Modeling and Analysis of Modern Fluid Problems

By Liancun Zheng and Xinxin Zhang

Modeling and Analysis of Modern Fluids helps researchers solve physical problems observed in fluid dynamics and related fields, such as heat and mass transfer, boundary layer phenomena, and numerical heat transfer. These problems are characterized by nonlinearity and large system dimensionality, and ‘exact’ solutions are impossible to provide using the conventional mixture of theoretical and analytical analysis with purely numerical methods.

To solve these complex problems, this work provides a toolkit of established and novel methods drawn from the literature across nonlinear approximation theory. It covers Padé approximation theory, embedded-parameters perturbation, Adomian decomposition, homotopy analysis, modified differential transformation, fractal theory, fractional calculus, fractional differential equations, as well as classical numerical techniques for solving nonlinear partial differential equations. In addition, 3D modeling and analysis are also covered in-depth.

• Systematically describes powerful approximation methods to solve nonlinear equations in fluid problems
• Includes novel developments in fractional order differential equations with fractal theory applied to fluids
• Features new methods, including Homotypy Approximation, embedded-parameter perturbation, and 3D models and analysis

• Language: English
• Publisher: Academic Press
• Release date: Apr 26, 2017
• ISBN: 9780128117590

Liancun Zheng

Liancun Zheng (University of Science and Technology, Beijing), is a Professor in Applied mathematics with interest in partial/ordinary differential equations, fractional differential equations, non-Newtonian fluids, viscoelastic fluids, micropolar fluids, nanofluids, heat and mass transfer, radioactive heat transfer, nonlinear boundary value problems, and numerical heat transfer. He has published more than 260 papers in international journals and 5 books (in Chinese) and has served as Editor or Guest Editor of International Journals on 10 occasions.

