Application of Perturbation Methods to Approximate the ... · Perturbation methods, also known as asymptotic, allow the simplification of complex mathematical problems. Use of perturbation

i

Application of Perturbation Methods to Approximate the Solutions to

Static and Non-linear Oscillatory Problems

by

William Thomas Royle

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

_________________________________________

Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, CT

December, 2011

(For Graduation May 2012)

ii

CONTENTS

Application of Perturbation Methods to Approximate the Solutions to Static and Non-

linear Oscillatory Problems .......................................................................................... i

LIST OF TABLES ............................................................................................................ iv

LIST OF FIGURES ........................................................................................................... v

ACKNOWLEDGMENT ................................................................................................. vii

ABSTRACT ................................................................................................................... viii

1. Introduction.................................................................................................................. 1

1.1 Background ........................................................................................................ 1

1.2 Project Scope ...................................................................................................... 3

2. Methodology ................................................................................................................ 5

2.1 Project Methodology .......................................................................................... 5

2.2 The Perturbation Method Explained with an Algebraic Equation ..................... 6

2.2.1 The Perturbation Method Applied to the Solution of an Algebraic

Equation ................................................................................................. 6

2.2.2 Exact Solution of the Algebraic Equation .............................................. 8

2.2.3 Perturbation Approximation Compared to Exact Solution .................... 8

2.2.4 Perturbation Approximation’s Small Parameter Sensitivity ................ 10

3. Results........................................................................................................................ 12

3.1 Brief Introduction to Non-dimensionalizing Differential Equations ............... 12

3.2 Linear Ordinary Differential Equation (Boundary Layer Problem) ................ 13

3.2.1 Perturbation Approximation................................................................. 14

3.2.2 Analytical Solution............................................................................... 17

3.2.3 Perturbation Approximation Compared to Analytical Solution........... 19

3.3 Unforced Duffing Equation.............................................................................. 20

3.3.1 Background .......................................................................................... 20

3.3.2 Regular Perturbation Approximation ................................................... 21

3.3.3 Poincare-Lindstedt Method .................................................................. 24

iii

3.3.4 Numerical Solution .............................................................................. 26


3.4 Van Der Pol Equation ...................................................................................... 34

3.4.1 Background .......................................................................................... 34

3.4.2 Regular Perturbation Approximation ................................................... 34

3.4.3 Poincare-Lindstedt Method .................................................................. 37

3.4.4 Multiple Scales Method ....................................................................... 38

3.4.5 Numerical Solution .............................................................................. 42


4. Conclusion ................................................................................................................. 51

References ........................................................................................................................ 53

A. Appendices ................................................................................................................ 54

A.1 Unforced Duffing Equation Numeric MAPLE Code ......................................... 55

A.2 Van Der Pol Equation Numeric MAPLE Code .................................................. 58

A.3 Numerical Value Tables for the Ordinary Differential Equation ....................... 61

A.4 Numerical Value Tables for the Duffing Equation ............................................. 62

A.5 Numerical Value Tables for the Van Der Pol Equation ..................................... 66

iv

LIST OF TABLES

Table 1: Perturbation and Exact Solutions to the Algebraic Equation .............................. 9

Table 2: Analytical Values Determined for the Ordinary Differential Equation ............ 19

Table 3: Perturbation and Exact Solutions to the Ordinary Differential Equation .......... 61

Table 4: Perturbation and Numerical Values Determined for the Unforced Duffing

Equation (ε=.01) .............................................................................................................. 62


Equation (ε=.05) .............................................................................................................. 64

Table 6: Perturbation and Numerical Values Determined for the Van Der Pol Equation

(ε=.01) .............................................................................................................................. 66

Table 7: Perturbation and Numerical Values Determined for the Van Der Pol Equation

(ε=.05) .............................................................................................................................. 68

v

LIST OF FIGURES

Figure 1: Comparative Solutions Plots for the Algebraic Equation .................................. 9

Figure 2: Perturbation Percent Error Plots for the Algebraic Equation ........................... 10

Figure 3: Comparative Solutions Plots for the Algebraic Equation as ε >1 .................... 11

Figure 4: Boundary Condition Visualization for Linear Ordinary Differential Equation 15

Figure 5: Comparative Solutions Plots for the Ordinary Differential Equation .............. 19

Figure 6: Regular Perturbation Percent Error Plot for the Ordinary Differential Equation

......................................................................................................................................... 20

Figure 7: Regular Perturbation versus Numeric Solution for Unforced Duffing Equation

(ε=.01) .............................................................................................................................. 27

Figure 8: Regular Perturbation versus Numeric Solution Percent Error Plot for Unforced

Duffing Equation (ε=.01) ................................................................................................. 28

Figure 9: Poincare-Lindstedt versus Numeric Solution for Unforced Duffing Equation

(ε=.01) .............................................................................................................................. 28

Figure 10: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for Unforced


Figure 11: Regular Perturbation versus Numeric Solution for Unforced Duffing Equation

(ε=.05) .............................................................................................................................. 30

Figure 12: Regular Perturbation versus Numeric Solution Percent Error Plot for

Unforced Duffing Equation (ε=.05)................................................................................. 31

Figure 13: Poincare-Lindstedt versus Numeric Solution for Unforced Duffing Equation

(ε=.05) .............................................................................................................................. 32

Figure 14: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for Unforced


Figure 15: Regular Perturbation versus Numeric Solution for Van Der Pol Equation

(ε=.01) .............................................................................................................................. 43

Figure 16: Regular Perturbation versus Numeric Solution Absolute Error Plot for Van

Der Pol Equation (ε=.01) ................................................................................................. 44

Figure 17: Multiple Scales versus Numeric Solution for Van Der Pol Equation (ε=.01) 45

Figure 18: Multiple Scales versus Numeric Solution Absolute Error Plot for Van Der Pol

Equation (ε=.01) .............................................................................................................. 46

vi

Figure 19: Perturbation versus Numeric Solution for Van Der Pol Equation (ε=.05) .... 47

Figure 20: Perturbation versus Numeric Solution Absolute Error Plot for Van Der Pol

Equation (ε=.05) .............................................................................................................. 48

Figure 21: Multiple Scales versus Numeric Solution for Van Der Pol Equation (ε=.05) 49

Figure 22: Figure 18: Multiple Scales versus Numeric Solution Absolute Error Plot for

Van Der Pol Equation (ε=.05) ......................................................................................... 50

vii

ACKNOWLEDGMENT

I would like to thank my family for their support over the course of my graduate study

especially during this final project. I would also like to thank the faculty and staff at

Rensselaer for their excellent education program. I would like to especially thank

Professor Gutierrez-Miravete for advising me throughout the duration of the project and

for making the cohort program a success. Additionally, I thank General Dynamics

Electric Boat Corporation and my work supervisor Thomas Lambert for supporting me

throughout my degree. I would like to thank one of my dearest friends and co-workers

Bernard Nasser Jr. for encouraging me to further my education by attending Rensselaer.

Finally my deepest thanks go to Jerold Lewandowski for spending countless time

mentoring me throughout my educational experience at Rensselaer.

viii

ABSTRACT

The purpose of this project is to learn and apply perturbation theory in order to

approximate solutions to engineering problems which would otherwise be intractable

through the use of traditional analytical methods. The report first outlines the technique

of perturbation theory with the aid of an algebraic equation. An introduction is provided

in the technique of non-dimensionalizing differential equations and how the ε term is

developed. Perturbation theory will then be applied to a linear ordinary differential

equation boundary layer problem. The boundary layer problem demonstrates the

technique required to match inner and outer solutions as well as the technique used to

develop a composite solution. Next, approximate solutions for several variations of a

non-linear mass spring dampener systems using various perturbation methods were

determined. The unforced Duffing and the Van Der Pol equations were investigated.

When regular perturbation approximations result with secular terms, a perturbation

approximation without the presence of secular terms will be developed through the use

of special perturbation methods; namely the Poincare-Lindstedt and Multiple Scales

methods. All problems investigated are also solved analytically or numerically as and

compared and contrasted to the approximations found through the use of perturbation

theory.

1

1. Introduction

1.1 Background

Perturbation methods, also known as asymptotic, allow the simplification of

complex mathematical problems. Use of perturbation theory will allow approximate

solutions to be determined for problems which cannot be solved by traditional analytical

methods. Second order ordinary linear differential equations are solved by engineers and

scientists routinely. However in many cases, real life situations can require much more

difficult mathematical models, such as non-linear differential equations.

Numerical methods used on a computer of today are capable of solving extremely

complex mathematical problems; however, they are not perfect. The numerical methods

of today can still run into a multitude of problems ranging from diverging solutions to

tracking wrong solutions. Numerical methods on a computer do not provide much

insight to the engineers or scientists running them. Perturbation theory can offer an

alternative approach to solving certain types of problems. Solving problems analytically

often helps an engineer or scientist to understand a physical problem better, and may

help improve future procedures and designs used to solve their problems. Also, in a time

where there are tough economic circumstances, it is not unreasonable to consider that

future employers may prefer to rely on human ingenuity over the necessity of

continually purchasing expensive software package licenses to solve problems in which

analytical approximations can be made.

