Ebook: Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory

The triumphant vindication of bold theories-are these not the pride and justification of our life's work? -Sherlock Holmes, The Valley of Fear Sir Arthur Conan Doyle The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Our objective is to help young and also establiShed scientists and engineers to build the skills necessary to analyze equations that they encounter in their work. Our presentation is aimed at developing the insights and techniques that are most useful for attacking new problems. We do not emphasize special methods and tricks which work only for the classical transcendental functions; we do not dwell on equations whose exact solutions are known. The mathematical methods discussed in this book are known collectively as­ 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. Thus, we concentrate on the most fruitful aspect of applied analysis; namely, obtaining the answer. We stress care but not rigor. To explain our approach, we compare our goals with those of a freshman calculus course. A beginning calculus course is considered successful if the students have learned how to solve problems using calculus.

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This book gives 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. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute- force numerical methods may not converge to useful solutions. The presentation is aimed at teaching the insights that are most useful in approaching new problems; it avoids special methods and tricks that work only for particular problems, such as the traditional transcendental functions. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader 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.

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This book gives 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. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute- force numerical methods may not converge to useful solutions. The presentation is aimed at teaching the insights that are most useful in approaching new problems; it avoids special methods and tricks that work only for particular problems, such as the traditional transcendental functions. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader 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.
Content:
Front Matter....Pages i-xiv
Front Matter....Pages 1-2
Ordinary Differential Equations....Pages 3-35
Difference Equations....Pages 36-57
Front Matter....Pages 59-60
Approximate Solution of Linear Differential Equations....Pages 61-145
Approximate Solution of Nonlinear Differential Equations....Pages 146-204
Approximate Solution of Difference Equations....Pages 205-246
Asymptotic Expansion of Integrals....Pages 247-316
Front Matter....Pages 317-318
Perturbation Series....Pages 319-367
Summation of Series....Pages 368-416
Front Matter....Pages 417-417
Boundary-Layer Theory....Pages 419-483
WKB Theory....Pages 484-543
Multiple-Scale Analysis....Pages 544-576
Back Matter....Pages 577-593