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)
i
CONTENTS
Application of Perturbation Methods to Approximate the Solutions to Static and Nonlinear 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
3.3
3.2.1
Perturbation Approximation ................................................................ 14
3.2.2
Analytical Solution .............................................................................. 17
3.2.3
Perturbation Approximation Compared to Analytical Solution .......... 19
Unforced Duffing Equation.............................................................................. 20
3.3.1
Background .......................................................................................... 20
3.3.2
Regular Perturbation Approximation ................................................... 21
3.3.3
Poincare-Lindstedt Method .................................................................. 24
ii
3.4
3.3.4
Numerical Solution .............................................................................. 26
3.3.5
Perturbation Approximation Compared to Analytical Solution .......... 26
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
3.4.6
Perturbation Approximation Compared to Analytical 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
iii
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
Table 5: Perturbation and Numerical Values Determined for the Unforced Duffing
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
iv
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
Duffing Equation (ε=.01) ................................................................................................. 29
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
Duffing Equation (ε=.05) ................................................................................................. 33
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
v
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
vi
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.
vii
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.
viii
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 nondimensionalizing 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.
1
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:
𝜕𝜌 𝜕(𝜌𝑢) 𝜕(𝜌𝑣) 𝜕(𝜌𝑤)
+
+
+
=0
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
[1-1-1]
The Navier Stokes equations for a Newtonian fluid with constant density and viscosity in
Cartesian coordinates is as follows:
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑝
𝜕 2𝑢 𝜕 2𝑢 𝜕 2𝑢
𝜌( +𝑢
+𝑣
+𝑤 )=−
+ µ [ 2 + 2 + 2 ] + 𝜌𝑔𝑥
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑧
[1-1-2]
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑝
𝜕 2𝑣 𝜕 2𝑣 𝜕 2𝑣
𝜌( +𝑢
+𝑣
+𝑤 )=−
+ µ [ 2 + 2 + 2 ] + 𝜌𝑔𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑧
[1-1-3]
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑝
𝜕 2𝑤 𝜕 2𝑤 𝜕 2𝑤
𝜌(
+𝑢
+𝑣
+𝑤
)=−
+ µ [ 2 + 2 + 2 ] + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
[1-1-4]
Assuming a steady, constant density and viscosity, and two dimensional flow, the
continuity and Navier stokes equations reduce to the following:
𝜕𝑢 𝜕𝑣
+
=0
𝜕𝑥 𝜕𝑦
[1-1-5]
𝜌 (𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑝
𝜕 2𝑢 𝜕 2𝑢
+𝑣 )=−
+ µ [ 2 + 2]
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜕𝑥
𝜕𝑦
[ 1-1-6]
𝜌 (𝑢
𝜕𝑣
𝜕𝑣
𝜕𝑝
𝜕 2𝑣 𝜕 2𝑣
+𝑣 )=−
+ µ [ 2 + 2]
𝜕𝑥
𝜕𝑦
𝜕𝑦
𝜕𝑥
𝜕𝑦
[1-1-7]
Equation [1-1-4] is totally eliminated.
2
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 nondimensionalizing 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.
3
The following is a list of the problems to be solved:
Algebraic Equation [1]
𝑥 2 + 𝜀𝑥 − 1 = 0
[1-2-1]
This has relevance because it is a simple example in which to introduce perturbation
theory.
Linear Ordinary Differential Equation [1]
𝜀
𝑑2 𝑥
𝑑𝑥
+2
+ 2𝑥 = 0
2
𝑑𝑡
𝑑𝑡
[1-2-2]
This describes a linear mass spring dampener oscillatory problem.
Unforced Duffing Equation [2]
𝑑2 𝑥
+ 𝑥 + 𝜀𝛼𝑥 3 = 0
𝑑𝑡 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]
𝑑2𝑥
𝑑𝑥
2
+
𝑥
+
𝜀(𝑥
−
1)
(
)=0
𝑑𝑡 2
𝑑𝑡
This represents a non-linear “stick” oscillatory problem.
4
[ 1-2-4]
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 nonlinear 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.
5
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
“𝜀” must be on the order of 0 < 𝜀 << 1. The perturbation method is most easily
understood through a simple algebraic equation. First, equation [1-2-1] is reintroduced as
seen below:
𝑥 2 + 𝜀𝑥 − 1 = 0
2.2.1
[1-2-1]
The Perturbation Method Applied to the Solution of an Algebraic Equation
Leading Order Solution:
Since the primary assumption of the perturbation method is that 𝜀 is very small, the
most obvious way to approximate a solution to [1-2-1] is to set 𝜀 = 0. This reduces to:
𝑥2 = 1
[2-2-1]
Solving for 𝑥, yields the leading order roots:
𝑥 = ±1
[2-2-2]
1st Order Solution:
Assuming δ(x) is some correction factor, the second solution approximation is as seen
below:
𝑥 = ± 1 + δ(x)
[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 𝜀 as 𝜀
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:
6
(1 + δ(x))(1 + δ(x)) + 𝜀(1 + δ(x)) − 1 = 0
[2-2-3]
Expanding [2-2-3] yields:
δ(x)2 + 2δ(x) + 1 + 𝜀 + 𝜀δ(x) − 1 = 0
[2-2-3]
Since both δ(x) and 𝜀 are small numbers, their products are extremely small. Using
an order of magnitude analysis, δ(x)2 and 𝜀δ(x) 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 [22-3] for δ(x) yields:
δ(x) = −
ε
2
Substitution of δ(x) back into [2-2-2] for the positive root yields:
ε
𝑥 = 1+−
2
[2-2-4]
[2-2-4]
2nd Order Solution:
Continuing with the positive root solution, the 3rd solution approximation is
assumed to be:
ε
𝑥 = 1 + − + β(x)
2
Substitution of the positive root seen in [2-2-5] into [1-2-1] yields:
ε
ε
ε
( 1 − + β(x)) ( 1 − + β(x)) + ε ( 1 − + β(x)) − 1 = 0
2
2
2
[2-2-5]
[2-2-6]
Expanding [2-2-6] yields:
ε
ε ε2 ε
ε
ε
1 − + β(x) − + − β(x) + β(x) − β(x) + β(x)2 + ε − + εβ(x) − 1
2
2 4 2
2
2
[2-2-7]
Again since both β(x) and 𝜀 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 β(x) yielding:
β(x) =
ε2
8
[2-2-8]
Substitution of β(x) back into [2-2-5] for the yields the 3rd positive root
approximation:
7
ε ε2
𝑥 = 1+− +
2 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:
−𝜀 ± √𝜀 2 + 4
2
2.2.3
[2-2-10]
Perturbation Approximation Compared to Exact Solution
The 1st term, 2nd term, and 3rd 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:
% 𝐸𝑟𝑟𝑜𝑟 = [
(𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 − 𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒)
] ∗ 100
𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒
[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].
8
Table 1: Perturbation and Exact Solutions to the Algebraic Equation
Small
Parameter
ε
0
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
10
Exact
Positive
Root
1
1
0.999995
0.99995
0.9995
0.995012
0.951249
0.618034
0.09902
Perturbation 1st
Perturbation 2nd
Term
Term
Positive
%Error
Positive
%Error
Root
R1
Root
R1
1
0
1
0
1
5E-05
1 -1.3E-11
1
0.0005 0.999995 -1.3E-09
1
0.005 0.99995 -1.3E-07
1 0.050012
0.9995 -1.3E-05
1 0.50125
0.995 -0.00126
1 5.124922
0.95 -0.13132
1 61.8034
0.5 -19.0983
1 909.902
-4 -4139.61
Perturbation 3rd
Term
Positive
%Error
Root
R1
1
0
1
0
0.999995
0
0.99995 1.11E-14
0.9995 7.89E-13
0.995013 7.85E-09
0.95125
8.2E-05
0.625 1.127124
8.5 8484.167
Plots of the perturbation approximation and exact solutions for 0 ≤ ε ≤ 1 can be
seen in Figure: 1.
1
Exact
Calculated Root
0.99
0.98
Perturbation
1st Term
0.97
Perturbation
2nd Term
0.96
0.95
Perturbation
3rd Term
0.94
0
0.02
0.04
0.06
ε
0.08
0.1
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
9
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 𝑥. 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.
0.01
-0.01 0
0.02
0.04
0.06
0.08
0.1
% Error
-0.03
Perturbation
2nd Term
-0.05
-0.07
-0.09
Perturbation
3rd Term
-0.11
-0.13
-0.15
ε
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.