The first step required to start the implementation of perturbation theory non-

dimensionalizing of the governing equation. Once the equation is non-dimensionalized,

perturbation theory requires taking advantage of a “small” parameter that appears in an

equation. This parameter, usually denoted “ε” is on the order of 0 < ε << 1.

Next, through educated assumptions on the order of magnitude of terms, a rough

approximate solution is determined through the use of logical elimination of low

impacting terms. The perturbation method then solves this reduced “outer problem”.

Next an “inner solution” is constructed to satisfy the other constraints of the problem. A

composite solution is obtained through a matching process.

2

Once a rough approximate solution is found, a “correction factor” may then be

determined using an order of magnitude analysis. While “correction factors” can be used

repeatedly, it is important to note, only a limited accuracy may be obtained through

perturbation theory. Correction terms may eventually result in a perturbation

approximation which diverges. This is unlike a series solution, which converges to the

answer as the number of terms goes to infinity.

To help understand conceptually the mechanics of perturbation, the following

example commonly known to most graduate level students is utilized. The equation of

continuity in Cartesian Coordinates is as follows:

[1-1-1]

The Navier Stokes equations for a Newtonian fluid with constant density and viscosity in

Cartesian coordinates is as follows:

[1-1-2]

[1-1-3]

[1-1-4]

Assuming a steady, constant density and viscosity, and two dimensional flow, the

continuity and Navier stokes equations reduce to the following:

[1-1-5]

[ 1-1-6]

[1-1-7]

Equation [1-1-4] is totally eliminated.

3

These equations are often used to model flow in boundary layer regions. Often

times, these equations are further simplified by engineers and scientist depending on the

physics of the problem being solved. This simplification can be performed by an order of

magnitude analysis. For example, the velocity in the vertical plane may be extremely

small compared to the velocity in the horizontal direction, therefore terms that carry the

vertical velocity term will be reduced to zero. While the vertical velocity may not be

exactly zero, this assumption will introduce some error into an eventual approximation.

The problem can be further simplified in this manor until an analytical solution is

obtainable. The mechanics of perturbation theory follows this same methodology

allowing analytical approximations to be found for equations which would otherwise be

impossible to solve without the use of a computer.

1.2 Project Scope

This objective of this project is to study, learn and introduce the perturbation

method with the support of simple algebraic equations. The process of non-

dimensionalizing prior to the start of developing a perturbation approximation will also

be addressed.

Once the perturbation method is introduced, it will be used to develop a set of

approximate solutions for an ordinary differential equation (boundary layer problem),

the Duffing equation and the Van Der Pol equation. Advanced perturbation methods will

be used to eliminate the burden of secular terms that appear in the devolvement of any

regular perturbation approximations.

The solutions obtained from the perturbation approximation are then compared to

analytical or numerical solutions obtained from the same problems throughout the study.

This allows confirmation of the correct application of the perturbation method, and for

the solutions to be compared and contrasted.

4

The following is a list of the problems to be solved:

Algebraic Equation [1]

[1-2-1]

This has relevance because it is a simple example in which to introduce perturbation

theory.

Linear Ordinary Differential Equation [1]

[1-2-2]

This describes a linear mass spring dampener oscillatory problem.

Unforced Duffing Equation [2]

[1-2-3]

Where α is consider to be a constant. This is a model of a non-linear restoration force

type problem.

Van Der Pol Equation [Reference 3]

[ 1-2-4]

This represents a non-linear “stick” oscillatory problem.

5

2. Methodology

2.1 Project Methodology

A polynomial algebraic equation will be solved using the traditional quadratic formula.

Next, solutions for the same equation will be approximated following the techniques of

perturbation theory. This will be done to develop the understanding of the methodology

required.

An analytical solution can be found for the ordinary linear differential equation by

using traditional methods for solving ordinary differential equations; however numerical

solutions will be required for the Duffing and Van Der Pol equations since they are non-

linear differential equations.

Microsoft Excel™ will be used to graph and compare analytical/numerical solutions to

the approximate solutions obtained through the use of perturbation theory. Maplesoft’s

MAPLE™ will be used to find numerical solutions as needed.

Sometimes during the development of a perturbation approximation, secular terms may

appear causing the perturbation approximation to diverge from the actual solution as time

increases. Secular terms are terms that grow as the approximation progresses without bound.

For these problems the Poincare-Lindstedt method will be used to develop perturbation

approximations without influence of secular terms. If the Poincare-Lindstedt method is

unable to eliminate all of the secular terms, the Multiple Scales method will be utilized.

6

2.2 The Perturbation Method Explained with an Algebraic

Equation

Perturbation methods find approximate solutions to problems by taking advantage

of a small parameter that appears in the initial problem. This parameter, usually denoted

“ ” must be on the order of 0 < << 1. The perturbation method is most easily

understood through a simple algebraic equation. First, equation [1-2-1] is reintroduced as

seen below:

[1-2-1]

2.2.1 The Perturbation Method Applied to the Solution of an Algebraic Equation

Leading Order Solution:

Since the primary assumption of the perturbation method is that is very small, the

most obvious way to approximate a solution to [1-2-1] is to set . This reduces to:

[2-2-1]

Solving for , yields the leading order roots:

[2-2-2]

1st Order Solution:

Assuming δ(x) is some correction factor, the second solution approximation is as seen

below:

[2-2-2]

It is important to note that the correction factor that is applied should always be smaller

than the leading term. Upon substitution of the leading order solution plus a correction,

δ(x), into the governing differential equation δ(x) is determined to be of the order as

goes to zero.

Since this is a second degree polynomial equation, it is known that there are two

roots. Both roots are determined through perturbation the same way by substitution of

[2-2-2] into [1-2-1]. This calculation will further develop the positive root. Substitution

of the positive root seen in [2-2-2] into [1-2-1] yields:

7

[2-2-3]

Expanding [2-2-3] yields:

[2-2-3]

Since both and are small numbers, their products are extremely small. Using

an order of magnitude analysis, and are eliminated from [2-2-3]. These

extremely small terms are known as higher order terms (HOTs). In perturbation

nomenclature these HOTs are often abbreviated as “…” since they carry little

significance to the solution resulting in often elimination. Solving the remainder for [2-

2-3] for yields:

[2-2-4]

Substitution of back into [2-2-2] for the positive root yields:

[2-2-4]

2nd

Order Solution:

Continuing with the positive root solution, the 3rd

solution approximation is

assumed to be:

[2-2-5]

Substitution of the positive root seen in [2-2-5] into [1-2-1] yields:

[2-2-6]

Expanding [2-2-6] yields:

[2-2-7]

Again since both and are small numbers, their products are extremely small.

These HOTs are eliminated from [2-2-7]. [2-2-7] is then used to solve for yielding:

[2-2-8]

Substitution of back into [2-2-5] for the yields the 3rd

positive root

approximation:

8

[2-2-9]

It is important to note that each correction term is smaller than that of the preceding

term. Larger correction terms can be an indication that either an algebraic error has

occurred, or that a mistake could have occurred during the elimination of the HOTs.

2.2.2 Exact Solution of the Algebraic Equation

Since this is a second degree polynomial, obviously the quadratic formula can be

used to determine the exact roots. The exact roots to [1-2-1] are:

[2-2-10]

2.2.3 Perturbation Approximation Compared to Exact Solution

The 1st term, 2

nd term, and 3

rd term perturbation approximations obtained in section 2.2.1

were compared to the exact solution determined in 2.2.2. Percent error was calculated

for each perturbation approximation. Percent Error was determined by the following

formula:

[2-2-11]

The actual value was taken to be the root solved by use of the quadratic formula.

Table 1 below was developed by utilizing equations [2-2-1], [2-2-4], [2-2-9], [2-2-10]

and [2-2-11].

9

Table 1: Perturbation and Exact Solutions to the Algebraic Equation

Small Parameter

Exact Perturbation 1st

Term Perturbation 2nd

Term Perturbation 3rd

Term

ε Positive

Root Positive

Root %Error

R1 Positive

Root %Error

R1 Positive

Root %Error

R1

0 1 1 0 1 0 1 0

0.000001 1 1 5E-05 1 -1.3E-11 1 0

0.00001 0.999995 1 0.0005 0.999995 -1.3E-09 0.999995 0

0.0001 0.99995 1 0.005 0.99995 -1.3E-07 0.99995 1.11E-14

0.001 0.9995 1 0.050012 0.9995 -1.3E-05 0.9995 7.89E-13

0.01 0.995012 1 0.50125 0.995 -0.00126 0.995013 7.85E-09

0.1 0.951249 1 5.124922 0.95 -0.13132 0.95125 8.2E-05

1 0.618034 1 61.8034 0.5 -19.0983 0.625 1.127124

10 0.09902 1 909.902 -4 -4139.61 8.5 8484.167

Plots of the perturbation approximation and exact solutions for 0 ε 1 can be

seen in Figure: 1.

Figure 1: Comparative Solutions Plots for the Algebraic Equation

Note that for any given value of ε the accuracy of the perturbation approximation

increases with the amount of corrections that were determined. The exact, 2nd

term and

3rd

term approximations are nearly indistinguishable at this magnification. Perturbations

approximations, unlike a typical series expansion, do not necessarily always become

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.02 0.04 0.06 0.08 0.1

Cal

cula

ted

Ro

ot

ε

Exact

Perturbation 1st Term

Perturbation 2nd Term

Perturbation 3rd Term

10

more precise as additional terms are added to the approximation. Perturbation solutions

are developed in powers of ε (in the limit as ε goes to zero), whereas series solutions are

developed in powers of . This distinction leads to differences in solution convergence.