Book preview

Mathematics in Science and Engineering

Modeling and Analysis of Modern Fluid Problems

Editors

Liancun Zheng

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, China

Xinxin Zhang

School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing, China

Series Editor

Goong Chen

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1. Introduction

1.1. Basic Ideals of Analytical Methods

1.2. Review of Analytical Methods

1.3. Fractal Theory and Fractional Viscoelastic Fluid

1.4. Numerical Methods

1.5. Modeling and Analysis for Modern Fluid Problems

1.6. Outline

Chapter 2. Embedding-Parameters Perturbation Method

2.1. Basics of Perturbation Theory

2.2. Embedding-Parameter Perturbation

2.3. Marangoni Convection

2.4. Marangoni Convection in a Power Law Non-Newtonian Fluid

2.5. Marangoni Convection in Finite Thickness

2.6. Summary

Chapter 3. Adomian Decomposition Method

3.1. Introduction

3.2. Nonlinear Boundary Layer of Power Law Fluid

3.3. Power Law Magnetohydrodynamic Fluid Flow Over a Power Law Velocity Wall

3.4. Marangoni Convection Over a Vapor–Liquid Surface

3.5. Summary

Chapter 4. Homotopy Analytical Method

4.1. Introduction

4.2. Flow and Radiative Heat Transfer of Magnetohydrodynamic Fluid Over a Stretching Surface

4.3. Flow and Heat Transfer of Nanofluids Over a Rotating Disk

4.4. Mixed Convection in Power Law Fluids Over Moving Conveyor

4.5. Magnetohydrodynamic Thermosolutal Marangoni Convection in Power Law Fluid

4.6. Summary

Chapter 5. Differential Transform Method

5.1. Introduction

5.2. Magnetohydrodynamics Falkner–Skan Boundary Layer Flow Over Permeable Wall

5.3. Unsteady Magnetohydrodynamics Mixed Flow and Heat Transfer Along a Vertical Sheet

5.4. Magnetohydrodynamics Mixed Convective Heat Transfer With Thermal Radiation and Ohmic Heating

5.5. Magnetohydrodynamic Nanofluid Radiation Heat Transfer With Variable Heat Flux and Chemical Reaction

5.6. Summary

Chapter 6. Variational Iteration Method and Homotopy Perturbation Method

6.1. Review of Variational Iteration Method

6.2. Fractional Diffusion Problem

6.3. Fractional Advection-Diffusion Equation

6.4. Review of Homotopy Perturbation Method

6.5. Unsteady Flow and Heat Transfer of a Power Law Fluid Over a Stretching Surface

6.6. Summary

Chapter 7. Exact Analytical Solutions for Fractional Viscoelastic Fluids

7.1. Introduction

7.2. Fractional Maxwell Fluid Flow Due to Accelerating Plate

7.3. Helical Flows of Fractional Oldroyd-B Fluid in Porous Medium

7.4. Magnetohydrodynamic Flow and Heat Transfer of Generalized Burgers' Fluid

7.5. Slip Effects on Magnetohydrodynamic Flow of Fractional Oldroyd-B Fluid

7.6. The 3D Flow of Generalized Oldroyd-B Fluid

7.7. Summary

Chapter 8. Numerical Methods

8.1. Review of Numerical Methods

8.2. Heat Transfer of Power Law Fluid in a Tube With Different Flux Models

8.3. Heat Transfer of the Power Law Fluid Over a Rotating Disk

8.4. Maxwell Fluid With Modified Fractional Fourier's Law and Darcy's Law

8.5. Unsteady Natural Convection Heat Transfer of Fractional Maxwell Fluid

8.6. Fractional Convection Diffusion With Cattaneo–Christov Flux

8.7. Summary

Index

Copyright

Academic Press is an imprint of Elsevier

125 London Wall, London EC2Y 5AS, United Kingdom

525 B Street, Suite 1800, San Diego, CA 92101-4495, United States

50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom

Copyright © 2017 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

ISBN: 978-0-12-811753-8

ISSN: 0076-5392

For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice Janco

Senior Acquisition Editor: Graham Nisbet

Editorial Project Manager: Susan Ikeda

Production Project Manager: Vijayaraj Purushothaman

Designer: Alan Studholme

Typeset by TNQ Books and Journals

Preface

A significant feature of modern science and engineering technology is that we are witnessing revolutionary changes regarding problems involved in containing enhanced degrees of nonlinearity and the complexity of system dimensions. Such problems are becoming increasingly prominent.

With the rapid development of computational science and technology, numerical methods of partial differential equations have become a branch of numerical analysis that studies numerical solutions to partial differential equations. However, numerical methods have limitations. For example, when a nonlinear problem is singular or has multiple solutions, a numerical method may not converge. Sometimes it may provide only limited or even misleading information; i.e., numerical methods cannot completely replace analytical methods. There has been good progress in the development of analytical methods for solving nonlinear ordinary/partial/fractional differential equations. Many empirical studies have shown that the analytical method can be used not only for local analysis but also to help us understand global behavior, offer insights into the structure of problems being studied, and solve real-world problems.

The purpose of this book is to provide a useful approach to modeling and analyzing methods for modern fluid problems arising in scientific research and engineering technology. This book can be used as a reference for researchers working in applied mathematics, fluid mechanics, energy, the environment, power, metallurgy, the chemical industry, water resources and hydropower, and engineering technology. It can also be used as a textbook for a graduate-level course in applied mathematics, mechanics, and engineering.

Most of the contents of this book are taken from the authors’ and their coauthors’ publications. The authors thank Xuehui Chen, Yan Zhang, Chunying Ming, Xinhui Si, Jing Zhu, Botong Li, Yaqing Liu, Chunrui Li, Chaoli Zhang, Jize Sui, Jinhu Zhao, Lin Liu, Fangfang Zhao, Lijun Wang, Zhenlin Guo, Chunxia Chen, and Chengru Jiao for their contributions to this book. During the writing of this book, we obtained enthusiastic help from Professor Buxuan Wang (Tsinghua University, Beijing, China, Members of the Chinese Academy of Sciences) and Professor Goong Chen (Texas A&M University, College Station, TX). They offered many valuable suggestions. We express our sincerest thanks to Professors Buxuan Wang and Goong Chen for their assistance and encouragement.

The authors wish to thank the National Natural Science Foundation of China (Grant Nos. 51,076,012, 51,276,014, and 51,476,191) for supporting our investigations.

Finally, we are very grateful for the excellent cooperation's of Editorial Project Manager Susan Ikeda, Project Manager Vijayaraj Purushothaman, and Senior Acquisitions Editor Graham Nisbet, during the preparation of the manuscript at Elsevier.

Owing to the limitations of our knowledge, this book will inevitably have shortcomings and deficiencies. We sincerely hope our readers will offer timely remarks and comments to improve the quality of the book.

Liancun Zheng     School of Mathematics and Physics,University of Science and Technology Beijing,Beijing, China

Xinxin Zheng     School of Mathematics and Physics,University of Science and Technology Beijing,Beijing, China

Chapter 1

Introduction

L. Zheng and X. Zhang

Abstract

Nature is full of many nonlinear and random phenomena. The study of nonlinear phenomena is a matter of natural sciences, engineering, and even social economic problems. Many scientists and engineers spend a lot of time and effort to study mathematical modeling and solving methods for these problems. This chapter presents an introduction to modeling, the development of analytical methods for modern fluid problems, and the outline of this book.