10
9
Exact
7
5
Calculated Root
Perturbation
1st Term
3
1
Perturbation
2nd Term
-1 0
2
4
6
10
Perturbation
3rd Term
-3
-5
8
ε
Figure 3: Comparative Solutions Plots for the Algebraic Equation as ε >1
11
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.
𝑚
𝑑2𝑥
𝑑𝑥
+𝑐
+ 𝑘𝑥 = 0
2
𝑑𝑡
𝑑𝑡
[3-1-1]
Here 𝑚 denotes the mass of the block, 𝑐 is the viscous friction coefficient of the
dampener, and 𝑘 is the spring coefficient. Since this equation will become nondimensionalized, the starting units can be either all SI or all English.
Assuming that:
𝑥 = 𝑥̂𝐿
[3-1-2]
𝑡 = 𝑡̂𝜏
[3-1-3]
𝑡
𝜏
[3-1-4]
And
Therefore:
𝑡̂ =
Where 𝑥̂ and 𝑡̂ are non-dimensionalized values and 𝐿 and 𝜏 are dimensionalized
variables. It follows that utilizing the chain rule the first derivative of a function with
respect to t is:
𝑑
𝑑𝑡̂ 𝑑
1𝑑
=
=
𝑑𝑡 𝑑𝑡 𝑑𝑡̂ 𝜏 𝑑𝑡̂
[3-1-5]
And the second derivative of some function with respect to t is:
𝑑2
𝑑 𝑑
1 𝑑2
=
=
𝑑𝑡 2 𝑑𝑡 𝑑𝑡 𝜏 2 𝑑𝑡̂ 2
[3-1-6]
Substituting [3-1-6] and [3-1-5] into equation [3-1-1] yields:
𝑚
1 𝑑2
1𝑑
𝑥̂𝐿 + 𝑐
𝑥̂𝐿 + 𝑘𝑥̂𝐿 = 0
2
2
𝜏 𝑑𝑡̂
𝜏 𝑑𝑡̂
Dividing [3-1-7] through by k and L yields:
12
[3-1-7]
𝑚 𝑑2
𝑐 𝑑
𝑥
̂
+
𝑥̂ + 𝑥̂ = 0
𝑘𝜏 2 𝑑𝑡̂ 2
𝑘𝜏 𝑑𝑡̂
Since the goal is to remove the dimensions for all the coefficients, let:
𝑐
𝜏=
𝑘
[3-1-8]
[3-1-9]
And substituting [3-1-9] into [3-1-8] simplifies to:
𝑚𝑘 𝑑 2
𝑑
𝑥̂ + 𝑥̂ + 𝑥̂ = 0
2
2
𝑐 𝑑𝑡̂
𝑑𝑡̂
[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:
ε=
𝑚𝑘
𝑐2
[3-1-11]
And substituting [3-1-11] into [3-1-10] yields:
ε
𝑑2
𝑑
𝑥̂ + 𝑥̂ + 𝑥̂ = 0
2
𝑑𝑡̂
𝑑𝑡̂
[3-1-12]
Note from inspection of equation [3-1-11] that there is combination of
parameters that form ε. 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. Re13
introducing the linear ordinary differential equation [1-2-2] as seen below with “𝑦” as
the dependent variable and “𝑥” as the independent variable:
𝜀
𝑑2 𝑦
𝑑𝑦
+2
+ 2𝑦 = 0
2
𝑑𝑥
𝑑𝑥
[1-2-2]
The initial conditions used to solve this problem as follows:
𝑦(0) = 0
[3-2-1]
𝑦(1) = 1
[3-2-2]
And
3.2.1
Perturbation Approximation
Determining the “Outer Solution”:
Setting 𝜀 = 0 reduces equation [1-2-2] to:
2
𝑑𝑦
+ 2𝑦 = 0
𝑑𝑥
[3-2-3]
Guessing the solution:
𝑦 = 𝑒 𝜆𝑥
[3-2-4]
𝜆+1=0
[3-2-5]
Substituting [3-2-4] into [3-2-3] yields
Since 𝜆 = -1, the general form solution of the differential equation is:
𝑦 = 𝑐1 𝑒 𝜆𝑥 + 𝑐2 𝑒 𝜆𝑥
[3-2-6]
Since there is only one root, equation [3-2-6] simplifies to:
𝑦 = 𝑐1 𝑒 −𝑥
[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 𝑐1 in equation [3-2-7] results in
𝑐1=0, which is firstly a trivial solution, but also would violate the initial problem as
shown in Figure: (4).
14
Y(X)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Initial
Conditions
0
0.2
0.4
0.6
0.8
1
X
Figure 4: Boundary Condition Visualization for Linear Ordinary Differential
Equation
Figure: 4 shows that 𝑦 is a positive value,
𝑑𝑦
𝑑𝑥
is a positive value, and
𝑑2 𝑦
𝑑𝑥 2
is also a
positive value (since the function is concave up). If this was true, then:
𝜀
𝑑2 𝑦
𝑑𝑦
+2
+ 2𝑦 ≠ 0
2
𝑑𝑥
𝑑𝑥
[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 𝑐3 in equation [3-2-7]
results in:
𝑐1 = 𝑒 1
[3-2-9]
Substitution of [3-2-9] into [3-2-7] yields the following “outer solution”:
𝑦𝑜𝑢𝑡𝑒𝑟 = 𝑒𝑒 −𝑥
[3-2-10]
Determining the “Inner Solution”:
To determine the inner solution, magnification at 𝑥 = 0 is required. Letting:
𝑥
𝜂= 𝛼
𝜀
15
[3-2-11]
Utilizing the chain rule on equation [3-2-11] follows:
𝑑
𝑑𝜂 𝑑
1 𝑑
=
= 𝛼
𝑑𝑥 𝑑𝑥 𝑑𝜂 𝜀 𝑑𝜂
[3-2-12]
𝑑2
𝑑 𝑑
1 𝑑2
=
(
)
=
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝜀 2𝛼 𝑑𝜂2
[3-2-13]
And:
Substitution of equations [3-2-12] and [3-2-13] into equation [1-2-2] yields:
𝜀 𝑑 2 𝑦 2 𝑑𝑦
+
+ 2𝑦 = 0
𝜀 2𝛼 𝑑𝜂2 𝜀 𝛼 𝑑𝜂
[3-2-14]
Assuming that the first two terms balance and solving for 𝛼 follows:
𝜀 1−2𝛼 = −𝜀 𝛼
[3-2-15]
𝛼=1
[3-2-16]
Simplifying to solve for 𝛼 yields:
The two terms that were assumed to balance were:
𝑑2𝑦
𝑑𝑦
+2
=0
2
𝑑𝜂
𝑑𝜂
[3-2-17]
It is then solved by guessing the general solution:
𝑦 = 𝑒 𝜔𝜂
[3-2-18]
Substituting equation [3-2-18] into [3-2-17] and simplifying yields:
𝜔(𝜔 + 2) = 0
[3-2-19]
Since 𝜔 =-2 and 0, the general solution of the equation takes the form:
𝑦 = 𝐴 + 𝐵𝑒 𝜔𝜂
[3-2-20]
Using the remaining boundary condition in equation [3-2-1] and solving for 𝐴 yields:
𝐴 = −𝐵
[3-2-21]
Substitution of equation [3-2-21] into equation [3-2-20] yields the following “inner
solution”:
𝑦𝑖𝑛𝑛𝑒𝑟 = −𝐵 + 𝐵𝑒 −2𝜂
16
[3-2-22]
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
(𝑦𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 ). To match the solution, the limit as the outer solution approaches 𝑥 → 0 is
set equal o the limit as the inner solution approaches 𝜂 → ∞:
lim( 𝑒𝑒 −𝑥 ) = lim ( −𝐵 + 𝐵𝑒 −2𝜂 )
[3-2-23]
−𝑒 1 = 𝐵
[3-2-24]
𝑥→0
𝜂→∞
This reduces to:
Equation [3-2-24] is not only used to determine the value of 𝐵, but it also determines the
common solution of the limits of the inner and outer solution ( 𝑦𝐶𝑜𝑚𝑚𝑜𝑛 ).
