CARL M.BENDER, STEVEN A.ORSZAG – ADVANCED MATH METHODS FOR SCIENTISTS

There is a self-contained presentation of the methods for obtaining approximate analytical solutions to differential and difference equations. Special methods and tricks that only work for specific problems are avoided in the text. It only assumes a limited familiarity with differential equations and complex variables for graduates and advanced undergraduates. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and concludes with an exposition of global asymptotic methods. The discussion stresses care rather than rigor and uses well-chosen examples to teach readers how to tackle applied mathematician 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 detailing the properties of special functions.

The book’s main purpose is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations. Our goal is to help young scientists and engineers build the skills necessary to analyze equations that they encounter in their work. Insights and techniques that are most useful for attacking new problems are the focus of our presentation. Special methods and tricks that only work for the classical transcendental functions are not emphasized. The mathematical methods discussed in this book are called asymptotic and perturbative analysis. These are the most useful and powerful methods for finding approximate solutions to equations, but they are difficult to justify rigorously. Obtaining the answer is the most fruitful aspect of applied analysis. We stress care, but not rigor. We compare our goals with those of a freshman class. The students have learned how to solve problems in the course.

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