Keywords

Adomian decomposition; Approximate solution; Differential transformation; Fractional viscoelastic fluid; Homotopy analysis; Nonlinear differential equations; Perturbation theory

1.1. Basic Ideals of Analytical Methods

1.1.1. Analytical Methods

Most problems in science and engineering are in essence nonlinear and can be modeled by nonlinear differential equations. However, because of the complexity of nonlinear differential equations, so far we have not found a suitable analytical method to arrive at general solutions for various nonlinear equations. For these nonlinear equations, most cannot be solved analytically and can be solved only by the approximate method. Many basic properties commonly used in linear equations, such as the superposition principle of solutions, are no longer held (Li, 1989; Wang, 1993; Zheng et al., 2003; Zheng and Zhang, 2013).

There is good progress in the development of approximate analytical methods to solve nonlinear partial/ordinary differential equations. Many approximate analytical methods have been proposed to solve nonlinear ordinary equations, partial equations, fractional differential equations (FDEs), and integral equations. The commonly used methods are perturbation, Adomian decomposition, homotopy analysis, variational iteration, and differential transformation. Based on those methods, this book presents some useful approaches to modeling and analytical methods for modern fluid problems.

To understand the essence of this book better, a short review of the concepts of Taylor series and Fourier series are presented here, which are the bases for exploring approximate analytical methods for solving nonlinear partial differential equations.

1.1.1.1. Taylor Series

Suppose f(x)  ∈  Cn[a,b], that f(n+1)(x) exists on [a,b], x0  ∈  [a,b]. For every x  ∈  [a,b], there exists a number ξ  =  ξ(x) between x and x0, such that

(1.1)

where

(1.2)

is called the nth Taylor polynomial of f(x) about x0  ∈  [a,b], and

is called the nth Lagrange remainder term (or truncation error) associated with Pn(x).

The approximate error absolute value is |Rn(x)|  =  |f(x)  −  Pn(x)|.

If |Rn(x)| decreases monotonously with an increase in n, one can improve the degree of approximation by increasing the number of items in the polynomial. If f(x) has derivatives of all orders in the vicinity of xholds, in terms of polynomial (1.1), we can obtain the infinite series

(1.3)

Series (1.3) is called the Taylor series of f(x) at the point x  =  x0.

In the case of x0  =  0, the Taylor polynomial is often called a Maclaurin polynomial and the Taylor series is called a Maclaurin series.

The term truncation error refers to the error involved in using a truncated, or finite, summation to approximate the sum of an infinite series.

1.1.1.2. Fourier Series

Let f(x) be a periodic function of 2l. If f(x) satisfies these conditions:

1. it is continuous or has only a finite number of the first kind of discontinuity points in a period

2. it has at most a finite number of extreme points in a period

then f(x) is able to expand into Fourier series

(1.4)

when x is the continuous point, the series converges to f(x); when x is the discontinuity point, the series converges to

(1.5)

There is a relationship between coefficients a0, a1, b1, …, and the function of f(x)

(1.6)

The coefficients determined by Eq. (1.6) are known as the Fourier coefficient of the function f(x). The triangle series defined by the Fourier coefficient is called the Fourier series.

It can be seen that the essence of the Taylor series and the Fourier series is to select a suitable set of base functions (BFs)

(1.7)

to express the unknown function f(x) as:

(1.8)

The coefficients are determined by the selected basis functions and the given function f(x) itself.

When the BF is selected by 1, x, x², …, xnwe obtain the Fourier series.

It is obvious that the expressions of the different approximate series will be obtained when we select different sets of BFs. Therefore, the essence of the analytical approach to deal with the nonlinear problem is based on the following ideas:

;

2. Express unknown functions φ(x) as the combination of the BFs

3. Determine the coefficients such that it makes it possible to approximate the unknown solution function to the problems.

1.1.2. Padé Approximation

The Padé approximation method was developed in the process of studying the Taylor series expansion. The coefficients of Taylor series expansion and the value of the function are not only a profound mathematical problem but also an important practical problem (Baker, 1975; Baker and Graves-Morris, 1996; Thukral, 1999; Xu, 1990). Their investigation is based on the basis of mathematical analysis and physical and biological sciences (Coope and Graves-Morris, 1993; Graves-Morris, 1990, 1994; Graves-Morris and Jenkins, 1989). The interpretation criterion is that if the Taylor series expansion converges absolutely, it uniquely establishes the value of the function, which is differentiable an arbitrary number of times. On the other hand, if a function is differentiable an arbitrary number of times, it uniquely defines the Taylor series expansion, as shown in the subsequent theorem.