𝑦𝐶𝑜𝑚𝑚𝑜𝑛 = 𝑒 1
[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:
𝑦𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = 𝑦𝑖𝑛𝑛𝑒𝑟 + 𝑦𝑜𝑢𝑡𝑒𝑟 − 𝑦𝐶𝑜𝑚𝑚𝑜𝑛
[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:
𝑦𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = −𝑒𝑒
3.2.2
−2𝑥
𝜀1
+ 𝑒𝑒 −𝑥
[3-2-27]
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:
𝑦 = 𝑒 𝑅𝑥
[3-2-28]
And substituting equation [3-2-28] into equation [1-2-2] and simplifying yields:
𝜀𝑅 2 + 2𝑅 + 2 = 0
[3-2-29]
Utilizing the quadratic equation roots 𝑅1 and 𝑅2 can be solved for:
𝑅1 =
−2 + √4 − 8𝜀
2𝜀
And
17
[3-2-30]
𝑅2 =
−2 − √4 − 8𝜀
2𝜀
[3-2-31]
Since both roots are real numbers, the general solution takes the form:
𝑦𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 = 𝑐1 𝑒 𝑅1 𝑥 + 𝑐2 𝑒 𝑅2𝑥
[3-2-32]
Substitution of equations [3-2-30] and [3-2-31] into equation [3-2-32] yields the
following:
𝑦𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 = 𝑐1 𝑒
−2+√4−8𝜀
𝑥
2𝜀
+ 𝑐2 𝑒
−2−√4−8𝜀
𝑥
2𝜀
[3-2-33]
Enforcement of the initial condition seen in equation [3-2-1] to equation [3-2-33] yields:
𝑐1 = −𝑐2
[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:
−𝑐2 𝑒
−2+√4−8𝜀
2𝜀
+ 𝑐2 𝑒
−2−√4−8𝜀
2𝜀
=1
[3-2-35]
Solving equation [3-2-35] for 𝑐2 and than using equation [3-2-34] to solve for 𝑐1 yields:
𝑐2 =
1
−2+√4−8𝜀
−𝑒 2𝜀
𝑐1 = − [
+𝑒
1
−2+√4−8𝜀
−𝑒 2𝜀
+
[3-2-36]
−2−√4−8𝜀
2𝜀
−2−√4−8𝜀
𝑒 2𝜀
]
[3-2-37]
Substitution of equations [3-2-36] and [3-2-37] into equation [3-2-33] yields the final
analytical solution:
1
𝑦𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 = − [
−𝑒
−2+√4−8𝜀
2𝜀
+𝑒
−2−√4−8𝜀
2𝜀
]𝑒
1
+
−𝑒
−2+√4−8𝜀
2𝜀
+𝑒
−2−√4−8𝜀
2𝜀
18
−2+√4−8𝜀
𝑥
2𝜀
[3-2-38]
−2−√4−8𝜀
𝑒 2𝜀 𝑥
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
ε
0.01
Analytical Roots
Root 1
-1.00505
Analytical Constants
Root 2
-198.99494
C1
2.73204
C2
-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 𝑥 increases. In Table 3, this boundary
layer can be seen for 𝑥 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.
3
2.5
Analytical
Roots
Y
2
1.5
Perturbation
1
0.5
0
0
0.2
0.4
0.6
X
0.8
1
1.2
Figure 5: Comparative Solutions Plots for the Ordinary Differential Equation
19
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.
0.2
0.1
% Error
0
-0.1 0
0.2
0.4
0.6
0.8
1
1.2
-0.2
-0.3
%Error
-0.4
-0.5
-0.6
X
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 “𝑦” as the dependent
variable and “𝑥” as the independent variable:
𝑑2𝑦
+ 𝑦 + 𝜀𝛼𝑦 3 = 0
𝑑𝑥 2
[1-2-3]
The initial conditions used to solve this problem are as follows:
𝑦(0) = 1
And
20
[3-3-1]
𝑑𝑦
(0) = 0
𝑑𝑥
3.3.2
[3-3-2]
Regular Perturbation Approximation
Leading Order Solution:
Setting 𝜀 = 0 reduces equation [1-2-3] to:
𝑑2𝑦
+𝑦=0
𝑑𝑥 2
[3-3-3]
𝑦 = 𝑒 𝜆𝑥
[3-3-4]
𝜆2 + 1 = 0
[3-3-5]
𝜆 = ±√−1
[3-3-6]
Guessing the solution:
Substituting [3-3-4] into [3-3-3] yields
Solving for:
The general form solution of the differential equation is:
𝑦 = 𝑐1 cos(𝑥) + 𝑐2 sin(𝑥)
[3-3-7]
The derivative of equation [3-3-7] with respect to 𝑥 is then:
𝑑𝑦
= −𝑐1 sin(𝑥) + 𝑐2 cos(𝑥)
𝑑𝑥
[3-3-8]
Using the boundary condition in equation [3-3-1] to solve for 𝑐1 in equation [3-3-7], and
boundary condition in equation [3-3-2] to solve for 𝑐2 in equation [3-3-8] results in:
𝑐1 = 1
[3-3-9]
𝑐2 = 0
[3-3-10]
And
Substitution of equations [3-3-9] and [3-3-10] into equation [3-3-7] yields:
𝑦 = cos(𝑥)
1st Order Solution:
21
[3-3-11]
Assuming δ(x) is some correction factor, the second solution approximation is as seen
below:
𝑦 = cos(𝑥) + 𝛿(𝑥)
[3-3-12]
The derivative of equation [3-3-12] with respect to 𝑥 is then:
𝑑𝑦
𝑑𝛿(𝑥)
= −sin(𝑥) +
𝑑𝑥
𝑑𝑥
[3-3-13]
The second derivative of equation [3-2-12] with respect to 𝑥 is then:
𝑑2𝑦
𝑑 2 𝛿(𝑥)
=
−cos(𝑥)
+
𝑑𝑥 2
𝑑𝑥 2
[3-3-14]
Using the boundary condition in equation [3-3-1] to solve for 𝛿(0) in equation [3-3-12],
and boundary condition in equation [3-3-2] to solve for
𝑑𝛿
𝑑𝑥
(0) in equation [3-3-13]
results in:
𝛿(0) = 0
[3-3-15]
𝑑𝛿
(0) = 0
𝑑𝑥
[3-3-16]
And
Substituting equation [3-3-14] and [3-3-12] into equation [1-2-3] and simplifying yields:
𝑑 2 𝛿(𝑥)
+ 𝛿(𝑥) + 𝜀𝛼(cos(𝑥) + 𝛿(𝑥))3 = 0
𝑑𝑥 2
[3-3-17]
Expanding (cos(𝑥) + 𝛿(𝑥))3 yields:
(cos(𝑥) + 𝛿(𝑥))3 = cos3 (𝑥) + 2δ(x)cos2 (𝑥) +
𝛿(𝑥)2 cos(𝑥) + δ(x)cos2 (𝑥) + 2𝛿(𝑥)2 cos(𝑥) + 𝛿(𝑥)3
[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:
𝑑2 𝛿(𝑥)
+ 𝛿(𝑥) = −𝜀𝛼 cos 3 (𝑥)
𝑑𝑥 2
[3-3-19]
𝛿(𝑥) = 𝜀𝛼𝛷(𝑥)
[3-3-20]
Letting:
And utilizing a combination of all the following common trigonometry identities:
22
cos2 (𝑥) + sin2 (𝑥) = 1
[3-3-21]
1 − cos(2𝑥)
2
[3-3-22]
sin2(𝑥) =
cos(𝑎 + 𝑏) = cos(𝑎) cos(𝑏) − sin(𝑎) sin(𝑏)
[3-3-23]
cos(𝑎 − 𝑏) = cos(𝑎) cos(𝑏) + sin(𝑎) sin(𝑏)
[3-3-24]
1
cos(𝑎) cos(𝑏) = [cos(𝑎 − 𝑏) + cos(𝑎 + 𝑏)]
2
[3-3-25]
cos3 (𝑥) can be expanded to:
3
1
cos3 (𝑥) = cos(𝑥) + cos(3𝑥)
4
4
Substitution of equation [3-3-26] into [3-3-19] and [3-3-20] yields:
𝑑2𝛷
3
1
+
𝛷
=
−
cos(𝑥)
+
cos(3𝑥)
𝑑𝑥 2
4
4
[3-3-26]
[3-3-27]
Solving for 𝛷 as a traditional ordinary differential equation through superposition:
𝛷 = 𝛷ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 + 𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
[3-3-28]
Noting that general solution takes the same form as equation [3-2-7], yields:
𝛷ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 = 𝑐3 cos(𝑥) + 𝑐4 sin(𝑥)
[3-3-29]
Guessing the particular solution:
𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 = 𝐴𝑥 cos(𝑥) + 𝐵𝑥 sin(𝑥) + 𝐶 cos(3𝑥)
[3-3-30]
The second derivatives of equation [3-3-28] with respect to 𝑥 are then:
𝑑2 𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
= −2𝐴 sin(𝑥) − 𝐴𝑥 cos(𝑥) + 2𝐵 cos(𝑥) − 𝐵𝑥 sin(𝑥)
𝑑𝑥 2
[3-3-31]
− 9𝐶 cos(3𝑥)
Substitution of equations [3-3-30] and [3-3-31] into equation [3-3-27] and solving for
