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Differential and Difference Equations: A Comparison of Methods of Solution

Autor Leonard C. Maximon
en Limba Engleză Hardback – 26 apr 2016
This book, intended for researchers andgraduate students in physics, applied mathematics and engineering, presents adetailed comparison of the important methods of solution for lineardifferential and difference equations - variation of constants, reduction oforder, Laplace transforms and generating functions - bringing out thesimilarities as well as the significant differences in the respective analyses.Equations of arbitrary order are studied, followed by a detailed analysis forequations of first and second order. Equations with polynomial coefficients areconsidered and explicit solutions for equations with linear coefficients aregiven, showing significant differences in the functional form of solutions ofdifferential equations from those of difference equations. An alternativemethod of solution involving transformation of both the dependent andindependent variables is given for both differential and difference equations.A comprehensive, detailed treatment of Green’s functions and the associatedinitial and boundary conditions is presented for differential and differenceequations of both arbitrary and second order. A dictionary of differenceequations with polynomial coefficients provides a unique compilation of secondorder difference equations obeyed by the special functions of mathematicalphysics. Appendices augmenting the text include, in particular, a proof ofCramer’s rule, a detailed consideration of the role of the superpositionprincipal in the Green’s function, and a derivation of the inverse of Laplacetransforms and generating functions of particular use in the solution of secondorder linear differential and difference equations with linear coefficients.
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Specificații

ISBN-13: 9783319297354
ISBN-10: 331929735X
Pagini: 162
Ilustrații: XV, 162 p.
Dimensiuni: 155 x 235 x 11 mm
Greutate: 0.43 kg
Ediția:1st ed. 2016
Editura: Springer International Publishing
Colecția Springer
Locul publicării:Cham, Switzerland

Cuprins

Preface.
Introduction.
1 Operators.
2 Solution of homogeneous and inhomogeneous linearequations.
2.1 Variation of constants. 2.2 Reduction of order when one solution to thehomogeneous equation is known.
3 First order homogeneous and inhomogeneous linearequations.
4 Second-order homogeneous and inhomogeneous equations.
5 Self-adjoint linear equations.
6 Green’s function.
6.1 Differential equations. 6.2 Difference equations.
7 Generating function, z-transforms, Laplace transforms andthe solution of linear differential and difference equations.
7.1 Laplace transforms and the solution of linear differential equations  with constant coefficients. 7.2 Generatingfunctions and the solution of linear difference equations with constantcoefficient. 7.3 Laplace transforms and the solution of linear differentialequations with polynomial coefficients. 7.4 Alternative method for the solutionof homogeneous linear differential equations with linear coefficients.  7.5 Generating functions and the solution oflinear difference equations with polynomial coefficients. 7.6 Solution ofhomogeneous linear difference equations with linear coefficients.
8 Dictionary of difference equations with polynomialcoefficients.
Appendix A: Difference operator.
Appendix B: Notation.
Appendix C: Wronskian Determinant.
Appendix D: Casoratian Determinant.
Appendix E: Cramer’s Rule.
Appendix F: Green’s function and the Superpositionprinciple. Appendix G: Inverse Laplace transforms and InverseGenerating functions.
Appendix H: Hypergeometric function.
Appendix I:  ConfluentHypergeometric function.
Appendix J. Solutions of the second kind.
Bibliography.

Notă biografică

LeonardMaximon is Research Professor of Physics in the Department of Physics at TheGeorge Washington University and Adjunct Professor in the Department of Physicsat Arizona State University. He has been an Assistant Professor in the GraduateDivision of Applied Mathematics at Brown University, a Visiting Professor atthe Norwegian Technical University in Trondheim, Norway, and a Physicist at theCenter for Radiation Research at the National Bureau of Standards. He is alsoan Associate Editor for Physics for the DLMF project and a Fellow of theAmerican Physical Society.
Maximonhas published numerous papers on the fundamental processes of quantumelectrodynamics and on the special functions of mathematical physics.

Textul de pe ultima copertă

This book, intended for researchers andgraduate students in physics, applied mathematics and engineering, presents adetailed comparison of the important methods of solution for lineardifferential and difference equations - variation of constants, reduction oforder, Laplace transforms and generating functions - bringing out thesimilarities as well as the significant differences in the respective analyses.Equations of arbitrary order are studied, followed by a detailed analysis forequations of first and second order. Equations with polynomial coefficients areconsidered and explicit solutions for equations with linear coefficients aregiven, showing significant differences in the functional form of solutions ofdifferential equations from those of difference equations. An alternativemethod of solution involving transformation of both the dependent andindependent variables is given for both differential and difference equations.A comprehensive, detailed treatment of Green’s functions and the associatedinitial and boundary conditions is presented for differential and differenceequations of both arbitrary and second order. A dictionary of differenceequations with polynomial coefficients provides a unique compilation of secondorder difference equations obeyed by the special functions of mathematicalphysics. Appendices augmenting the text include, in particular, a proof ofCramer’s rule, a detailed consideration of the role of the superpositionprincipal in the Green’s function, and a derivation of the inverse of Laplacetransforms and generating functions of particular use in the solution of secondorder linear differential and difference equations with linear coefficients.

Caracteristici

Provides a unique and detailed comparison of methods of solution of differential and difference equations in application to problems in physical sciences and engineering
Places emphasis on application of these methods to specific problems
a unique compendium of difference equations for the special functions of
mathematical physics
Includes supplementary material: sn.pub/extras