Advanced Mathematical Methods For scientists & engineers

Includes Index

I. Fundamentals. Ordinary differential equations --
Difference equations. II. Local analysis. Approximate solution of linear differential equations --
Approximate solution of nonlinear differential equations --
Approximate solution of difference equations --
Asymptotic expansion of integrals. III. Perturbation methods. Perturbation series --
Summation of series. IV. Global analysis. Boundary layer theory --
WKB theory --
Multiple-scale analysis.

A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.