coefficients 𝐴, 𝐵, and 𝐶 yeild:
23
𝐴=0
3
8
[3-3-33]
1
32
[3-3-34]
𝐵=−
𝐶=
[3-3-32]
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
1
𝛿(𝑥) = 𝜀𝛼 [𝑐3 cos(𝑥) + 𝑐4 sin(𝑥) + − 𝑥 sin(𝑥) + cos(3𝑥)]
8
32
[3-3-35]
Taking the derivative of equation [3-3-35] with respect to 𝑥 yields:
𝑑𝛿(𝑥)
3
3
= 𝜀𝛼 [−𝑐3 sin(𝑥) + 𝑐4 cos(𝑥) + − sin(𝑥) − 𝑥 cos(𝑥)
𝑑𝑥
8
8
3
− sin(3𝑥)]
32
[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 𝑐3 and 𝑐4 yields:
𝑐3 = −
1
32
[3-3-37]
And
𝑐4 = 0
[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:
𝑦 = cos(𝑥) + 𝜀𝛼 [−
3.3.3
1
3
1
cos(𝑥) − 𝑥 sin(𝑥) +
cos(3𝑥)]
32
8
32
[3-3-39]
Poincare-Lindstedt Method
Upon a more detailed inspection of the 1st order perturbation approximation
developed in equation [3-3-39], not that as 𝑥 increases to a large number, the magnitude
of the 1st order correction factor increases. As 𝑥 progresses the
24
3
8
𝑥 sin(𝑥) term (secular
term), even though multiplied by small number 𝜀, will eventually dominate the
approximation. This will limit the range of 𝑥 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 𝑥 increases, the Poincare-Lindstedt
method is used. Utilizing this method the frequency 𝑥 will be shifted which therefore
will reduce the error from the secular term. As 𝑥 continues to increase, more frequency
corrections need to be determined to further reduce error. Assuming Ϛ is the correction
to 𝑥, new variable 𝑧 is:
𝑧 = (1 + Ϛ)𝑥
[3-3-40]
Using the chain rule, the first and second derivatives of [3-3-40] with respect to 𝑥 are:
𝑑
𝑑𝑧 𝑑
𝑑
=
= (1 + Ϛ)
𝑑𝑥 𝑑𝑥 𝑑𝑧
𝑑𝑧
[3-3-41]
𝑑2
𝑑 𝑑
𝑑2
2
(1
=
=
+
Ϛ)
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝑧 2
[3-3-42]
And
Allowing 𝜀 → 0 reduces equation [1-2-3] to equation [3-3-3]. Substituting [3-3-42] into
[3-3-1] yields:
(Ϛ2 + 2Ϛ + 1)
𝑑2𝑦
+𝑦 =0
𝑑𝑧 2
[3-3-43]
Knowing that the shift Ϛ and Ϛ2 are very small, utilizing an order of magnitude analysis
equation [3-3-43] simplifies to:
𝑑2𝑦
+𝑦=0
𝑑𝑧 2
[3-3-44]
Following the same mathematical analysis as in section 3.3.2, the first PoincareLindstedt leading order solution is determined to be:
𝑦 = cos(𝑧)
[3-3-45]
Assuming θ(𝑧) is some correction factor, the second solution approximation is as seen
below:
𝑦 = cos(𝑧) + θ
The second derivative of equation [3-3-46] with respect to 𝑧 is:
25
[3-3-46]
𝑑2𝑦
𝑑2θ
= − cos(𝑧) + 2
𝑑𝑧 2
𝑑𝑧
[3-3-47]
Substitution of equations [3-3-42], [3-3-46] and [3-3-47] into equation [1-2-3] yields:
(Ϛ2
𝑑2 θ
+ 2Ϛ + 1) [− cos(𝑧) + 2 ] + cos(𝑧) + θ + 𝜀𝛼(cos(𝑧) + θ)3 = 0
𝑑𝑧
[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:
𝑑2 θ
− 2Ϛ cos(𝑧) + θ + 𝜀𝛼 cos3 (𝑧) = 0
𝑑𝑧 2
[3-3-49]
Expansion of cos 3 (𝑧) as performed in section 3.3.2 and rearrangement yields:
𝑑2θ
3
1
+ θ = 2Ϛ cos(𝑧) − 𝜀𝛼 [ cos(𝑧) + cos(3𝑧)]
2
𝑑𝑧
4
4
[3-3-50]
Solving for Ϛ in order to prevent the formation of the secular term yields:
Ϛ=
3
𝜀𝛼
8
[3-3-51]
Combining equations [3-3-40], [3-3-45] and [3-3-51] result in the Poincare-Lindstedt
approximation:
3
𝑦 = cos ([1 + 𝜀𝛼] 𝑥)
8
3.3.4
[3-3-52]
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.
3.3.5
Perturbation Approximation Compared to Analytical Solution
For oscillator solutions absolute error is used for comparison in lieu of percent
error. The absolute error is determined by the following relation:
𝐴𝑏𝑠𝑜𝑙𝑢𝑒 𝐸𝑟𝑟𝑜𝑟 = 𝑁𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 − 𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑡ℎ𝑜𝑑 𝑉𝑎𝑙𝑢𝑒
Case 1: ε = .01
26
[3-3-53]
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.
1.3
0.8
0.3
Y
Y Numerical
-0.2 0
50
100
150
200
Y Perturbation
-0.7
-1.2
X
Figure 7: Regular Perturbation versus Numeric Solution for Unforced Duffing
Equation (ε=.01)
It is important to note that as 𝑥 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.
27
0.25
Error
0.15
Absolute Error
(Perturbation
Numerical)
0.05
-0.05 0
50
100
150
200
-0.15
-0.25
X
Figure 8: Regular Perturbation versus Numeric Solution Percent Error Plot for
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.
1.3
0.8
Y Numerical
Y
0.3
-0.2 0
50
100
150
200
Y Lindstedt
-0.7
-1.2
X
Figure 9: Poincare-Lindstedt versus Numeric Solution for Unforced Duffing
Equation (ε=.01)
28
It is important to note that as 𝑥 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.
0.002
0.0015
0.001
Error
0.0005
0
-0.0005 0
50
100
-0.001
150
200
Absolute
Error
(Lindstedt
to
Numerical)
-0.0015
-0.002
X
Figure 10: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for
Unforced Duffing Equation (ε=.01)
It is important to note the Poincare-Lindstedt method percent error also increases
with 𝑥 as the with the regular perturbation approximation however the magnitude of the
percent error is as much as two orders of magnitude smaller. As 𝑥 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.
29
The regular perturbation approximation seen in Table 5 is plotted together with
the numerical solution that was obtained with MAPLE in Figure: 11 below.
4
3
2
1
Y Numerical
Y
0
-1
0
50
100
150
200
-2
Y
Perturbation
-3
-4
-5
X
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.
30
5
4
3
2
Error
1
0
-1 0
50
100
-2
-3
150
200
Absolute
Error
(Perturbation
Numerical)
-4
-5
X
Figure 12: Regular Perturbation versus Numeric Solution Percent Error Plot for
Unforced Duffing Equation (ε=.05)
The rapid error increase in the regular perturbation approximation is a result of
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 𝑥. The range of 𝑥 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.
31
1.5
1
Y
Numerical
Y
0.5
0
0
50
100
150
-0.5
200
Y
Lindstedt
-1
-1.5
X
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 PoincareLindstedt 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.
