Radhika / Iyengar / Rani Approximate Analytical Methods for Solving Ordinary Differential Equations

1. Auflage 2014
ISBN: 978-1-4665-8816-5
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 200 Seiten

Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently developed procedures, including the classical series solution method, diverse perturbation methods, pioneering asymptotic methods, and the latest homotopy methods.

The book is suitable not only for mathematicians and engineers but also for biologists, physicists, and economists. It gives a complete description of the methods without going deep into rigorous mathematical aspects. Detailed examples illustrate the application of the methods to solve real-world problems.

The authors introduce the classical power series method for solving differential equations before moving on to asymptotic methods. They next show how perturbation methods are used to understand physical phenomena whose mathematical formulation involves a perturbation parameter and explain how the multiple-scale technique solves problems whose solution cannot be completely described on a single timescale. They then describe the Wentzel, Kramers, and Brillown (WKB) method that helps solve both problems that oscillate rapidly and problems that have a sudden change in the behavior of the solution function at a point in the interval. The book concludes with recent nonperturbation methods that provide solutions to a much wider class of problems and recent analytical methods based on the concept of homotopy of topology.

Zielgruppe

Researchers, practitioners, and graduate and advanced undergraduate students in mathematics, engineering, biology, physics, and economics.

Autoren/Hrsg.

Radhika, T.S.L

Iyengar, T. K.V.

Iyengar, T.

Rani, T. Raja

Rani, T.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Introduction

Power Series Method
Introduction
Algebraic Method (Method of Undetermined Coefficients)
Solution at Ordinary Point of an Ordinary Differential Equation
Solution at a Singular Point (Regular) of an Ordinary
Differential Equation
Remarks on the Frobenius Solution at Irregular Singular Points
Taylor Series Method

Asymptotic Method
Introduction
Asymptotic Solutions at Irregular Singular Points at Infinity
Asymptotic Solutions of Perturbed Problems
Solutions to ODEs Containing a Large Parameter

Perturbation Techniques
Introduction
Regular Perturbation Theory
Singular Perturbation Theory

Method of Multiple Scales
Introduction
Method of Multiple Scales

WKB Theory
Introduction
WKB Approximation for Unperturbed Problems
WKB Approximation for Perturbed Problems

Nonperturbation Methods
Introduction
Lyapunov’s Artificial Small-Parameter Method
Delta Expansion Method
Adomian Decomposition Method

Homotopy Methods
Introduction
Homotopy Analysis Method
Homotopy Perturbation Method
Optimal Homotopy Analysis Method

Exercise Problems, Applications, and a Bibliography appear at the end of each chapter.