Differential and Difference Equations: A Comparison of Methods of Solution

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Springer, Apr 18, 2016 - Science - 162 pages

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.

 

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Contents

1 Operators
2
2 Solution of Homogeneous and Inhomogeneous Linear Equations
5
3 First Order Homogeneous and Inhomogeneous Linear Equations
23
4 Second Order Homogeneous and Inhomogeneous Equations
27
5 Selfadjoint Linear Equations
35
6 Greens Function
39
7 Generating Functions ZTransforms Laplace Transforms and the Solution of Linear Differential and Difference Equations
63
8 Dictionary of Difference Equations with Polynomial Coefficients
112
Appendix D Casoratian Determinant
137
Appendix E Cramers Rule
139
Appendix F Greens Function and the SuperpositionPrinciple
141
Appendix G Inverse Laplace Transforms and InverseGenerating Functions
144
Appendix H Hypergeometric Function
149
Appendix I Confluent Hypergeometric Functions
151
Appendix J Solutions of the Second Kind
155
Bibliography
159

Appendix A Difference Operator
125
Appendix B Notation
130
Appendix C Wronskian Determinant
133

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About the author (2016)

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.

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