32
0.05
0.04
0.03
Error
0.02
0.01
0
-0.01
0
50
100
150
200
Absolute
Error
(Lindstedt
to
Numerical)
-0.02
-0.03
-0.04
X
Figure 14: Poincare-Lindstedt versus Numeric Solution Percent Error Plot for
Unforced Duffing Equation (ε=.05)
The Poincare-Lindstedt method is able to provide a solution approximation that has
error two orders of magnitude small than the regular perturbation method.
33
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 “𝑦” as the
dependent variable and “𝑥” as the independent variable:
𝑑2 𝑦
𝑑𝑦
+ 𝑦 + 𝜀(𝑦 2 − 1) ( ) = 0
2
𝑑𝑥
𝑑𝑥
[1-2-4]
The initial conditions used to solve this problem are as follows:
𝑦(0) = 1
[3-4-1]
𝑑𝑦
(0) = 0
𝑑𝑥
[3-4-2]
And
3.4.2
Regular Perturbation Approximation
Leading Order Solution:
Setting 𝜀 = 0 reduces equation [1-2-4] to:
𝑑2𝑦
+𝑦=0
𝑑𝑥 2
[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 [33-1] 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:
𝑦 = cos(𝑥)
[3-4-4]
1st Order Solution:
Assuming δ(x) is some correction factor, the second solution approximation is as seen
below:
34
𝑦 = cos(𝑥) + 𝛿(𝑥)
[3-4-5]
The first and second derivatives of equation [3-4-5] with respect to 𝑥 are the
same as equations [3-3-13] and [3-3-14] in the Duffing equation section. The values of
𝛿(0) and
𝑑𝛿
𝑑𝑥
(0) 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:
𝑑 2 𝛿(𝑥)
𝑑𝛿(𝑥)
+
𝛿(𝑥)
−
𝜀
(sin(𝑥)
+
) (1 − (cos(𝑥) + 𝛿(𝑥))2 ) = 0
𝑑𝑥 2
𝑑𝑥
[3-4-6]
Expanding equation [3-4-6] and eliminating the HOTs terms results in the following the
remaining terms rewritten as
𝑑 2 𝛿(𝑥)
+ 𝛿(𝑥) = 𝜀[− sin(𝑥) + cos2 (𝑥) sin(𝑥)]
𝑑𝑥 2
[3-4-7]
𝛿(𝑥) = 𝜀𝛷(𝑥)
[3-4-8]
Letting:
Substitution of equations [3-4-8] and [3-3-21] into equation [3-4-7] yields:
𝑑2 𝛷(𝑥)
+ 𝛷(𝑥) = 𝜀 sin3 (𝑥)
𝑑𝑥 2
[3-4-9]
Performing the same type of trigonometric expansion as performed with equation [3-319] yields:
𝑑2 𝛷(𝑥)
3
1
+ 𝛷(𝑥) = − sin(𝑥) + sin(3𝑥)
2
𝑑𝑥
4
4
[3-4-10]
Solving for 𝛷 as a traditional ordinary differential equation through superposition:
𝛷 = 𝛷ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 + 𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
[3-4-11]
Noting that general solution takes the same form as equation [3-4-3], yields:
𝛷ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 = 𝑐3 cos(𝑥) + 𝑐4 sin(𝑥)
[3-4-12]
Guessing the particular solution:
𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 = 𝐴𝑥 sin(𝑥) + 𝐵𝑥 cos(𝑥) + 𝐶 sin(3𝑥)
The second derivatives of equation [3-4-13] with respect to 𝑥 are then:
35
[3-4-13]
𝑑2 𝛷𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
= 2𝐴 cos(𝑥) − 𝐴𝑥 sin(𝑥) − 2𝐵 sin(𝑥) − 𝐵𝑥 cos(𝑥)
𝑑𝑥 2
[3-4-14]
− 9𝐶 sin(3𝑥)
Substitution of equations [3-4-13] and [3-4-14] into equation [3-4-10] and solving for
coefficients 𝐴, 𝐵, and 𝐶 yield:
𝐴=0
3
8
[3-4-16]
1
32
[3-4-17]
𝐵=−
𝐶=−
[3-4-15]
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
1
𝛿(𝑥) = 𝜀𝛼 [𝑐3 cos(𝑥) + 𝑐4 sin(𝑥) + 𝑥 cos(𝑥) − sin(3𝑥)]
8
32
[3-4-18]
Taking the derivative of equation [3-4-18] with respect to 𝑥 yields:
𝑑𝛿(𝑥)
3
3
3
= 𝜀𝛼 [−𝑐3 sin(𝑥) + 𝑐4 cos(𝑥) + cos(𝑥) − 𝑥 sin(𝑥) − cos(3𝑥)]
𝑑𝑥
8
8
32
[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 𝑐3 and 𝑐4 yields:
𝑐3 = 0
[3-4-20]
And
𝑐4 = −
9
32
[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:
𝑦 = cos(𝑥) + 𝜀𝛼 [−
9
3
1
sin(𝑥) + 𝑥 cos(𝑥) −
sin(3𝑥)]
32
8
32
36
[3-4-22]
3.4.3
Poincare-Lindstedt Method
As seen with the Duffing equation, the development of the regular perturbation
3
approximation for the Van Der Pol equation results the secular term, (8 𝑥 cos(𝑥)),
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:
𝑦 = cos(𝑧) + θ(𝑧)
[3-3-46]
The first derivative of equation [3-3-46] with respect to 𝑧 is:
𝑦 = −sin(𝑧) +
dθ(z)
dz
[3-4-23]
The second derivative of equation [3-3-46] with respect to 𝑧 is identical as seen in
equation [3-3-47]:
𝑑2𝑦
𝑑 2 θ(z)
=
−
cos(𝑧)
+
𝑑𝑧 2
𝑑𝑧 2
[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:
(Ϛ2 + 2Ϛ + 1) [− cos(𝑧) +
𝑑 2 θ(z)
]
𝑑𝑧 2
− 𝜀(1 − (cos(𝑧) + 𝜃 (𝑧))2 )(1 + Ϛ) (−sin(𝑧) +
dθ(z)
)
dz
[3-4-24]
+ cos(𝑧) + θ(z) = 0
Expanding and eliminating the HOTs in equation [3-4-24] in the same manner as
performed in section 3.3.2 and simplification yields:
𝑑 2 θ(z)
− 2Ϛ cos(𝑧) + θ(z) − 𝜀 sin3 (𝑧) = 0
𝑑𝑧 2
Expansion of sin3 (𝑧) as performed in section 3.3.2 and rearrangement yields:
𝑑 2 θ(z)
3
1
+
θ(z)
=
2Ϛ
cos(𝑧)
+
𝜀
[
sin(𝑧)
−
sin(3𝑧)]
𝑑𝑧 2
4
4
37
[3-4-25]
[3-4-26]
Unlike the Duffing equation from section 3-3-3, there is no value for Ϛ 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 𝑥:
𝛹 = 𝜀𝑥
[3-4-27]
Therefore, when 𝑥 becomes large in relative magnitude, the magnitude of 𝛹 becomes
normal sized.