Engineers and scientists should be wary that distinct limitations exist with the accuracy

that can be achieved with perturbation approximations.

Plots of the percent error of the 2nd

and 3rd

term perturbation approximation for 0

ε 1 can be seen below in Figure: 2.

Figure 2: Perturbation Percent Error Plots for the Algebraic Equation

2.2.4 Perturbation Approximation’s Small Parameter Sensitivity

One of the major limitations of the perturbation method is that as the value of ε

approaches a number on the order of 1 or larger; the accuracy of the perturbation

approximation rapidly decreases. This can be seen clearly in Figure: 3 which was plotted

with data from Table 1 in Section 2.2.3.

-0.15

-0.13

-0.11

-0.09

-0.07

-0.05

-0.03

-0.01

0.01

0 0.02 0.04 0.06 0.08 0.1

% E

rro

r

ε



11

Figure 3: Comparative Solutions Plots for the Algebraic Equation as ε >1

-5

-3

-1

1

3

5

7

9

0 2 4 6 8 10

Cal

cula

ted

Ro

ot

ε

Exact

Perturbation 1st Term



12

3. Results

3.1 Brief Introduction to Non-dimensionalizing Differential

Equations

Non-dimensionalizing the equation is the first step required in perturbation

methods. To introduce how this is to be accomplished, the typical linear ordinary

differential equation from a mass spring dash-pot dampener system is introduced below.

[3-1-1]

Here denotes the mass of the block, is the viscous friction coefficient of the

dampener, and is the spring coefficient. Since this equation will become non-

dimensionalized, the starting units can be either all SI or all English.

Assuming that:

[3-1-2]

And

[3-1-3]

Therefore:

[3-1-4]

Where and are non-dimensionalized values and and are dimensionalized

variables. It follows that utilizing the chain rule the first derivative of a function with

respect to t is:

[3-1-5]

And the second derivative of some function with respect to t is:

[3-1-6]

Substituting [3-1-6] and [3-1-5] into equation [3-1-1] yields:

[3-1-7]

Dividing [3-1-7] through by k and L yields:

13

[3-1-8]

Since the goal is to remove the dimensions for all the coefficients, let:

[3-1-9]

And substituting [3-1-9] into [3-1-8] simplifies to:

[3-1-10]

For the perturbation method to work there needs to be a small parameter ε

introduced into the problem. The first term is selected to be written with ε since all terms

of [3-1-10] have a coefficient of 1. Letting:

[3-1-11]

And substituting [3-1-11] into [3-1-10] yields:

[3-1-12]

Note from inspection of equation [3-1-11] that there is combination of

parameters that form . Perturbation methods can be applied to equation [3-1-12] with

relatively low error if the mass or spring constant in [3-1-1] is relatively very small, or if

the viscous friction coefficient of the dampener is relatively high. All governing

equations evaluated in this project were given and investigated in non-dimensional form.

3.2 Linear Ordinary Differential Equation (Boundary Layer

Problem)

Even though it is relatively straightforward to obtain exact solutions to linear

second order ordinary differential equations, it is valuable to address that not all

perturbation problems can be solved exactly the same way. While the Duffing and Van

Der Pol problems discussed in this paper are non-linear equations which solutions are

intractable through normal analytical methods, equation [1-2-2] was specifically chosen

in order to introduce the technique of matching and composite solution development.

The method of determining a composite solution Equation [1-2-2] is notably similar to

the equation [3-1-12] which was non-dimensionalized in section 3-1 of this paper. Re-

14

introducing the linear ordinary differential equation [1-2-2] as seen below with “ ” as

the dependent variable and “ ” as the independent variable:

[1-2-2]

The initial conditions used to solve this problem as follows:

[3-2-1]

And

[3-2-2]

3.2.1 Perturbation Approximation

Determining the “Outer Solution”:

Setting reduces equation [1-2-2] to:

[3-2-3]

Guessing the solution:

[3-2-4]

Substituting [3-2-4] into [3-2-3] yields

[3-2-5]

Since = -1, the general form solution of the differential equation is:

[3-2-6]

Since there is only one root, equation [3-2-6] simplifies to:

[3-2-7]

With this solution, only one of the boundary conditions from the initial problem can

be enforced. Using equation [3-2-1] and solving for in equation [3-2-7] results in

=0, which is firstly a trivial solution, but also would violate the initial problem as

shown in Figure: (4).

15

Figure 4: Boundary Condition Visualization for Linear Ordinary Differential

Equation

Figure: 4 shows that is a positive value,

is a positive value, and

is also a

positive value (since the function is concave up). If this was true, then:

[3-2-8]

Equation [3-2-8] violates the initial problem in equation [1-2-2] and therefore

equation [3-2-1] is not the proper boundary condition for equation [3-2-7].

Using the boundary condition in equation [3-2-2] to solve for in equation [3-2-7]

results in:

[3-2-9]

Substitution of [3-2-9] into [3-2-7] yields the following “outer solution”:

[3-2-10]

Determining the “Inner Solution”:

To determine the inner solution, magnification at is required. Letting:

[3-2-11]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Y(X

)

X

Initial Conditions

16

Utilizing the chain rule on equation [3-2-11] follows:

[3-2-12]

And:

[3-2-13]

Substitution of equations [3-2-12] and [3-2-13] into equation [1-2-2] yields:

[3-2-14]

Assuming that the first two terms balance and solving for follows:

[3-2-15]

Simplifying to solve for yields:

[3-2-16]

The two terms that were assumed to balance were:

[3-2-17]

It is then solved by guessing the general solution:

[3-2-18]

Substituting equation [3-2-18] into [3-2-17] and simplifying yields:

[3-2-19]

Since =-2 and 0, the general solution of the equation takes the form:

[3-2-20]

Using the remaining boundary condition in equation [3-2-1] and solving for yields:

[3-2-21]

Substitution of equation [3-2-21] into equation [3-2-20] yields the following “inner

solution”:

[3-2-22]

17

Since two separate solutions, equations [3-2-10] and [3-2-22], have been

obtained; matching is required to be performed in order to develop a composite solution

( ). To match the solution, the limit as the outer solution approaches is

set equal o the limit as the inner solution approaches :

[3-2-23]

This reduces to:

[3-2-24]

Equation [3-2-24] is not only used to determine the value of , but it also determines the

common solution of the limits of the inner and outer solution ( ).

[3-2-25]

The composite solution is determined by combining the inner and outer solutions and by

shifting the solutions by removing the common solution:

[3-2-26]

Combining equations [3-2-10], [3-2-11], [3-2-22], [3-2-24], [3-2-25] and [3-2-26] and

simplifying yields the composite solution:

[3-2-27]

3.2.2 Analytical Solution

Since this is a second order linear ordinary differential equation, traditional

analytical methods can be used to find a solution. Guessing the solution:

[3-2-28]

And substituting equation [3-2-28] into equation [1-2-2] and simplifying yields:

[3-2-29]

Utilizing the quadratic equation roots and can be solved for:

[3-2-30]

And

18

[3-2-31]

Since both roots are real numbers, the general solution takes the form:

[3-2-32]

Substitution of equations [3-2-30] and [3-2-31] into equation [3-2-32] yields the

following:

[3-2-33]

Enforcement of the initial condition seen in equation [3-2-1] to equation [3-2-33] yields:

[3-2-34]

Substituting equation [3-2-34] into [3-2-33] and enforcing the initial condition seen in

equation [3-2-2] into equation [3-2-33] yields:

[3-2-35]

Solving equation [3-2-35] for and than using equation [3-2-34] to solve for yields:

[3-2-36]

[3-2-37]

Substitution of equations [3-2-36] and [3-2-37] into equation [3-2-33] yields the final

analytical solution:

[3-2-38]

19

3.2.3 Perturbation Approximation Compared to Analytical Solution

Letting ε = .01, values determined from equations [3-2-30], [3-2-31], [3-2-36],

and [3-2-37] are determined in Table 2 below:

Table 2: Analytical Values Determined for the Ordinary Differential Equation

Small Parameter

Analytical Roots Analytical Constants

ε Root 1 Root 2 C1 C2

0.01 -1.00505 -198.99494 2.73204 -2.73205

Data from Table 2 was used in conjunction with equations [2-2-11], [3-2-27],

and [3-2-38] to create the comparative data plots seen in Table 3. The formation of the

boundary layer become apparent upon the inspection of the roots in Table 2. Since Root

2 is large in magnitude compared to Root one, the influence of the solution dependent on

Root 2 on the total solution is quickly reduced as increases. In Table 3, this boundary

layer can be seen for values up to .023038. Table 3 can be found in Appendix A.3.

Plots of the composite perturbation approximation and the analytical solutions for

the linear ordinary differential equation can be seen below in Figure: 5.

Figure 5: Comparative Solutions Plots for the Ordinary Differential Equation

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2

Y

X

Analytical Roots

Perturbation

20

A plot of the composite perturbation approximation’s percent error compared to the

analytical solution for the linear ordinary differential equation can be seen below in

Figure: 6.