1.1.2.1. Weierstrass Approximation Theorem

Suppose f(x)  ∈  C[a,b]; then for each ε  >  0, there exists a polynomial P(x) with the property that

The proof of this theorem can be found in most elementary texts on real analysis.

In terms of the Weierstrass approximation theorem, it is possible to approximate a function as a polynomial by combining more and more BFs. However, this approach has undesirable limitations in practical operations.

Consider the following example:

(1.9)

It is easy to see that the Taylor series expression is not convergent for any value of x  >  0.5, even though for any value of 0  ≤  x  <  +∞, f(x.

The standard method is to use the original expression to develop a new Taylor series expression for f(x) from the old one by computing f(x) and its derivatives at a new point x0（0  <  x0  <  0.5, this new Taylor series expression, can be met in a larger range of x but it does not include x  =  ∞). In fact, x  =  ∞ can never be achieved, any progress in this direction is lengthy, and it will be tedious to attempt to meet the very lengthy range.

It is supposed that a variable substitution is introduced as x  =  w/(1  −  2w) or w  =  x/(1  +  2x); then

(1.10)

In this substitution, x  =  ∞ is transformed into w  =  0.5. The Taylor series expression of f(x(w)) converges into w  =  0.5, i.e., x  =  ∞. Therefore, the sequence initial estimate values for f(∞) are:

. In view of the initial variables of x, the successive approximations are expressed as:

They are all the rational fractional functions of variable x.

The Padé approximation is a method for approximating the value of a known function using special fractional functions. The main idea is to match a Taylor series expression as quickly as possible (Graves-Morris, 1994; Graves-Morris and Jenkins, 1989).

For example, for this problem, we introduce an approximate function of the form

(1.11)

The formula is bounded when x approaches infinity. When we approximate f(x) using the first three coefficients of terms of the Taylor series, we can obtain the following expression:

In view of this formula, when x  =  ∞, it has a value of 1.4. This result is better than any approximation results obtained previously. Similarly, we can compute a new approximate result as:

. Furthermore, if we continue to use this method to calculate the approximation, we find that the more coefficients are used, the better convergence result are obtained, which are shown as:

where the final value is obtained by 11 coefficients of Taylor series, which has an error of 10−⁸ compared with the exact value.

In the same way, we can arrive at an approximation for function f(x(w)), i.e., by substituting x  =  w/(1  −  2w) or w  =  x/(1  +  2x) into this formula, and we obtain:

The three results are obtained using the coefficients of the first, third, and fifth terms of the function f(x(w)).

When using w  =  0.5 (i.e., x  =  ∞) in this formula, we obtain the values of 1, 1.4, 41/29, … . Those values are identical with the results obtained using the Taylor series polynomial expansion of x. This invariance property is a general and important property of Padé approximation methods and is the basis of their ability to sum the x series in our example; it gives excellent results, such that we can obtain ideal results even up to the value of x  =  ∞.

1.1.2.2. Definition of Padé Approximants

Definition: The Padé approximant of a function A(x) is denoted as:

(1.12)

where PL(x) is a polynomial of the highest degree of L and QM(x) is a polynomial of the highest degree of M.

In view of the formal power series

(1.13)

we can determine the coefficients of polynomials PL(x) and QM(x) by the following equation

(1.14)

When the fraction with the numerator and denominator is multiplied by a nonzero constant the fractional values remain unchanged, we can define the standardized normalization condition as

(1.15)

Note that there are no public factors of the polynomial for functions PL(x) and QM(x).

If we express the coefficients of polynomial functions PL(x) and QM(x) as

(1.16)

In view of Eq. (1.16), multiplying Eq. (1.14) by QM(x), we can obtain the following linear systems of coefficients:

(1.17)

and we may define

(1.18)

Definition Padé approximants given here are different from the definition of classical approximants in several ways. First, the expression of symbols is the different in the classical definition:

(1.19)

However, some authors use the notation:

(1.20)

The formula employed here is used (Eq. 1.12) to avoid confusion. By convention, L denotes the degree of numerator and M denotes the degree of denominator and the following formula:

(1.21)

to express the sum and difference of those degrees of the numerator and the denominator.

