Perturbation Methods for Engineers and Scientists

✍ Scribed by Alan W. Bush

• Publisher: CRC Press;Routledge
• Year: 2017
• Tongue: English
• Leaves: 316
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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