Figure 6: Regular Perturbation Percent Error Plot for the Ordinary Differential

Equation

3.3 Unforced Duffing Equation

3.3.1 Background

The Duffing Oscillator is a differential equation that used to model non-linear

restoration force type problems. The Duffing Oscillator can be used to approximate the

physics of a pendulum problem [2].

Re-introducing the Duffing equation [1-2-3] as seen below with “ ” as the dependent

variable and “ ” as the independent variable:

[1-2-3]

The initial conditions used to solve this problem are as follows:

[3-3-1]

And

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1 1.2

% E

rro

r

X

%Error

21

[3-3-2]

3.3.2 Regular Perturbation Approximation



[3-3-3]

Guessing the solution:

[3-3-4]

Substituting [3-3-4] into [3-3-3] yields

[3-3-5]

Solving for:

[3-3-6]

The general form solution of the differential equation is:

[3-3-7]

The derivative of equation [3-3-7] with respect to is then:

[3-3-8]

Using the boundary condition in equation [3-3-1] to solve for in equation [3-3-7], and

boundary condition in equation [3-3-2] to solve for in equation [3-3-8] results in:

[3-3-9]

And

[3-3-10]


[3-3-11]

1st Order Solution:

22


below:

[3-3-12]

The derivative of equation [3-3-12] with respect to is then:

[3-3-13]

The second derivative of equation [3-2-12] with respect to is then:

[3-3-14]

Using the boundary condition in equation [3-3-1] to solve for in equation [3-3-12],

and boundary condition in equation [3-3-2] to solve for

in equation [3-3-13]

results in:

[3-3-15]

And

[3-3-16]

Substituting equation [3-3-14] and [3-3-12] into equation [1-2-3] and simplifying yields:

[3-3-17]

Expanding yields:

=

[3-3-18]

Eliminating the HOTs from equation [3-3-18], the remaining terms are substituted back

into equation [3-3-17], which is re-written as:

[3-3-19]

Letting:

[3-3-20]

And utilizing a combination of all the following common trigonometry identities:

23

[3-3-21]

[3-3-22]

[3-3-23]

[3-3-24]

[3-3-25]

can be expanded to:

[3-3-26]

Substitution of equation [3-3-26] into [3-3-19] and [3-3-20] yields:

[3-3-27]

Solving for as a traditional ordinary differential equation through superposition:

[3-3-28]

Noting that general solution takes the same form as equation [3-2-7], yields:

[3-3-29]

Guessing the particular solution:

[3-3-30]

The second derivatives of equation [3-3-28] with respect to are then:

[3-3-31]

Substitution of equations [3-3-30] and [3-3-31] into equation [3-3-27] and solving for

coefficients , , and yeild:

24

[3-3-32]

[3-3-33]

[3-3-34]

Combining equations [3-3-28], [3-3-29], [3-3-30], [3-3-32], [3-3-33], and [3-3-34] and

simplifying with equation [3-3-20] yields:

[3-3-35]

Taking the derivative of equation [3-3-35] with respect to yields:

[3-3-36]

Using the boundary conditions for equations [3-3-15] and [3-3-16] in equations [3-3-35]

and [3-3-36] and solving for and yields:

[3-3-37]

And

[3-3-38]

Combining equations [3-3-12], [3-3-35], [3-3-37], and [3-3-38] yields the 1st order

perturbation approximation is:

[3-3-39]

3.3.3 Poincare-Lindstedt Method

Upon a more detailed inspection of the 1st order perturbation approximation

developed in equation [3-3-39], not that as increases to a large number, the magnitude

of the 1st order correction factor increases. As progresses the

term (secular

25

term), even though multiplied by small number , will eventually dominate the

approximation. This will limit the range of in which the perturbation approximation

will be effective. In order to develop a perturbation approximation in which the negative

effect of the secular term can be minimized as increases, the Poincare-Lindstedt

method is used. Utilizing this method the frequency will be shifted which therefore

will reduce the error from the secular term. As continues to increase, more frequency

corrections need to be determined to further reduce error. Assuming is the correction

to , new variable is:

[3-3-40]

Using the chain rule, the first and second derivatives of [3-3-40] with respect to are:

[3-3-41]

And

[3-3-42]

Allowing reduces equation [1-2-3] to equation [3-3-3]. Substituting [3-3-42] into

[3-3-1] yields:

[3-3-43]

Knowing that the shift and are very small, utilizing an order of magnitude analysis

equation [3-3-43] simplifies to:

[3-3-44]

Following the same mathematical analysis as in section 3.3.2, the first Poincare-

Lindstedt leading order solution is determined to be:

[3-3-45]

Assuming θ is some correction factor, the second solution approximation is as seen

below:

[3-3-46]

The second derivative of equation [3-3-46] with respect to is:

26

[3-3-47]

Substitution of equations [3-3-42], [3-3-46] and [3-3-47] into equation [1-2-3] yields:

[3-3-48]

Expanding and eliminating the HOTs in equation [3-3-58] in the same manner as

performed in section 3.3.2 and simplification yields:

[3-3-49]

Expansion of as performed in section 3.3.2 and rearrangement yields:

[3-3-50]

Solving for in order to prevent the formation of the secular term yields:

[3-3-51]

Combining equations [3-3-40], [3-3-45] and [3-3-51] result in the Poincare-Lindstedt

approximation:

[3-3-52]

3.3.4 Numerical Solution

The numerical solution was obtained utilizing MAPLE’s built in Fehlberg fourth-

fifth order Runge-Kutta method with degree four interpolant. The MAPLE file used to

perform the numerical analysis can be seen attached in Appendix A.1.


For oscillator solutions absolute error is used for comparison in lieu of percent

error. The absolute error is determined by the following relation:

[3-3-53]

Case 1: ε = .01

27

Letting ε = .01 and α =1, equations [3-3-39], [3-3-52] and [3-3-53] as well as the

numerical solution developed in section 3.2.4 was used in order to produce Table 4 in

Appendix A.4.

The regular perturbation approximation seen in Table 4 is plotted together with

the numerical solution that was obtained with MAPLE in Figure: 7 below.

Figure 7: Regular Perturbation versus Numeric Solution for Unforced Duffing

Equation (ε=.01)

It is important to note that as increases, the tradition perturbation

approximation tends to rapidly increase in error with respect to the the numerical

solution. The absolute error plot of the regular perturbation versus the numerical solution

is seen in Figure: 8.

-1.2

-0.7

-0.2

0.3

0.8

1.3

0 50 100 150 200

Y

X

Y Numerical

Y Perturbation

28


Unforced Duffing Equation (ε=.01)

The rapid error increase in the regular perturbation approximation is a result of

the secular term in equation identified in [3-3-39].

The Poincare-Lindstedt perturbation approximation results shown in Table 4 are plotted

together with the numerical solution that was obtained with MAPLE in Figure: 9.

Figure 9: Poincare-Lindstedt versus Numeric Solution for Unforced Duffing

Equation (ε=.01)

-0.25

-0.15

-0.05

0.05

0.15

0.25

0 50 100 150 200

Erro

r

X

Absolute Error (Perturbation Numerical)

-1.2

-0.7

-0.2

0.3

0.8

1.3

0 50 100 150 200

Y

X

Y Numerical

Y Lindstedt

29

It is important to note that as increases, the Poincare-Lindstedt perturbation

approximation track the numerical solution far better than the regular perturbation

approximation. The absolute error plot of the Poincare-Lindstedt perturbation versus the

numeric solution is seen in Figure: 10.

Figure 10: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for


It is important to note the Poincare-Lindstedt method percent error also increases

with as the with the regular perturbation approximation however the magnitude of the

percent error is as much as two orders of magnitude smaller. As is continued to

progress the error of the Poincare-Lindstedt approximation can be reduced by further

correcting the frequency as needed.

Case 2: ε = .05

Letting ε = .05 and α =1, equations [3-3-39], [3-3-52] and [3-3-53] as well as the

numerical solution developed in section 3.2.4 was used in order to produce Table 5 in

Appendix A.4.

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0 50 100 150 200

Erro

r

X

Absolute Error (Lindstedt to Numerical)

30


the numerical solution that was obtained with MAPLE in Figure: 11 below.

Figure 11: Regular Perturbation versus Numeric Solution for Unforced Duffing

Equation (ε=.05)

Since ε has increased in size, the secular term found in the regular perturbation

approximation now dominates the solution faster than seen in Case 1. The absolute error

plot of the regular perturbation versus the numeric solution is seen in Figure: 12.

-5

-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 Y

X

Y Numerical

Y Perturbation

31




the secular term in equation identified in [3-3-39]. Since ε is now larger, the secular term

can influence the perturbation approximation faster. This causes a higher order of error

magnitude to appear in the approximation at the same values of . The range of in both

Cases 1 and 2 are identical.

The Poincare-Lindstedt perturbation approximation seen in Table 5 is plotted

together with the numerical solution that was obtained with MAPLE in Figure: 13

below.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 Err

or

X


32

Figure 13: Poincare-Lindstedt versus Numeric Solution for Unforced Duffing

Equation (ε=.05)

It is important to note that even with the increased ε value, the Poincare-

Lindstedt perturbation approximation track the numerical solution far better than the

regular perturbation approximation. The absolute error plot of the Poincare-Lindstedt

perturbation versus the numeric solution is seen in Figure: 14.

