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**Advanced mathematical methods for scientists and engineers I. Asymptotic methods and perturbation theory.
Reprint of the original 1978.**
*(English)*
Zbl 0938.34001

New York, NY: Springer. 593 p. DM 139.00; öS 1015.00; sFr. 126.50; £48.00; $ 69.95 (1999).

Reprint of the original review in Zbl 0417.34001.

From the authors’ preface: “The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that can not be solved exactly. Our objective is to help young and also established scientists and engineers to build the skills necessary to analyze equations. …We concentrate on …obtaining the answer. We stress care but not rigor. …This book is completely self-contained. …We are concerned only with functions of one variable. …The minimum prerequisites …are …calculus and an elementary knowledge of differential equations”. The book is divided into 4 parts.

Part 1 (Ch. 1-2) deals with some basic facts from the differential and difference equation.

Part 2 (Ch. 3-6) deals with the behavior of solutions of differential equations near singular points, including asymptotic analysis of the behavior near infinity, Stokes phenomenon and elements of the asymptotic series theory; among many examples with nonlinear equations are first Painlevé and Thomas-Fermi equations. There are some remarks about systems “with random behavior”, such as the Lorenz system. The two concluding chapters of this part deal with the approximation of solutions to difference equations and with asymptotic expansions of integrals (Laplace’s method, methods of stationary phase and steepest descents).

Part 3 (Ch. 7-8) deals with perturbation methods for solution of algebraic and differential equations, eigenvalue problems, patching and matching, singular perturbations. Ch. 8 deals with the summation of series (improvement of convergence, summation of divergent series, Padé approximation.

Part 4 (Ch. 9-11), Global analysis, deals with the boundary-layer theory, WKB approximation and the so-called multi-scaled analysis (some prescriptions for obtaining uniformly valid approximations). There are no theorems in this book, but there are lots of examples, exercises and problems. Many problems of practical interest are discussed in detail and it makes the book very useful for engineers, who can consider it as a kind of problem book with solutions fit for self-education. In the literature there are many books which deal with the same questions [A. Erdélyi, Asymptotic expansions (1956; Zbl 0070.29002); N. G. de Brujn, Asymptotic methods in analysis (1958; Zbl 0082.04202); J. Heading, An introduction to phase-integral methods (1962; Zbl 0115.07102); M. Wasow, Asymptotic expansions for ordinary differential equations (1965; Zbl 0133.35301)], cited by the authors, and many other books. Though the present book is not fit for people who want to learn the theory of asymptotic methods, it is a useful complement to the literature which can serve as a guide for people who want to obtain an approximate solution to a problem they encounter in their work.

From the authors’ preface: “The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that can not be solved exactly. Our objective is to help young and also established scientists and engineers to build the skills necessary to analyze equations. …We concentrate on …obtaining the answer. We stress care but not rigor. …This book is completely self-contained. …We are concerned only with functions of one variable. …The minimum prerequisites …are …calculus and an elementary knowledge of differential equations”. The book is divided into 4 parts.

Part 1 (Ch. 1-2) deals with some basic facts from the differential and difference equation.

Part 2 (Ch. 3-6) deals with the behavior of solutions of differential equations near singular points, including asymptotic analysis of the behavior near infinity, Stokes phenomenon and elements of the asymptotic series theory; among many examples with nonlinear equations are first Painlevé and Thomas-Fermi equations. There are some remarks about systems “with random behavior”, such as the Lorenz system. The two concluding chapters of this part deal with the approximation of solutions to difference equations and with asymptotic expansions of integrals (Laplace’s method, methods of stationary phase and steepest descents).

Part 3 (Ch. 7-8) deals with perturbation methods for solution of algebraic and differential equations, eigenvalue problems, patching and matching, singular perturbations. Ch. 8 deals with the summation of series (improvement of convergence, summation of divergent series, Padé approximation.

Part 4 (Ch. 9-11), Global analysis, deals with the boundary-layer theory, WKB approximation and the so-called multi-scaled analysis (some prescriptions for obtaining uniformly valid approximations). There are no theorems in this book, but there are lots of examples, exercises and problems. Many problems of practical interest are discussed in detail and it makes the book very useful for engineers, who can consider it as a kind of problem book with solutions fit for self-education. In the literature there are many books which deal with the same questions [A. Erdélyi, Asymptotic expansions (1956; Zbl 0070.29002); N. G. de Brujn, Asymptotic methods in analysis (1958; Zbl 0082.04202); J. Heading, An introduction to phase-integral methods (1962; Zbl 0115.07102); M. Wasow, Asymptotic expansions for ordinary differential equations (1965; Zbl 0133.35301)], cited by the authors, and many other books. Though the present book is not fit for people who want to learn the theory of asymptotic methods, it is a useful complement to the literature which can serve as a guide for people who want to obtain an approximate solution to a problem they encounter in their work.

Reviewer: A.G.Ramm (Manhattan)

### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |

40E05 | Tauberian theorems |