A important point in mathematics is that in the standardized condition, the new definition is clearly different from the classical definition in the normalization condition (Eq. 1.15). Frobenius (1881) and Padé (1892) required only that QM(x)  ≠  0, as illustrated in the following example:

(1.22)

For L  =  M  =  1, it is easy to see that

(1.23)

It satisfies

(1.24)

However, it does not satisfy Eq. (1.14). In fact, for this series, the expression [1/1] does not exist in the new definition.

1.1.2.3. Uniqueness Theorem of Padé Approximants

According to the definition of Frobenius (1881) and Padé (1892), we have the following theorem.

Theorem (uniqueness): If the power series of Padé approximants exists, [L/M] of Padé is unique.

Proof. If the theorem is not true, without a loss of generality we may assume that two Padé approximants exist, i.e., X(x)/Y(x) and U(x)/V(x), in which the degree of X and U is less than or equal to L, and the degree of Y and V is less than or equal to M, respectively. Therefore, the following formula holds:

(1.25)

Multiplying both sides of Eq. (1.25) by Y(x)V(x) yields

(1.26)

It is seen that the highest degree of polynomial on the left-hand of Eq. (1.26) is at most L  +  M, so both sides are identically zero. Because neither Y nor V is identically zero, we have

(1.27)

Therefore, by definition, X and Y, and U and V are relatively prime and Y(0)  =  V(0)  =  1.0. We can assert that the assumption of two Padé approximants is actually the same. The proof of theorem is completed.

This theorem is held whether or not the defining equations are nonsingular. If the equations are nonsingular, we can solve them directly and obtain:

(1.28)

1.1.2.4. Table of Padé Approximants

Frobenius (1881) initially expressed Padé approximants in an indexed array. Padé (1892) was the first to emphasize their importance and arrange the values of Padé approximants into a table

(1.29)

We can see from the table that the partial sums of the Taylor series occupy the first column of the table. This table transposes the original expressions of Padé (1892) and many subsequent workers.

Using this method to make Padé approximants to the function of ex, we can obtain:

(1.30)

(1.31)

(1.32)

(1.33)

For x  =  1, the values of approximants satisfy:

(1.34)

The last value has an error to the exact value of 10−⁸.

1.2. Review of Analytical Methods

1.2.1. Perturbation Method

Perturbation theory is closely related to the methods used in numerical analysis. The earliest use of what astrophysical scientists now call perturbation theory was to deal with otherwise unsolvable mathematical problems of celestial mechanics.

Perturbation theory was first devised to solve otherwise intractable problems in calculating the motions of planets in the solar system. The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led several notable 18th- and 19th-century mathematicians to extend and generalize the methods of perturbation theory. The main idea of the perturbation method is to express the problem by analyzing the small or large parameter. The first several terms (often one or two) can reveal the important features of the solution to the problem; subsequent steps give only a little correction. Thus, the theory of perturbation theory has been widely applied (Li, 1999; Wasow, 1994).

Lindstedt et al. first established the perturbation when they studied the problem of planetary orbit (Sheng and Gui, 1996). The perturbation method is a kind of asymptotic analysis method that can analyze the global behavior of the solution of a differential equation. Its advantage is that it is not only able to give the correct approximate solution; it can also give the analytical structure for an qualitative and quantitative analysis of physical problems. The advantage is that the numerical solution cannot be reached.

After World War II, the mechanics analysis method obtained an important development in the generalized variational principle and singular perturbation theory. These well-developed perturbation methods were adopted and adapted to solve new problems arising in 20th-century atomic and subatomic physics. Combined with computers, the generalized variational principle provided a broad working field, and perturbation theory provided an effective means for mechanics to enter the nonlinear fields (Grasman and Matkowsky, 1977; Grasman et al., 1978; Kreiss and Parter, 1974; Lin and Segel, 1974; Matkowsky, 1975; Pearson, 1968; Stuff, 1972).

1.2.1.1. Regular Perturbation and Singular Perturbation

Perturbation theory is a theory for solving approximate solutions with small parameters. A differential equation with small parameters is formulated as:

(1.35)

where Lε is a differential operator containing small parameters ε and Bε, j is the differential operators defined on the boundary.

The general procedure for solving the problem of Pε is:

1. Select an asymptotic series sequence {δn(ε)} in which {εn} is the most commonly used asymptotic sequence, satisfying

(1.36)

2. Expand the unknown solution according to asymptotic series sequence

(1.37)

3. Substitute the expanded formula into the original problem and compare the coefficients of δn(ε); one can obtain the recursive equation and boundary conditions as

(1.38)

4. The solutions can be obtained by solving these equations, in which u0 is the solution of degeneration problem (ε  =  0).