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200

Y

X

Y Numerical

Y Lindstedt

33

Figure 14: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for


The Poincare-Lindstedt method is able to provide a solution approximation that has

error two orders of magnitude small than the regular perturbation method.

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 50 100 150 200

Erro

r

X

Absolute Error (Lindstedt to Numerical)

34

3.4 Van Der Pol Equation

3.4.1 Background

The Van Der Pol oscillator is a model of a non-conservative energy system. The

Van Der Pol equation can be used to model stick-oscillations, aero-elastic flutter and

biological oscillatory phenomena [Reference 2].

Re-introducing the Van Der Pol equation [1-2-4] as seen below with “ ” as the

dependent variable and “ ” as the independent variable:

[1-2-4]

The initial conditions used to solve this problem are as follows:

[3-4-1]

And

[3-4-2]

3.4.2 Regular Perturbation Approximation



[3-4-3]

This is the same equation as equation [3-3-3] in the unforced Duffing equation

section, and the boundary conditions in equations [3-4-1] and [3-4-2] are the same as [3-

3-1] and [3-3-2]. Therefore the development of the leading order solution for [3-4-3] is

identical to that of [3-3-3]. Refer to section 3-3 for more information.

The leading order solution is determined to be:

[3-4-4]

1st Order Solution:


below:

35

[3-4-5]

The first and second derivatives of equation [3-4-5] with respect to are the

same as equations [3-3-13] and [3-3-14] in the Duffing equation section. The values of

and

are also determined identically as seen in the Duffing equation section

equations [3-3-15] and [3-3-16].

Substituting equation [3-3-14] and [3-3-12] into equation [1-2-4] and simplifying yields:

[3-4-6]

Expanding equation [3-4-6] and eliminating the HOTs terms results in the following the

remaining terms rewritten as

[3-4-7]

Letting:

[3-4-8]


[3-4-9]

Performing the same type of trigonometric expansion as performed with equation [3-3-

19] yields:

[3-4-10]

Solving for as a traditional ordinary differential equation through superposition:

[3-4-11]

Noting that general solution takes the same form as equation [3-4-3], yields:

[3-4-12]

Guessing the particular solution:

[3-4-13]

The second derivatives of equation [3-4-13] with respect to are then:

36

[3-4-14]

Substitution of equations [3-4-13] and [3-4-14] into equation [3-4-10] and solving for

coefficients , , and yield:

[3-4-15]

[3-4-16]

[3-4-17]

Combining equations [3-4-11], [3-4-12], [3-4-13], [3-4-15], [3-4-16], and [3-4-17] and

simplifying with equation [3-4-8] yields:

[3-4-18]

Taking the derivative of equation [3-4-18] with respect to yields:

[3-4-19]

Using the boundary conditions for equations [3-3-15] and [3-3-16] in equations [3-4-18]

and [3-4-19] and solving for and yields:

[3-4-20]

And

[3-4-21]

Combining equations [3-4-5], [3-4-18], [3-4-20], and [3-4-21] yields the 1st order

perturbation approximation is:

[3-4-22]

37

3.4.3 Poincare-Lindstedt Method

As seen with the Duffing equation, the development of the regular perturbation

approximation for the Van Der Pol equation results the secular term, (

),

appearing. The Poincare-Lindstedt method was utilized in attempt to develop a

perturbation approximation for the Van Der Pol equation without the hindrance of

secular terms.

Guessing the same shift used during the Poincare-Lindstedt section of the Duffing

equation as seen in equation [3-3-40], and carrying out identical analysis of the leading

order solution using equation [1-2-4] in lieu of [1-2-3] allows the development of the

same leading order solution as equation [3-3-46] rewritten as:

[3-3-46]

The first derivative of equation [3-3-46] with respect to is:

[3-4-23]

The second derivative of equation [3-3-46] with respect to is identical as seen in

equation [3-3-47]:

[3-3-47]

Substitution of equations [3-3-42], [3-3-46], [3-3-47], and [3-4-23] into equation [1-2-4]

yields:

[3-4-24]

Expanding and eliminating the HOTs in equation [3-4-24] in the same manner as

performed in section 3.3.2 and simplification yields:

[3-4-25]

Expansion of as performed in section 3.3.2 and rearrangement yields:

38

[3-4-26]

Unlike the Duffing equation from section 3-3-3, there is no value for in which

prevention of the formation of the secular terms can be obtained. The Poincare-Lindstedt

method approximation is therefore unable to alleviate the unwanted effects of a secular

term.

3.4.4 Multiple Scales Method

Since shifting the frequency of the solution through the use of the Poincare-

Lindstedt method has failed to yield a perturbation approximation for the Van Der Pol

equation, the next attempt to eliminate the unwanted effects of secular terms by utilizing

the Multiple Scales method. The Multiple Scales method introduces a new variable Ψ,

that forms the following relation to ε and :

[3-4-27]

Therefore, when becomes large in relative magnitude, the magnitude of becomes

normal sized.


The first derivative of function with respect to is:

[3-4-28]

Substitution of equation [3-4-27] into [3-4-28] yields:

[3-4-29]

The second derivative of function with respect to with substitution of equation [3-4-

27] is:

[3-4-30]


[3-4-31]

39

Simplification of equation [3-4-31] through the elimination of 2nd

and higher order terms

in ε:

[3-4-32]

Setting would results in the leading order problem reminisnt of the solution seen

in the ordinary differential equation in section 3-4-2, however the coefficients are now

unknown functions of due to the partial derivatives. Therefore the adjusted leading

order solution becomes:

[3-4-33]

Rewriting equation [3-4-34] yields:

[3-4-34]

The first derivative of equation [3-4-34] is:

[3-4-35]

Using the boundary condition seen in equation [3-4-1] in conjunction with equation [3-

4-34] and the boundary condition seen in equation [3-4-2] with conjunction with

equation [3-4-35] yields:

[3-4-36]

And

[3-4-37]

It should be noted there is a degree of non-uniqueness associated with equations [3-4-34]

and [3-4-35]. Equations [3-4-36] and [3-4-37] are assumed to satisfy the solution. These

values are to be carried through the remainder of the calculation. If the calculation was to

fail, the assumed values of [3-4-36] and [3-4-37] need to be re-determined.

1st Order Solution:


below:

[3-4-38]

The first derivative of equation [3-4-38] with respect to is:

40

[3-4-39]

The second derivative of equation [3-4-38] with respect to is:

[3-4-40]

The derivative of equation [3-4-38] with respect to once and once is:

[3-4-41]

Substitution of equations [3-4-38], [3-4-39], and [3-4-40] into equation [3-4-32] yields:

( , )2*

[3-4-42]

Expansion, simplification, and elimination of HOTS in equation [3-4-42] yields:

[3-4-43]

Using equation [3-4-43] for and

yields:

[3-4-44]

And

41

[3-4-45]

Noting that both equations [3-4-37] and [3-4-45] are equal to zero. is determined

to be a constant 0.

Simplifying equation [3-4-44] yields:

[3-4-46]

Separation of variables of equation [3-4-46] results in:

[3-4-47]

Using practical fraction decomposition the left side of equation [3-4-47] and setting it

equal to the right side, then integrating once results in:

[3-4-48]

Where is a constant of integration. Using the following log properties:

[3-4-49]

[3-4-50]

And

[3-4-51]

Equation [3-4-48] reduces to:

[3-4-52]

Solving equation [3-4-51] for yields:

[3-4-53]

Substituting equations and [3-4-27] and [3-4-53] into equation [3-4-35] and the

observation that is determined to be a constant yields the following:

[3-4-54]

Using the boundary condition in equation [3-4-1] and equation [3-4-54], can be

determined to be:

42

[3-4-55]

Substitution of equations [3-4-27], [3-4-55] into [3-4-54] yields:

[3-4-56]

3.4.5 Numerical Solution

As before, the numerical solution was obtained utilizing MAPLE’s built in

Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. The

MAPLE file used to perform the numerical analysis can be seen attached in Appendix

A.2.


Case 1: ε = .01

Letting ε = .01 equations [3-4-22], [3-4-56] and [3-3-53] as well as the numerical

solution developed in section 3.4.5 was used in order to produce Table 6 in Appendix

A.5

The regular perturbation approximation seen in Table 6 is plotted together with the

numerical solution that was obtained with MAPLE in Figure: 15 below.

43

Figure 15: Regular Perturbation versus Numeric Solution for Van Der Pol

Equation (ε=.01)

The absolute error plot of the regular perturbation versus the numerical solution

is seen in Figure: 16.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200

Y

X

Y Numerical

Y Perturbation

44

Figure 16: Regular Perturbation versus Numeric Solution Absolute Error Plot for

Van Der Pol Equation (ε=.01)

Once again as seen in the Duffing equation section, the regular perturbation

approximation’s absolute error increases as becomes large due to the secular term in

the regular perturbation approximation.

The Multiple Scales perturbation approximation seen in Table 6 is plotted

together with the numerical solution that was obtained with MAPLE in Figure: 17.

