the dependent variable; The Green's function of the general second order differential equation; Normalisation of second order problems; Riccati's differential equations; Periodic solutions; Approximate solution of a differential equation of second order; The joining of regions; Bessel functions and the hypergeometric series; Asymptotic properties of __________ in the complex domain; Asymptotic expressions of ____ for large values of Behaviour of ____________ along the imaginary axis; The Bessel functions of the order _________; Jump conditions for the transition "exponential-periodic"; Jump conditions for the transition "periodic-exponential"; Amplitude and phase in the periodic domain; Eigenvalue problems; Hermite's differential equations; Bessel's differential equation; The substitute functions in the transitory range; Tabulation of the four substitute functions; Increased accuracy in the transition domain; Eigensolutions reducible to the hypergeometric series; The ultraspherical polynomials; The Legendre polynomials; The Laguerre polynomials; The exact amplitude equation; Sturm-Liouville problems and the calculus of variations; Chapter 8: Boundary Value Problems. Introduction; Inhomogeneous boundary conditions; The method of the "separation of variables"; The potential equation of the plane; The potential equation in three dimensions; Vibration problems; The problem of the vibrating string; The analytical nature of hyperbolic differential operators; The heat flow equation; Minimum problems with constraints; Integral equations in the service of boundary value problems; The conservation laws of mechanics; Unconventional boundary value problems; The eigenvalue as a limit point; Variational motivation of the parasitic spectrum; Examples for the parasitic spectrum; Physical boundary conditions; A universal approach to the theory of boundary value problems; Chapter 9: Numerical Solution of Trajectory Problems. Introduction; Differential equations in normal form; Trajectory problems; Local expansions; The method of undetermined coefficients; Lagrangian interpolation in terms of double points; Extrapolations of maximum efficiency; Extrapolations of minimum round-off; Estimation of the truncation error; End-point extrapolation; Mid-point interpolations; The problem of starting values; The accumulation of truncation errors; The method of Gaussian quadrature; Global integration by Chebyshev polynomials; Numerical aspects of the method of global integration; The method of global correction; Appendix; Index.

ISBN: 9780898713701

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