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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.
List of contents
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.
About the author
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.
Summary
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.