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 50 100 150 200 Err

or

X


45

Figure 17: Multiple Scales versus Numeric Solution for Van Der Pol Equation

(ε=.01)

It is important to note that as increases, the Multiple Scales perturbation

approximation tracks the numerical solution far better than the regular perturbation

approximation. The absolute error plot of the Multiple Scales perturbation versus the

numeric solution is seen in Figure: 18.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200

Y

X

Y Numerical

Y Multiple Scales

46

Figure 18: Multiple Scales versus Numeric Solution Absolute Error Plot for Van

Der Pol Equation (ε=.01)

The Multiple Scale method is able to provide a solution approximation that has


Case 2: ε = .05

Letting ε = .05 equations [3-4-22], [3-3-56] and [3-3-53] as well as the numerical

solution developed in section 3.4.5 was used in order to produce Table 7 in Appendix

A.5.

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0 50 100 150 200 Err

or

X

Absolute Error (Multiple Scales to Numerical)

47


the numerical solution that was obtained with MAPLE in Figure: 19.

Figure 19: Perturbation versus Numeric Solution for Van Der Pol Equation (ε=.05)

Since ε has increased in size, the secular term found in the regular perturbation

approximation now dominates the solution faster than seen in Case 1. The absolute error

plot of the regular perturbation versus the numerical solution is seen in Figure: 20.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200

Y

X

Y Numerical

Y Perturbation

48

Figure 20: Perturbation versus Numeric Solution Absolute Error Plot for Van Der

Pol Equation (ε=.05)


the secular term in equation identified in [3-4-22]. Since ε is now larger, the secular term

can influence the perturbation approximation faster. This causes a higher order of error

magnitude to appear in the solution at the same values of . The range of in both Cases

1 and 2 are identical.

The Multiple Scale perturbation approximation seen in Table 7 is plotted together

with the numerical solution that was obtained with MAPLE in Figure: 21.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 50 100 150 200 Err

or

X


49

Figure 21: Multiple Scales versus Numeric Solution for Van Der Pol Equation

(ε=.05)

It is important to note that especially with the increased ε value, the Multiple

Scale perturbation approximation tracks the numerical solution far better than the regular

perturbation approximation. The absolute error plot of the Poincare-Lindstedt

perturbation versus the numeric solution is seen in Figure: 22.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 50 100 150 200

Y

X

Y Numerical

Y Multiple Scales

50

Figure 22: Figure 23: Multiple Scales versus Numeric Solution Absolute Error Plot

for Van Der Pol Equation (ε=.05)

The Multiple Scales method is able to provide a solution approximation that has


-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 50 100 150 200 Err

or

X

Absolute Error (Multiple Scales to Numerical)

51

4. Conclusion

The intent of the work reported in this paper was to demonstrate and convey the

idea of using perturbation methods to solve some selected engineering and mathematical

problems.

This paper first explained the theory of finding approximate solutions through the

use of perturbation methods through a simple algebraic example. Error of first, second,

and third order perturbation corrections were compared. The sensitivity of perturbation

approximations accuracy as ε increases was compared to the exact solution determined

through the use of the quadratic equation.

Next, a brief introduction into the process of non-dimensionalizing an ordinary

linear differential equation was discussed. The differential equation selected can be used

to model the physics of a typical mass spring dampener problem. This non-

dimensionalization allowed for the formation of ε, and was shown that non-

dimensionalization of the problem allowed the development of a single equation to

represent multiple physical parameter variations.

A similar linear second order ordinary differential equation was solved using

perturbation methods. Due to the location of ε in the differential equation, the equation

resulted in a specific subset known as a boundary layer problem. In order to enforce

both boundary conditions, the perturbation approximation developed an inner and outer

solution. Then, through the use of matching, a single composite solution was

determined. The perturbation approximation was compared to the exact analytical

solution obtained through normal application of differential equation theory.

A regular perturbation approximation was then developed for the unforced

Duffing equation. The regular perturbation approximation resulted in a secular term

being present. In order to develop a approximation without a secular term, the Poincare-

Lindstedt method was used to shift the frequency of the perturbation approximation.

Both of these approximations were compared to a numerical solution which was

obtained through the use of MAPLE for two different values of ε. While both the regular

perturbation approximation and the Poincare-Lindstedt methods tracked the numerical

solution with low error at low values of , the Poincare-Lindstedt method had

significantly lower error as values of increased.

52

Finally, a regular perturbation approximation was then developed for the Van

Der Pol equation. The regular perturbation approximation resulted in a secular term

being present. In order to develop a approximation without a secular term, the Poincare-

Lindstedt method was attempted. The Poincare-Lindstedt was unable to eliminate all the

terms that would result in secular term being present in a perturbation approximation.

The Multiple Scales method was then used to introduce a new variable which is

dependent on ε and . This new variable allowed the successful elimination of secular

terms from appearing in a perturbation approximation. Both the regular perturbation

approximation and the Multiple Scales method approximations were compared to a

numerical solution which was obtained through the use of MAPLE for two different

values of ε. While both the regular perturbation approximation and the Multiple Scales

methods tracked the numerical solution with low error at low values of , the Multiple

Scales method had significantly lower error as values of increased.

53

References

[1] “Introduction to Perturbation Methods”. M.H Holmes; Springer; 1995

[2] “Lecture Notes on Nonlinear Vibrations”; Richard Rand; 2005

[3] “Introduction to Singular Perturbation Methods Nonlinear Oscillations”; A; Aceves,

N.Ercolani, C.Jones, J. Lega & J. Moloney; 1994

Additional Reading:

[4] “Transport Phenomenan”; Second Edition; Bird, Stewart and Lightfoot; John

Wiley& Sons; 2007

[5] “Perturbation Methods”; Ali Nayfeh; John Wiley& Sons; 1973

[6] “Perturbation Theory & Stability Analysis” University of Twente; T. Weinhart, A

Singh, A.R. Thornton; May 17, 2010

[7] “Some Asymptotic Methods for Strongly Nonlinear Equations”; Ji-Huan He; 2006

54

A. Appendices

55

A.1 Unforced Duffing Equation Numeric MAPLE Code

56

57

58

A.2 Van Der Pol Equation Numeric MAPLE Code

59

60

61

A.3 Numerical Value Tables for the Ordinary Differential Equation

Table 3: Perturbation and Exact Solutions to the Ordinary Differential Equation

X Y Analytical Y Composite %Error

0 0 0 -

0.011519 2.424557348 2.415660385 -0.36695

0.023038 2.641622008 2.629258016 -0.46805

0.034907 2.635230466 2.622507364 -0.48281

0.069813 2.546917667 2.534980398 -0.46869

0.10472 2.4591161 2.448021737 -0.45115

0.139626 2.374339022 2.364043877 -0.4336

0.174533 2.292484598 2.282946823 -0.41605

0.20944 2.213452074 2.204631754 -0.39849

0.244346 2.137144166 2.129003234 -0.38093

0.279253 2.063466944 2.055969103 -0.36336

0.314159 1.992329715 1.985440363 -0.34579

0.349066 1.923644915 1.917331067 -0.32822

0.383972 1.857327997 1.851558218 -0.31065

0.418879 1.79329733 1.788041666 -0.29307

0.453786 1.731474094 1.726704009 -0.27549

0.488692 1.671782192 1.667470503 -0.25791

0.523599 1.614148144 1.610268966 -0.24032

0.558505 1.558501008 1.555029691 -0.22273

0.593412 1.504772286 1.501685365 -0.20514

0.628319 1.452895841 1.450170984 -0.18755

0.663225 1.402807816 1.400423771 -0.16995

0.698132 1.354446556 1.352383105 -0.15235

0.733038 1.307752533 1.305990444 -0.13474

0.767945 1.262668268 1.261189255 -0.11713

0.802851 1.219138265 1.217924943 -0.09952

0.837758 1.177108943 1.176144786 -0.08191

0.872665 1.136528565 1.135797871 -0.06429

0.907571 1.09734718 1.096835032 -0.04667

0.942478 1.059516559 1.059208789 -0.02905

0.977384 1.022990133 1.022873291 -0.01142

1.012291 0.987722942 0.987784259 0.006208

1.047198 0.953671574 0.953898935 0.023841

1.082104 0.920794114 0.921176026 0.041476

1.117011 0.889050091 0.889575656 0.059115

1.151917 0.858400431 0.859059317 0.076757

1.186824 0.828807407 0.829589821 0.094402

62

A.4 Numerical Value Tables for the Duffing Equation


Equation (ε=.01)

X Y Perturbation Y Lindstedt Y Numerical Absolute Error (Lindstedt vs Numerical)

Absolute Error (Perturbation vs Numerical)