Leading Order Solution:
The first derivative of function 𝑦 with respect to 𝑥 is:
𝑑𝑦
𝜕𝑦 𝑑𝛹 𝜕𝑦
(𝑥, 𝛹) =
+
𝑑𝑥
𝜕𝑥 𝑑𝑥 𝜕𝛹
[3-4-28]
Substitution of equation [3-4-27] into [3-4-28] yields:
𝑑𝑦
𝜕𝑦
𝜕𝑦
(𝑥, 𝛹) =
+𝜀
𝑑𝑥
𝜕𝑥
𝜕𝛹
[3-4-29]
The second derivative of function 𝑦 with respect to 𝑥 with substitution of equation [3-427] is:
𝑑2𝑦 𝜕 2𝑦
𝜕 2𝑦
𝜕 2𝑦
2
=
+
2𝜀
+
𝜀
𝑑𝑥 2 𝜕𝑥 2
𝜕𝑥𝜕𝛹
𝜕𝛹 2
[3-4-30]
Substitution of equations [3-4-29] and [3-4-30] into equation [1-2-4] yields:
𝜕 2𝑦
𝜕 2𝑦
𝜕 2𝑦
𝜕𝑦
𝜕𝑦
2
+
2𝜀
+
𝜀
+ 𝑦 − 𝜀(𝑦 2 − 1) ( + 𝜀
)=0
2
2
𝜕𝑥
𝜕𝑥𝜕𝛹
𝜕𝛹
𝜕𝑥
𝜕𝛹
[3-4-31]
Simplification of equation [3-4-31] through the elimination of 2nd and higher order terms
in ε:
38
𝜕 2𝑦
𝜕 2𝑦
𝜕𝑦
2
+
2𝜀
+
𝑦
−
𝜀(𝑦
−
1)
(
)=0
𝜕𝑥 2
𝜕𝑥𝜕𝛹
𝜕𝑥
[3-4-32]
Setting 𝜀 = 0 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 𝛹 due to the partial derivatives. Therefore the adjusted leading
order solution becomes:
𝑦 = 𝐴(𝛹) cos(𝑥) + 𝐵(𝛹) sin(𝑥)
[3-4-33]
Rewriting equation [3-4-34] yields:
𝑦 = 𝐶(𝛹) cos(𝑥 + 𝐷(𝛹))
[3-4-34]
The first derivative of equation [3-4-34] is:
𝑦 = −𝐶(𝛹) sin(𝑥 + 𝐷(𝛹))
[3-4-35]
Using the boundary condition seen in equation [3-4-1] in conjunction with equation [34-34] and the boundary condition seen in equation [3-4-2] with conjunction with
equation [3-4-35] yields:
𝐶(0) = 1
[3-4-36]
𝐷(0) = 0
[3-4-37]
And
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:
Assuming δ(x) is some correction factor, the second solution approximation is as seen
below:
𝑦 = 𝐶(𝛹) cos(𝑥 + 𝐷(𝛹)) + 𝛿 (𝛹, 𝑥)
[3-4-38]
The first derivative of equation [3-4-38] with respect to 𝑥 is:
𝜕𝑦
𝜕𝛿(𝛹, 𝑥)
= −𝐶(𝛹) sin(𝑥 + 𝐷(𝛹)) +
𝜕𝑥
𝜕𝑥
39
[3-4-39]
The second derivative of equation [3-4-38] with respect to 𝑥 is:
𝜕 2𝑦
𝑑𝜕 2 𝛿(𝛹, 𝑥)
=
−𝐶(𝛹)
cos(𝑥
+
𝐷(𝛹))
+
𝜕𝑥 2
𝜕𝑥 2
[3-4-40]
The derivative of equation [3-4-38] with respect to 𝑥 once and 𝛹 once is:
𝜕 2𝑦
𝜕𝐶(𝛹)
𝜕𝛿(𝛹, 𝑥)
𝑑𝐷(𝛹)
=−
sin(𝑥 + 𝐷(𝛹)) +
− 𝐶(𝛹)
cos(𝑥 + 𝐷(𝛹))
𝜕𝑥𝜕𝛹
𝜕𝛹
𝜕𝛹𝜕𝑥
𝑑𝛹
[3-4-41]
Substitution of equations [3-4-38], [3-4-39], and [3-4-40] into equation [3-4-32] yields:
𝑑2 𝛿(𝛹, 𝑥)
+ 𝛿(𝛹, 𝑥)
𝑑𝑥 2
𝑑𝐶(𝛹)
𝑑𝐷(𝛹)
= −2𝜀 [
sin(𝑥 + 𝐷(𝛹)) + 𝐶(𝛹)
cos(𝑥 + 𝐷(𝛹))
𝑑𝛹
𝑑𝛹
+
𝜕𝛿(𝛹, 𝑥)
]
𝜕𝑥
[3-4-42]
−[1 − (𝐶 2 (𝛹) cos2 (𝑥 + 𝐷(𝛹)) + 2𝛿(𝛹, 𝑥)𝐶(𝛹) cos(𝑥 + 𝐷(𝛹)) +
𝛿(𝛹, 𝑥)2 ]*
[𝜀𝐶(𝛹) sin(𝑥 + 𝐷(𝛹)) +
𝜕𝛿(𝛹, 𝑥)
]
𝜕𝑥
Expansion, simplification, and elimination of HOTS in equation [3-4-42] yields:
𝜕 2 𝛿(𝛹, 𝑥)
+ 𝛿(𝛹, 𝑥)
𝜕𝑥 2
= −𝜀 (2
𝑑𝐶(𝛹) 1
+ 𝐶(𝛹)3 − 𝐶(𝛹)) sin(𝑥 + 𝐷(𝛹))
𝑑𝛹
4
− 2𝜀𝐶(𝛹)
[3-4-43]
𝑑𝐷(𝛹)
cos(𝑥 + 𝐷(𝛹))
𝑑𝛹
𝜀
+ 𝐶(𝛹)3 sin(3𝑥 + 3𝐷(𝛹))
4
Using equation [3-4-43] for 𝐶(𝛹) and
2
𝐷(𝛹)
𝑑𝑥
yields:
𝑑𝐶(𝛹) 1
+ 𝐶(𝛹)3 − 𝐶(𝛹) = 0
𝑑𝛹
4
[3-4-44]
And
𝑑𝐷(𝛹)
=0
𝑑𝛹
40
[3-4-45]
Noting that both equations [3-4-37] and [3-4-45] are equal to zero. 𝐷(𝛹) is determined
to be a constant 0.
Simplifying equation [3-4-44] yields:
2
𝑑𝐶(𝛹)
1
+ (𝐶(𝛹)2 − 4) 𝐶(𝛹) = 0
𝑑𝛹
4
[3-4-46]
Separation of variables of equation [3-4-46] results in:
𝑑𝑐
𝑑𝑐
𝑑𝑐
1
+
+
= 𝑑𝛹
4𝐶 8(𝐶 − 2) 8(2 + 𝐶) 8
[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:
1
1
1
1
ln(𝐶) − ln(2 − 𝐶) − ln(2 + 𝐶) = 𝛹 + 𝑘
4
8
8
8
Where 𝑘 is a constant of integration. Using the following log properties:
𝑎
ln(𝑎) − ln(𝑏) = ln( )
𝑏
[3-4-48]
[3-4-49]
ln(𝑎) + ln(𝑏) = ln(𝑎𝑏)
[3-4-50]
aln(𝑏) = ln(ba )
[3-4-51]
1
𝐶2
1
ln (
)= 𝛹+𝑘
2
8
4−𝐶
8
[3-4-52]
And
Equation [3-4-48] reduces to:
Solving equation [3-4-51] for 𝑐 yields:
𝐶=
2
√𝑘𝑒 𝛹 + 1
[3-4-53]
Substituting equations and [3-4-27] and [3-4-53] into equation [3-4-35] and the
observation that 𝐷(𝛹) is determined to be a constant yields the following:
𝑦=
2
√𝑘𝑒 𝛹 + 1
cos(𝑥)
[3-4-54]
Using the boundary condition in equation [3-4-1] and equation [3-4-54], 𝑘 can be
determined to be:
𝑘=3
41
[3-4-55]
Substitution of equations [3-4-27], [3-4-55] into [3-4-54] yields:
𝑦=
3.4.5
2
√3𝑒 𝜀𝑥 + 1
cos(𝑥)
[3-4-56]
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.
3.4.6
Perturbation Approximation Compared to Analytical Solution
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.
42
2
1.5
Y
Numerical
1
Y
0.5
0
0
50
100
-0.5
150
200
Y
Perturbati
on
-1
-1.5
-2
X
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.
43
0.05
0.04
0.03
0.02
Error
0.01
0
-0.01
0
50
100
-0.02
150
200
Absolute
Error
(Perturbation
Numerical)
-0.03
-0.04
-0.05
X
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 𝑥 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.
44
2
1.5
1
Y
Numerical
Y
0.5
0
0
50
100
-0.5
150
200
Y Multiple
Scales
-1
-1.5
-2
X
Figure 17: Multiple Scales versus Numeric Solution for Van Der Pol Equation
(ε=.01)
It is important to note that as 𝑥 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.
45
0.005
0.004
0.003
0.002
Error
0.001
0
-0.001
0
50
100
150
-0.002
Absolute Error
(Multiple
200
Scales to
Numerical)
-0.003
-0.004
-0.005
X
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
error two orders of magnitude small than the regular perturbation method.
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.
46
The regular perturbation approximation seen in Table 7 is plotted together with
the numerical solution that was obtained with MAPLE in Figure: 19.
5
4
3
Y Numerical
2
Y
1
0
-1
0
50
100
150
200
Y
Perturbation
-2
-3
-4
-5
X
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.