In most practical physical problems, the solutions obtained are often compared with the experimental and numerical results, so as to judge the validity of the asymptotic expansion. If these steps can be used to obtain the uniformly valid solution in the region, the perturbation problem is called the regular perturbation problem in the region; otherwise it is known as the singular perturbation problem.

In a certain accuracy range, we can use the first few terms of the asymptotic expansion to approximate the original solution of u(x; ε). Therefore, the perturbation method is widely used in practical problems. It makes up for the shortcomings of the pure numerical method and has the characteristics of the analytical solution, so it is called the semianalytical method.

1.2.1.2. Asymptotic Matching Method

The matching asymptotic expansion method is the main method used in dealing with the singular perturbation boundary layer problem. The boundary layer was first proposed by L. Prandtl in 1904 (presented at the third International Congress of Mathematicians in Heidelberg, Germany). It simplifies the Navier–Stokes equations of fluid flow by dividing the flow field into two parts: one is the part near the surface in the immediate vicinity of a bounding surface, called the boundary layer, where the effects of viscosity are significant. It is dominated by viscosity and creates most of the drag experienced by the boundary body. Another part is outside the boundary layer, where viscosity can be neglected without significant effects on the solution. Therefore, Prandtl derived the boundary layer equation and the solutions for two parts. Then the two solutions link up, leading to the boundary layer technique, or the method of matching, i.e., the Prandtl matching principle.

1.2.1.3. Poincare–Lighthill–Kuo Method

In general in nonlinear problems, the period or frequency is no longer a constant but is commonly associated with amplitude. When one makes asymptotic expansion a function, one should simultaneously make asymptotic expansion for the frequency, wave velocity, characteristic value, or coordinates, such as the independent variable of some parameters. The secular term elimination is used to determine whether the independent variable is perturbation expansion.

In this way, the uniformly valid asymptotic solution of the problem is obtained, which is the basic ideas of the Poincare–Lighthill–Kuo (PLK) method. Because the variables should make scale transform in this method, the PLK method is also called the coordinate deformation method.

Some Chinese scientists made important contributions to the development of perturbation theory. In 1948, Professor Qian Wei-Chang created a synthetic expansion method to solve the large deflection problem of a circular plate; good results were obtained. Guo Yong-Huai generalized the Poincaré–Lighthill method to the boundary layer problem of viscous flow in 1953. In 1956, Qian Xue-Sen (LLS. Tsien) pointed out the importance of this method and called it the PLK method. Lin Jia-Qiao in 1954 proposed the characteristics theory, which is usually referred to as an analytical method for the problem of hyperbolic differential equations, and which provides an effective way to study nonlinear wave problems. Many scholars have made contributions in this field (Kang and Gui, 1996; Li, 1999; Stuff, 1972).

1.2.1.4. Average Method

The average method originated from Vanderpol in 1926, in the study of the self-excited oscillation of the circuit. The main approach is to use the amplitude and phase characteristics of nonlinear vibration to derive the approximate equation, which is satisfied by the method of the average value in one period. The method is also called the KBM method; it is an important tool to study the periodic solution of a differential equation.

1.2.1.5. Multiple Scales Method

The multiple scales method has been the fastest development and most widely used singular perturbation method. For singular perturbation problems in solving nonlinear vibration or boundary layer theory, the solution of the independent variable is not necessarily the same order of magnitude everywhere, such as in nonlinear vibration problems, amplitude is often slow change the amount of, and as the fluid in the vicinity of the wall bounded the flow speed is fast changing the amount of. Therefore, it is necessary to deal with the independent variables in multiple scales to describe these kinds of mathematical and physical problems.

Professor Lin Jia-Qiao demonstrated that if the multiple scales are chosen properly, the zeroth-order asymptotic approximation of the solution will give the true solution to the problem. The multiple scales method contains problems that can be solved by means of the asymptotic matching method, PLK method, average method, and so on, but it is difficult to choose the form for multiple scales variables. It is necessary to have some practical knowledge and experience when using the method of multiple scales.

While pursuing a doctorate at the University of Science and Technology Beijing (USTB), Professor Zheng’s doctoral student, Yan Zhang, proposed a novel technique, i.e., the embedding-parameters perturbation method. The basic idea of this method is by introducing special parameters transformation for both independent and dependent variables, we can embed