0 1 1 1 0 0

3.926991 -0.696251833 -0.69661748 -0.696195777 0.000421703 5.60552E-05

7.853982 -0.029452431 -0.029448173 -0.029347606 0.000100567 0.000104825

11.78097 0.73790386 0.737645704 0.737164133 -0.000481572 -0.000739727

15.70796 -1 -0.99826561 -0.998270494 -4.88334E-06 0.001729506

19.63495 0.654599805 0.653172843 0.652830896 -0.000341947 -0.001768909

23.56194 0.088357293 0.088242371 0.087942462 -0.000299909 -0.000414832

27.48894 -0.779555888 -0.776115199 -0.775592601 0.000522598 0.003963287

31.41593 1 0.993068457 0.993084824 1.63671E-05 -0.006915176

35.34292 -0.612947777 -0.607462493 -0.607218292 0.000244201 0.005729485

39.26991 -0.147262156 -0.146730474 -0.146236211 0.000494263 0.001025944

43.1969 0.821207915 0.81189252 0.811346481 -0.000546039 -0.009861434

47.12389 -1 -0.984426568 -0.984461257 -3.46884E-05 0.015538743

51.05088 0.57129575 0.55964499 0.559516144 -0.000128846 -0.011779606

54.97787 0.206167018 0.204709603 0.204028918 -0.000680684 -0.002138099

58.90486 -0.862859943 -0.844853565 -0.84430102 0.000552546 0.018558923

62.83185 1 0.97236992 0.972430178 6.02572E-05 -0.027569822

66.75884 -0.529643722 -0.509886202 -0.509889556 -3.35462E-06 0.019754166

70.68583 -0.26507188 -0.261978638 -0.261122264 0.000856375 0.003949616

74.61283 0.90451197 0.874884 0.874340809 -0.00054319 -0.030171161

78.53982 -1 -0.956940336 -0.957033969 -9.36335E-05 0.042966031

82.46681 0.487991695 0.458358731 0.458510253 0.000151522 -0.029481442

86.3938 0.323976742 0.318338928 0.317320245 -0.001018683 -0.006656497

90.32079 -0.946163998 -0.901879654 -0.901360556 0.000519098 0.044803441

94.24778 1 0.938191336 0.938326825 0.000135489 -0.061673175

98.17477 -0.446339667 -0.405241314 -0.405555365 -0.000314051 0.040784302

102.1018 -0.382881605 -0.37359497 -0.372429655 0.001165315 0.01045195

106.0288 0.987816025 0.925746887 0.925265415 -0.000481472 -0.06255061

109.9557 -1 -0.916187957 -0.916374533 -0.000186576 0.083625467

113.8827 0.40468764 0.350718205 0.351207602 0.000489397 -0.053480038

117.8097 0.441786467 0.427555093 0.42626083 -0.001294264 -0.015525637

121.7367 -1.029468053 -0.946402908 -0.945971377 0.000431531 0.083496676

125.6637 1 0.891006524 0.891254275 0.000247751 -0.108745725

63

129.5907 -0.363035612 -0.294978531 -0.29565392 -0.000675389 0.067381692

133.5177 -0.500691329 -0.480032122 -0.478628474 0.001403648 0.022062855

137.4447 1.07112008 0.963776066 0.963405484 -0.000370582 -0.107714597

141.3717 -1 -0.862734386 -0.863054266 -0.00031988 0.136945734

145.2987 0.321383585 0.238215642 0.239085462 0.000869821 -0.082298122

149.2257 0.559596191 0.530844026 0.529351867 -0.001492159 -0.030244325

153.1526 -1.112772108 -0.977806097 -0.977506394 0.000299703 0.135265714

157.0796 1 0.831469612 0.831873314 0.000403702 -0.168126686

161.0066 -0.279731557 -0.180626435 -0.181696578 -0.001070143 0.098034979

164.9336 -0.618501054 -0.579814548 -0.57825575 0.001558798 0.040245303

168.8606 1.154424136 0.988444334 0.988224316 -0.000220018 -0.16619982

172.7876 -1 -0.797320654 -0.797820567 -0.000499914 0.202179433

176.7146 0.23807953 0.122410675 0.123684165 0.00127349 -0.114395364

180.6416 0.677405916 0.626773822 0.625170868 -0.001602954 -0.052235048

184.5686 -1.196076163 -0.995653875 -0.995521381 0.000132495 0.200554782

188.4956 1 0.760405966 0.761015097 0.000609131 -0.238984903

192.4226 -0.196427502 -0.0637703 -0.065247244 -0.001476945 0.131180258

196.3495 -0.736310778 -0.671558955 -0.669934738 0.001624217 0.06637604

64


Equation (ε=.05)

X Y Perturbation Y Lindstedt Y Numerical Absolute Error (Lindstedt vs Numerical)

Absolute Error (Perturbation vs

Numerical)

0 1 1 1 0 0

3.926991 -0.652832038 -0.653172843 -0.651510889 0.001661954 0.001321149

7.853982 -0.147262156 -0.146730474 -0.144319338 0.002411136 0.002942818

11.78097 0.861092176 0.844853565 0.842120293 -0.002733273 -0.018971883

15.70796 -1 -0.956940336 -0.957372181 -0.000431845 0.042627819

19.63495 0.4445719 0.405241314 0.406783672 0.001542358 -0.037788228

23.56194 0.441786467 0.427555093 0.421193484 -0.00636161 -0.020592983

27.48894 -1.069352314 -0.963776066 -0.961865209 0.001910857 0.107487105

31.41593 1 0.831469612 0.833423339 0.001953726 -0.166576661

35.34292 -0.236311763 -0.122410675 -0.128666376 -0.006255701 0.107645387

39.26991 -0.736310778 -0.671558955 -0.663478817 0.008080138 0.072831961

43.1969 1.277612451 0.999698819 0.999876475 0.000177657 -0.277735976

47.12389 -1 -0.634393284 -0.639385186 -0.004991902 0.360614814

51.05088 0.028051625 -0.170961889 -0.159940417 0.011021472 -0.187992042

54.97787 1.030835089 0.85772861 0.850611969 -0.007116641 -0.180223121

58.90486 -1.485872589 -0.949528181 -0.952639425 -0.003111244 0.533233164

62.83185 1 0.382683432 0.392271396 0.009587963 -0.607728604

66.75884 0.180208513 0.44961133 0.435506458 -0.014104872 0.255297945

70.68583 -1.325359401 -0.970031253 -0.966117774 0.003913479 0.359241626

74.61283 1.694132727 0.817584813 0.824514735 0.006929922 -0.869617992

78.53982 -1 -0.09801714 -0.112975562 -0.014958422 0.887024438

82.46681 -0.388468651 -0.689540545 -0.675287168 0.014253376 -0.286818518

86.3938 1.619883712 0.998795456 0.999497083 0.000701627 -0.620386629

90.32079 -1.902392864 -0.615231591 -0.627095178 -0.011863587 1.275297686

94.24778 1 -0.195090322 -0.175529348 0.019560974 -1.175529348

98.17477 0.596728788 0.870086991 0.858892521 -0.01119447 0.262163732

102.1018 -1.914408023 -0.941544065 -0.947661839 -0.006117774 0.966746184

106.0288 2.110653002 0.359895037 0.377656432 0.017761395 -1.73299657

109.9557 -1 0.471396737 0.449718518 -0.021678219 1.449718518

113.8827 -0.804988926 -0.97570213 -0.970125687 0.005576443 -0.165136761

117.8097 2.208932335 0.803207531 0.815394368 0.012186837 -1.393537966

121.7367 -2.31891314 -0.073564564 -0.097250226 -0.023685663 2.221662914

125.6637 1 -0.707106781 -0.686932352 0.020174429 -1.686932352

129.5907 1.013249064 0.997290457 0.998861592 0.001571135 -0.014387472

65

133.5177 -2.503456646 -0.595699304 -0.61464376 -0.018944456 1.888812886

137.4447 2.527173278 -0.21910124 -0.191082216 0.028019024 -2.718255494

141.3717 -1 0.881921264 0.866959421 -0.014961843 1.866959421

145.2987 -1.221509201 -0.932992799 -0.942440494 -0.009447695 0.279068707

149.2257 2.797980957 0.336889853 0.362942244 0.026052391 -2.435038713

153.1526 -2.735433415 0.492898192 0.463825972 -0.02907222 3.199259387

157.0796 1 -0.98078528 -0.973887501 0.00689778 -1.973887501

161.0066 1.429769339 0.788346428 0.806064396 0.017717969 -0.623704943

164.9336 -3.092505268 -0.049067674 -0.081494299 -0.032426624 3.01101097

168.8606 2.943693553 -0.724247083 -0.698411033 0.02583605 -3.642104586

172.7876 -1 0.995184727 0.997969843 0.002785117 1.997969843

176.7146 -1.638029477 -0.575808191 -0.602033989 -0.026225797 1.035995488

180.6416 3.38702958 -0.24298018 -0.206595202 0.036384977 -3.593624782

184.5686 -3.151953691 0.893224301 0.874810338 -0.018413963 4.026764029

188.4956 1 -0.923879533 -0.936976536 -0.013097004 -1.936976536

192.4226 1.846289615 0.31368174 0.348132316 0.034450576 -1.498157298

196.3495 -3.681553891 0.514102744 0.477825188 -0.036277556 4.159379079

66

A.5 Numerical Value Tables for the Van Der Pol Equation

Table 6: Perturbation and Numerical Values Determined for the Van Der Pol

Equation (ε=.01)

Y Perturbation Y Multiple

Scales Y Numerical

Absolute Error (Multiple Scales vs

Numerical)


Numerical)