47
2.5
2
1.5
1
Error
0.5
0
-0.5
0
50
100
150
-1
200
Absolute
Error
(Perturbation
Numerical)
-1.5
-2
-2.5
X
Figure 20: Perturbation versus Numeric Solution Absolute Error Plot for Van Der
Pol Equation (ε=.05)
The rapid error increase in the regular perturbation approximation is a result of
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 𝑥. The range of 𝑥 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.
48
2.5
2
1.5
Y
Numerical
1
Y
0.5
0
-0.5
0
50
100
150
200
Y Multiple
Scales
-1
-1.5
-2
-2.5
X
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.
49
0.03
0.02
Error
0.01
0
0
50
100
150
-0.01
Absolute Error
200 (Multiple
Scales to
Numerical)
-0.02
-0.03
X
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
error two orders of magnitude small than the regular perturbation method.
50
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 nondimensionalization allowed for the formation of ε, and was shown that nondimensionalization 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 PoincareLindstedt 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 𝑥, the Poincare-Lindstedt method had
significantly lower error as values of 𝑥 increased.
51
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 PoincareLindstedt 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 𝑥. 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 𝑥, the Multiple
Scales method had significantly lower error as values of 𝑥 increased.
52
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
53
A. Appendices
54
A.1 Unforced Duffing Equation Numeric MAPLE Code
55
56
57
A.2 Van Der Pol Equation Numeric MAPLE Code
58
59
60
A.3 Numerical Value Tables for the Ordinary Differential Equation
Table 3: Perturbation and Exact Solutions to the Ordinary Differential Equation
X
0
0.011519
0.023038
0.034907
0.069813
0.10472
0.139626
0.174533
0.20944
0.244346
0.279253
0.314159
0.349066
0.383972
0.418879
0.453786
0.488692
0.523599
0.558505
0.593412
0.628319
0.663225
0.698132
0.733038
0.767945
0.802851
0.837758
0.872665
0.907571
0.942478
0.977384
1.012291
1.047198
1.082104
1.117011
1.151917
1.186824
Y Analytical
0
2.424557348
2.641622008
2.635230466
2.546917667
2.4591161
2.374339022
2.292484598
2.213452074
2.137144166
2.063466944
1.992329715
1.923644915
1.857327997
1.79329733
1.731474094
1.671782192
1.614148144
1.558501008
1.504772286
1.452895841
1.402807816
1.354446556
1.307752533
1.262668268
1.219138265
1.177108943
1.136528565
1.09734718
1.059516559
1.022990133
0.987722942
0.953671574
0.920794114
0.889050091
0.858400431
0.828807407
Y Composite
0
2.415660385
2.629258016
2.622507364
2.534980398
2.448021737
2.364043877
2.282946823
2.204631754
2.129003234
2.055969103
1.985440363
1.917331067
1.851558218
1.788041666
1.726704009
1.667470503
1.610268966
1.555029691
1.501685365
1.450170984
1.400423771
1.352383105
1.305990444
1.261189255
1.217924943
1.176144786
1.135797871
1.096835032
1.059208789
1.022873291
0.987784259
0.953898935
0.921176026
0.889575656
0.859059317
0.829589821
61
%Error
-0.36695
-0.46805
-0.48281
-0.46869
-0.45115
-0.4336
-0.41605
-0.39849
-0.38093
-0.36336
-0.34579
-0.32822
-0.31065
-0.29307
-0.27549
-0.25791
-0.24032
-0.22273
-0.20514
-0.18755
-0.16995
-0.15235
-0.13474
-0.11713
-0.09952
-0.08191
-0.06429
-0.04667
-0.02905
-0.01142
0.006208
0.023841
0.041476
0.059115
0.076757
0.094402
A.4 Numerical Value Tables for the Duffing Equation
Table 4: Perturbation and Numerical Values Determined for the Unforced Duffing
Equation (ε=.01)
X
Y Perturbation
Y Lindstedt
Y Numerical
Absolute Error
(Lindstedt vs
Numerical)
0
3.926991
7.853982
11.78097
15.70796
19.63495
23.56194
27.48894
31.41593
35.34292
39.26991
43.1969
47.12389
51.05088
54.97787
58.90486
62.83185
66.75884
70.68583
74.61283
78.53982
82.46681
86.3938
90.32079
94.24778
98.17477
102.1018
106.0288
109.9557
113.8827
117.8097
121.7367
125.6637
1
-0.696251833
-0.029452431
0.73790386
-1
0.654599805
0.088357293
-0.779555888
1
-0.612947777
-0.147262156
0.821207915
-1
0.57129575
0.206167018
-0.862859943
1
-0.529643722
-0.26507188
0.90451197
-1
0.487991695
0.323976742
-0.946163998
1
-0.446339667
-0.382881605
0.987816025
-1
0.40468764
0.441786467
-1.029468053
1
1
-0.69661748
-0.029448173
0.737645704
-0.99826561
0.653172843
0.088242371
-0.776115199
0.993068457
-0.607462493
-0.146730474
0.81189252
-0.984426568
0.55964499
0.204709603
-0.844853565
0.97236992
-0.509886202
-0.261978638
0.874884
-0.956940336
0.458358731
0.318338928
-0.901879654
0.938191336
-0.405241314
-0.37359497
0.925746887
-0.916187957
0.350718205
0.427555093
-0.946402908
0.891006524
1
-0.696195777
-0.029347606
0.737164133
-0.998270494
0.652830896
0.087942462
-0.775592601
0.993084824
-0.607218292
-0.146236211
0.811346481
-0.984461257
0.559516144
0.204028918
-0.84430102
0.972430178
-0.509889556
-0.261122264
0.874340809
-0.957033969
0.458510253
0.317320245
-0.901360556
0.938326825
-0.405555365
-0.372429655
0.925265415
-0.916374533
0.351207602
0.42626083
-0.945971377
0.891254275
0
0.000421703
0.000100567
-0.000481572
-4.88334E-06
-0.000341947
-0.000299909
0.000522598
1.63671E-05
0.000244201
0.000494263
-0.000546039
-3.46884E-05
-0.000128846
-0.000680684
0.000552546
6.02572E-05
-3.35462E-06
0.000856375
-0.00054319
-9.36335E-05
0.000151522
-0.001018683
0.000519098
0.000135489
-0.000314051
0.001165315
-0.000481472
-0.000186576
0.000489397
-0.001294264
0.000431531
0.000247751
62
Absolute Error
(Perturbation
vs Numerical)
0
5.60552E-05
0.000104825
-0.000739727
0.001729506
-0.001768909
-0.000414832
0.003963287
-0.006915176
0.005729485
0.001025944
-0.009861434
0.015538743
-0.011779606
-0.002138099
0.018558923
-0.027569822
0.019754166
0.003949616
-0.030171161
0.042966031
-0.029481442
-0.006656497
0.044803441
-0.061673175
0.040784302
0.01045195
-0.06255061
0.083625467
-0.053480038
-0.015525637
0.083496676
-0.108745725
129.5907
133.5177
137.4447
141.3717
145.2987
149.2257
153.1526
157.0796
161.0066
164.9336
168.8606
172.7876
176.7146
180.6416
184.5686
188.4956
192.4226
196.3495
-0.363035612
-0.500691329
1.07112008
-1
0.321383585
0.559596191
-1.112772108
1
-0.279731557
-0.618501054
1.154424136
-1
0.23807953
0.677405916
-1.196076163
1
-0.196427502
-0.736310778
-0.294978531
-0.480032122
0.963776066
-0.862734386
0.238215642
0.530844026
-0.977806097
0.831469612
-0.180626435
-0.579814548
0.988444334
-0.797320654
0.122410675
0.626773822
-0.995653875
0.760405966
-0.0637703
-0.671558955
-0.29565392
-0.478628474
0.963405484
-0.863054266
0.239085462
0.529351867
-0.977506394
0.831873314
-0.181696578
-0.57825575
0.988224316
-0.797820567
0.123684165
0.625170868
-0.995521381
0.761015097
-0.065247244
-0.669934738
63
-0.000675389
