Differential and Difference Equations (eBook)

A Comparison of Methods of Solution

Leonard C. Maximon (Autor)

eBook Download: PDF

2016 | 1st ed. 2016
XV, 162 Seiten
Springer International Publishing (Verlag)
978-3-319-29736-1 (ISBN)

Infos

This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green's functions and the associated initial and boundary conditions is presented for differential and difference equations of both arbitrary and second order. A dictionary of difference equations with polynomial coefficients provides a unique compilation of second order difference equations obeyed by the special functions of mathematical physics. Appendices augmenting the text include, in particular, a proof of Cramer's rule, a detailed consideration of the role of the superposition principal in the Green's function, and a derivation of the inverse of Laplace transforms and generating functions of particular use in the solution of second order 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.

Inhalt

Preface.

Introduction.

1 Operators.

2 Solution of homogeneous and inhomogeneous linear
equations.
2.1 Variation of constants. 2.2 Reduction of order when one solution to the
homogeneous equation is known.

3 First order homogeneous and inhomogeneous linear
equations.

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 and
the solution of linear differential and difference equations.
7.1 Laplace transforms and the solution of linear differential equations  with constant coefficients. 7.2 Generating
functions and the solution of linear difference equations with constant
coefficient. 7.3 Laplace transforms and the solution of linear differential
equations with polynomial coefficients. 7.4 Alternative method for the solution
of homogeneous linear differential equations with linear coefficients.  7.5 Generating functions and the solution of
linear difference equations with polynomial coefficients. 7.6 Solution of
homogeneous linear difference equations with linear coefficients.

8 Dictionary of difference equations with polynomial
coefficients.

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 Superposition
principle.

Appendix G: Inverse Laplace transforms and Inverse
Generating functions.

Appendix H: Hypergeometric function.

Appendix I:  Confluent
Hypergeometric function.

Appendix J. Solutions of the second kind.

Bibliography.

Produkt-Details

• Erscheint lt. Verlag: 18.4.2016
• Zusatzinfo: XV, 162 p.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Casoratian determinant • Classical hypergeometric functions • Confluent hypergeometric functions • Difference Equations • Green’s function and the superposition principle • Green’s function and the superposition principle • Method of variation of constants • Ordinary differential equations • Proof of Cramer's rule • Solution of linear difference equations • Solution of linear differential equations • Special functions of mathematical physics • Wronskian determinant
• ISBN-10: 3-319-29736-8 / 3319297368
• ISBN-13: 978-3-319-29736-1 / 9783319297361