Differential and Difference Equations
Springer International Publishing (Verlag)
978-3-319-29735-4 (ISBN)
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.
Leonard Maximon is Research Professor of Physics in the Department of Physics at The George Washington University and Adjunct Professor in the Department of Physics at Arizona State University. He has been an Assistant Professor in the Graduate Division of Applied Mathematics at Brown University, a Visiting Professor at the Norwegian Technical University in Trondheim, Norway, and a Physicist at the Center for Radiation Research at the National Bureau of Standards. He is also an Associate Editor for Physics for the DLMF project and a Fellow of the American Physical Society. Maximon has published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics.
Preface.- Introduction.- 1 Operators.- 2 Solution ofhomogeneous and inhomogeneous linear equations.- 3 First order homogeneous andinhomogeneous linear equations.- 4 Second-order homogeneous and inhomogeneousequations.- 5 Self-adjoint linear equations.- 6 Green's function.- 7 Generatingfunction, z-transforms, Laplace transforms and the solution of lineardifferential and difference equations.- 8 Dictionary of difference equationswith polynomial coefficients.- Appendix A: Difference operator.- Appendix B:Notation.- Appendix C: Wronskian Determinant.- Appendix D: CasoratianDeterminant.- Appendix E: Cramer's Rule.- Appendix F: Green's function and theSuperposition principle.- Appendix G: Inverse Laplace transforms and InverseGenerating functions.- Appendix H: Hypergeometric function.- Appendix I: Confluent Hypergeometric function.- AppendixJ. Solutions of the second kind.- Bibliography.
| Erscheinungsdatum | 08.10.2016 |
|---|---|
| Zusatzinfo | XV, 162 p. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika | |
| Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik | |
| Schlagworte | Appl.Mathematics/Computational Methods of Engineer • Casoratian determinant • Classical hypergeometric functions • Confluent hypergeometric functions • Difference and Functional Equations • Difference Equations • Green's function and the superposition principle • Green’s function and the superposition principle • Mathematical Applications in the Physical Sciences • Mathematical methods in physics • Method of variation of constants • Ordinary differential equations • Physics and Astronomy • Proof of Cramer's rule • Solution of homogenous and inhomogeneous linear equations • Solution of linear difference equations • Solution of linear differential equations • Special functions of mathematical physics • Wronskian determinant |
| ISBN-10 | 3-319-29735-X / 331929735X |
| ISBN-13 | 978-3-319-29735-4 / 9783319297354 |
| Zustand | Neuware |
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