Perturbation Methods for Engineers and Scientists

Description

The subject of perturbation expansions is a powerful analytical technique which can be applied to problems which are too complex to have an exact solution - for example, calculating the drag of an aircraft in flight. These techniques can be used in place of complicated numerical solutions. In some areas such as boundary layers it provides the essential ideas of scaling of regions of rapid change which must be understood before an appropriate discretization can be constructed. The book is aimed at students in applied mathematics, engineering, industrial mathematics, fluid mechanics and computational mechanics. Content: Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface 1: Introduction to Perturbation Expansions 1.1 Model problem. Resisted motion of a particle A perturbation expansion The exact solution 1.2 Roots of polynomials Regular and singular perturbation expansions Exercises 1.3 Initial value problems Exercises 1.4 Expansions involving the independent variable Convergent and divergent series Asymptotic series Optimum truncation rule The error function Comparison with a convergent series Exercises 2: Asymptotics 2.1 Order symbols Gauge functions L'Hospital's ruleTaylor's formula with remainder Properties of order symbols Exercises Behavior of logarithmic and exponential functions 2.2 Asymptotic sequences and expansions Exercise Uniqueness Maclaurin series are asymptotic expansions 2.3 Uniform and nonuniform expansions Region of nonuniformity Exercises 2.4 Sources of nonuniformity Infinite domains Small parameter multiplying the highest derivative 3 Strained Coordinates 3.1 The Lindstedt-Poincare technique Model problem 3.2 Duifing's equation The pendulum A mass and spring oscillator Solutions of Duffing's equation using the Lindstedt-Poincare techniqueExercise 3.3 Lighthill's technique Renormalization Shift in the singularity of a differential equation Exercise Application to nonlinear oscillators An example of the failure of renormalization Exercises 3.4 Flow past an aerofoil The complex potential The parabolic aerofoil Comparison with the exact solution The elliptic aerofoil Comparison with the exact solution 4: Multiple Scales 4.1 Second order systems The phase plane Nonlinear autonomous systems and limit cycles 4.2 Limitation of renormalization Perturbations which cause a time-changing amplitude4.3 The method of multiple scales Time scales Van der Pol oscillator Exercises 4.4 Surface roughness effects in lubricated bearings Reynolds equation Surface roughness Average Reynolds equation Effect of roughness on the average pressure distribution 4.5 The method of averaging: the Krylov-Bogoliubov technique 5: Boundary layers 5.1 Model problem The stretched variable and inner expansion Prandtl's matching condition The composite expansion 5.3 Boundary layer thickness and the principle of least degeneracy A boundary layer of 0{^e)An interior boundary layer 5.2 Boundary layer location The general linear equation Exercises 5.4 Higher order matching Exercises 5.5 Nonlinear examples Exercises 5.6 Practical applications Peclet and Reynolds numbers Flow between parallel planes with heat transfer Heat transfer rate The finite difference solution Exercise Estimation of boundary layer thickness The Prandtl number Exercise A remark on turbulence The boundary integral equation method Prandtl's momentum boundary layer equations The partial differential equation method