1 1 1 0 0

-0.715310079 -0.717544628 -0.715308572 0.002236056 1.50724E-06

-0.0025 8.54548E-15 -0.002534072 -0.002534072 -3.40716E-05

0.740555511 0.738556114 0.740870018 0.002313904 0.000314507

-1.058904862 -1.059416551 -1.059416984 -4.32976E-07 -0.000512122

0.756962107 0.759718006 0.757347418 -0.002370588 0.000385311

0.0025 -2.61588E-14 0.002605122 0.002605122 0.000105122

-0.782207538 -0.780991622 -0.783449645 -0.002458023 -0.001242107

1.117809725 1.119573047 1.119573761 7.14061E-07 0.001764037

-0.798614134 -0.802336314 -0.799817226 0.002519089 -0.001203091

-0.0025 4.87368E-14 -0.002680175 -0.002680175 -0.000180175

0.823859566 0.823709742 0.826326049 0.002616307 0.002466483

-1.176714587 -1.180010766 -1.180011592 -8.26118E-07 -0.003297005

0.840266162 0.845068189 0.842386545 -0.002681644 0.002120383

0.0025 -7.35352E-14 0.002759142 0.002759142 0.000259142

-0.865511593 -0.866366918 -0.869155381 -0.002788463 -0.003643788

1.235619449 1.240236618 1.240237356 7.37134E-07 0.004617907

-0.88191819 -0.887560558 -0.884703033 0.002857525 -0.002784844

-0.0025 9.14918E-14 -0.0028415 -0.0028415 -0.0003415

0.907163621 0.908603521 0.911576962 0.002973441 0.004413341

-1.294524311 -1.299739054 -1.299739495 -4.41485E-07 -0.005215184

0.923570217 0.929450432 0.926405093 -0.003045339 0.002834876

0.0025 -1.10718E-13 0.002926298 0.002926298 0.000426298

-0.948815648 -0.950056575 -0.953225986 -0.003169411 -0.004410338

1.353429174 1.358006599 1.358006535 -6.38997E-08 0.004577362

-0.965222245 -0.970378338 -0.967135468 0.003242871 -0.001913223

-0.0025 1.50835E-13 -0.003012053 -0.003012053 -0.000512053

0.990467676 0.990373647 0.99374757 0.003373923 0.003279894

-1.412334036 -1.41454768 -1.414546908 7.72049E-07 -0.002212873

1.006874272 1.010002383 1.006555088 -0.003447296 -0.000319184

0.0025 -1.73126E-13 0.0030967 0.0030967 0.0005967

-1.032119703 -1.029226772 -1.032810299 -0.003583527 -0.000690595

1.471238898 1.468910053 1.468908387 -1.66624E-06 -0.002330511

67

-1.0485263 -1.048011733 -1.044356655 0.003655078 0.004169645

-0.0025 2.17602E-13 -0.003178281 -0.003178281 -0.000678281

1.073771731 1.066325186 1.070119773 0.003794588 -0.003651958

-1.53014376 -1.520698125 -1.520695408 2.71691E-06 0.009448352

1.090178327 1.084138308 1.080275913 -0.003862395 -0.009902415

0.0025 -2.64316E-13 0.003254214 0.003254214 0.000754214

-1.115423758 -1.101425731 -1.105428917 -0.004003186 0.009994841

1.589048623 1.569586644 1.569582752 -3.89276E-06 -0.019465871

-1.131830355 -1.118165687 -1.114100668 0.004065018 0.017729686

-0.0025 3.13001E-13 -0.003321966 -0.003321966 -0.000821966

1.157075786 1.134340085 1.138545217 0.004205132 -0.018530569

-1.647953485 -1.615329667 -1.615324513 5.15366E-06 0.032628972

1.173482382 1.149934542 1.145675353 -0.004259188 -0.027807029

0.0025 -3.16855E-13 0.003378976 0.003378976 0.000878976

-1.198727814 -1.164938344 -1.169334777 -0.004396433 0.029393037

1.706858347 1.657764312 1.657757855 -6.45626E-06 -0.049100492

-1.21513441 -1.179344367 -1.174903203 0.004441164 0.040231207

-0.0025 3.67492E-13 -0.003423445 -0.003423445 -0.000923445

68

Table 7: Perturbation and Numerical Values Determined for the Van Der Pol

Equation (ε=.05)

X Y Perturbation Y Multiple

Scales Y Numerical

Absolute Error (Multiple Scales vs

Numerical)


Numerical)

0 1 1 1 0 0

3.926991 -0.748123272 -0.759718006 -0.747966033 0.011751972 0.000157239

7.853982 -0.0125 9.54311E-15 -0.013403983 -0.013403983 -0.000903983

11.78097 0.874350428 0.866366918 0.880398134 0.014031216 0.006047706

15.70796 -1.294524311 -1.299739054 -1.299528534 0.00021052 -0.005004223

19.63495 0.95638341 0.970378338 0.954246161 -0.016132177 -0.002137249

23.56194 0.0125 -3.46252E-14 0.015498069 0.015498069 0.002998069

27.48894 -1.082610566 -1.066325186 -1.085391266 -0.01906608 -0.0027807

31.41593 1.589048623 1.569586644 1.56907201 -0.000514635 -0.019976613

35.34292 -1.164643548 -1.149934542 -1.128744127 0.021190415 0.03589942

39.26991 -0.0125 7.11143E-14 -0.017147211 -0.017147211 -0.004647211

43.1969 1.290870703 1.218955321 1.242782828 0.023827506 -0.048087875

47.12389 -1.883572934 -1.764777172 -1.764004837 0.000772334 0.119568096

51.05088 1.372903685 1.273265719 1.24846984 -0.024795879 -0.124433846

54.97787 0.0125 -1.11311E-13 0.017093211 0.017093211 0.004593211

58.90486 -1.499130841 -1.314329515 -1.340725129 -0.026395613 0.158405712

62.83185 2.178097245 1.881740046 1.880847285 -0.00089276 -0.29724996

66.75884 -1.581163823 -1.344415243 -1.318537991 0.025877251 0.262625832

70.68583 -0.0125 1.38149E-13 -0.015052885 -0.015052885 -0.002552885

74.61283 1.707390979 1.365939003 1.39240169 0.026462687 -0.314989289

78.53982 -2.472621556 -1.943389019 -1.942499258 0.000889761 0.530122298

82.46681 1.789423961 1.381071207 1.35620113 -0.024870076 -0.43322283

86.3938 0.0125 -1.63382E-13 0.011594317 0.011594317 -0.000905683

90.32079 -1.915651117 -1.391578068 -1.416362673 -0.024784605 0.499288444

94.24778 2.767145868 1.973582897 1.972767831 -0.000815066 -0.794378037

98.17477 -1.997684099 -1.398809787 -1.376194256 0.022615531 0.621489843

102.1018 -0.0125 2.15621E-13 -0.007354759 -0.007354759 0.005145241

106.0288 2.123911254 1.403757128 1.425900721 0.022143593 -0.698010534

109.9557 -3.061670179 -1.987824615 -1.987115154 0.000709461 1.074555025

113.8827 2.205944236 1.407127574 1.387399208 -0.019728366 -0.818545028

117.8097 0.0125 -2.39125E-13 0.002739235 0.002739235 -0.009760765

121.7367 -2.332171392 -1.409417186 -1.428444397 -0.019027211 0.903726995

125.6637 3.35619449 1.994421102 1.993825342 -0.00059576 -1.362369148

129.5907 -2.414204374 -1.410969541 -1.394422714 0.016546827 1.01978166

69

133.5177 -0.0125 2.90529E-13 0.002044676 0.002044676 0.014544676

137.4447 2.54043153 1.412020647 1.427706849 0.015686202 -1.112724681

141.3717 -3.650718801 -1.997450568 -1.996965526 0.000485042 1.653753275

145.2987 2.622464512 1.412731717 1.399495979 -0.013235738 -1.222968532

149.2257 0.0125 -3.41747E-13 -0.006901328 -0.006901328 -0.019401328

153.1526 -2.748691667 -1.413212462 -1.425451914 -0.012239452 1.323239753

157.0796 3.945243113 1.998836407 1.998454034 -0.000382373 -1.946789079

161.0066 -2.830724649 -1.413537354 -1.403666262 0.009871092 1.427058387

164.9336 -0.0125 3.9285E-13 0.011788486 0.011788486 0.024288486

168.8606 2.956951805 1.413756859 1.422497587 0.008740728 -1.534454218

172.7876 -4.239767424 -1.999469221 -1.999179232 0.000289989 2.240588192

176.7146 3.038984787 1.413905133 1.407417697 -0.006487436 -1.63156709

180.6416 0.0125 -3.87055E-13 -0.016688209 -0.016688209 -0.029188209

184.5686 -3.165211943 -1.414005279 -1.419219222 -0.005213943 1.745992721

188.4956 4.534291735 1.999757945 1.999549071 -0.000208874 -2.534742664

192.4226 -3.247244925 -1.414072913 -1.410973045 0.003099868 1.83627188

196.3495 -0.0125 4.38051E-13 0.021592419 0.021592419 0.034092419

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Application of Perturbation Methods to Approximate the ... · Perturbation methods, also known as asymptotic, allow the simplification of complex mathematical problems. Use of perturbation