Linear Differential Operators

Erscheinungsjahr 1987
ISBN: 978-0-89871-370-1
Verlag: Society for Industrial and Applied Mathematics

Buch, Englisch, Band 18, 582 Seiten, Format (B × H): 152 mm x 228 mm, Gewicht: 780 g

Reihe: Classics in Applied Mathematics

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.

Autoren/Hrsg.

Lanczos, Cornelius

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

Preface; Bibliography; 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; 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 Fejér'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; 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; 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; 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