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.