0.001403648
-0.000370582
-0.00031988
0.000869821
-0.001492159
0.000299703
0.000403702
-0.001070143
0.001558798
-0.000220018
-0.000499914
0.00127349
-0.001602954
0.000132495
0.000609131
-0.001476945
0.001624217
0.067381692
0.022062855
-0.107714597
0.136945734
-0.082298122
-0.030244325
0.135265714
-0.168126686
0.098034979
0.040245303
-0.16619982
0.202179433
-0.114395364
-0.052235048
0.200554782
-0.238984903
0.131180258
0.06637604
Table 5: Perturbation and Numerical Values Determined for the Unforced Duffing
Equation (ε=.05)
X
Y Perturbation
Y Lindstedt
0
3.926991
7.853982
11.78097
15.70796
19.63495
23.56194
27.48894
31.41593
35.34292
39.26991
43.1969
47.12389
51.05088
54.97787
58.90486
62.83185
66.75884
70.68583
74.61283
78.53982
82.46681
86.3938
90.32079
94.24778
98.17477
102.1018
106.0288
109.9557
113.8827
117.8097
121.7367
125.6637
129.5907
1
-0.652832038
-0.147262156
0.861092176
-1
0.4445719
0.441786467
-1.069352314
1
-0.236311763
-0.736310778
1.277612451
-1
0.028051625
1.030835089
-1.485872589
1
0.180208513
-1.325359401
1.694132727
-1
-0.388468651
1.619883712
-1.902392864
1
0.596728788
-1.914408023
2.110653002
-1
-0.804988926
2.208932335
-2.31891314
1
1.013249064
1
-0.653172843
-0.146730474
0.844853565
-0.956940336
0.405241314
0.427555093
-0.963776066
0.831469612
-0.122410675
-0.671558955
0.999698819
-0.634393284
-0.170961889
0.85772861
-0.949528181
0.382683432
0.44961133
-0.970031253
0.817584813
-0.09801714
-0.689540545
0.998795456
-0.615231591
-0.195090322
0.870086991
-0.941544065
0.359895037
0.471396737
-0.97570213
0.803207531
-0.073564564
-0.707106781
0.997290457
Y Numerical
1
-0.651510889
-0.144319338
0.842120293
-0.957372181
0.406783672
0.421193484
-0.961865209
0.833423339
-0.128666376
-0.663478817
0.999876475
-0.639385186
-0.159940417
0.850611969
-0.952639425
0.392271396
0.435506458
-0.966117774
0.824514735
-0.112975562
-0.675287168
0.999497083
-0.627095178
-0.175529348
0.858892521
-0.947661839
0.377656432
0.449718518
-0.970125687
0.815394368
-0.097250226
-0.686932352
0.998861592
64
Absolute Error
(Lindstedt vs
Numerical)
Absolute Error
(Perturbation vs
Numerical)
0
0.001661954
0.002411136
-0.002733273
-0.000431845
0.001542358
-0.00636161
0.001910857
0.001953726
-0.006255701
0.008080138
0.000177657
-0.004991902
0.011021472
-0.007116641
-0.003111244
0.009587963
-0.014104872
0.003913479
0.006929922
-0.014958422
0.014253376
0.000701627
-0.011863587
0.019560974
-0.01119447
-0.006117774
0.017761395
-0.021678219
0.005576443
0.012186837
-0.023685663
0.020174429
0.001571135
0
0.001321149
0.002942818
-0.018971883
0.042627819
-0.037788228
-0.020592983
0.107487105
-0.166576661
0.107645387
0.072831961
-0.277735976
0.360614814
-0.187992042
-0.180223121
0.533233164
-0.607728604
0.255297945
0.359241626
-0.869617992
0.887024438
-0.286818518
-0.620386629
1.275297686
-1.175529348
0.262163732
0.966746184
-1.73299657
1.449718518
-0.165136761
-1.393537966
2.221662914
-1.686932352
-0.014387472
133.5177
137.4447
141.3717
145.2987
149.2257
153.1526
157.0796
161.0066
164.9336
168.8606
172.7876
176.7146
180.6416
184.5686
188.4956
192.4226
196.3495
-2.503456646
2.527173278
-1
-1.221509201
2.797980957
-2.735433415
1
1.429769339
-3.092505268
2.943693553
-1
-1.638029477
3.38702958
-3.151953691
1
1.846289615
-3.681553891
-0.595699304
-0.21910124
0.881921264
-0.932992799
0.336889853
0.492898192
-0.98078528
0.788346428
-0.049067674
-0.724247083
0.995184727
-0.575808191
-0.24298018
0.893224301
-0.923879533
0.31368174
0.514102744
-0.61464376
-0.191082216
0.866959421
-0.942440494
0.362942244
0.463825972
-0.973887501
0.806064396
-0.081494299
-0.698411033
0.997969843
-0.602033989
-0.206595202
0.874810338
-0.936976536
0.348132316
0.477825188
65
-0.018944456
0.028019024
-0.014961843
-0.009447695
0.026052391
-0.02907222
0.00689778
0.017717969
-0.032426624
0.02583605
0.002785117
-0.026225797
0.036384977
-0.018413963
-0.013097004
0.034450576
-0.036277556
1.888812886
-2.718255494
1.866959421
0.279068707
-2.435038713
3.199259387
-1.973887501
-0.623704943
3.01101097
-3.642104586
1.997969843
1.035995488
-3.593624782
4.026764029
-1.936976536
-1.498157298
4.159379079
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
1
-0.715310079
-0.0025
0.740555511
-1.058904862
0.756962107
0.0025
-0.782207538
1.117809725
-0.798614134
-0.0025
0.823859566
-1.176714587
0.840266162
0.0025
-0.865511593
1.235619449
-0.88191819
-0.0025
0.907163621
-1.294524311
0.923570217
0.0025
-0.948815648
1.353429174
-0.965222245
-0.0025
0.990467676
-1.412334036
1.006874272
0.0025
-1.032119703
1.471238898
Y Multiple
Scales
Y Numerical
1
-0.717544628
8.54548E-15
0.738556114
-1.059416551
0.759718006
-2.61588E-14
-0.780991622
1.119573047
-0.802336314
4.87368E-14
0.823709742
-1.180010766
0.845068189
-7.35352E-14
-0.866366918
1.240236618
-0.887560558
9.14918E-14
0.908603521
-1.299739054
0.929450432
-1.10718E-13
-0.950056575
1.358006599
-0.970378338
1.50835E-13
0.990373647
-1.41454768
1.010002383
-1.73126E-13
-1.029226772
1.468910053
1
-0.715308572
-0.002534072
0.740870018
-1.059416984
0.757347418
0.002605122
-0.783449645
1.119573761
-0.799817226
-0.002680175
0.826326049
-1.180011592
0.842386545
0.002759142
-0.869155381
1.240237356
-0.884703033
-0.0028415
0.911576962
-1.299739495
0.926405093
0.002926298
-0.953225986
1.358006535
-0.967135468
-0.003012053
0.99374757
-1.414546908
1.006555088
0.0030967
-1.032810299
1.468908387
66
Absolute Error
(Multiple
Scales vs
Numerical)
0
0.002236056
-0.002534072
0.002313904
-4.32976E-07
-0.002370588
0.002605122
-0.002458023
7.14061E-07
0.002519089
-0.002680175
0.002616307
-8.26118E-07
-0.002681644
0.002759142
-0.002788463
7.37134E-07
0.002857525
-0.0028415
0.002973441
-4.41485E-07
-0.003045339
0.002926298
-0.003169411
-6.38997E-08
0.003242871
-0.003012053
0.003373923
7.72049E-07
-0.003447296
0.0030967
-0.003583527
-1.66624E-06
Absolute Error
(Perturbation vs
Numerical)
0
1.50724E-06
-3.40716E-05
0.000314507
-0.000512122
0.000385311
0.000105122
-0.001242107
0.001764037
-0.001203091
-0.000180175
0.002466483
-0.003297005
0.002120383
0.000259142
-0.003643788
0.004617907
-0.002784844
-0.0003415
0.004413341
-0.005215184
0.002834876
0.000426298
-0.004410338
0.004577362
-0.001913223
-0.000512053
0.003279894
-0.002212873
-0.000319184
0.0005967
-0.000690595
-0.002330511
-1.0485263
-0.0025
1.073771731
-1.53014376
1.090178327
0.0025
-1.115423758
1.589048623
-1.131830355
-0.0025
1.157075786
-1.647953485
1.173482382
0.0025
-1.198727814
1.706858347
-1.21513441
-0.0025
-1.048011733
2.17602E-13
1.066325186
-1.520698125
1.084138308
-2.64316E-13
-1.101425731
1.569586644
-1.118165687
3.13001E-13
1.134340085
-1.615329667
1.149934542
-3.16855E-13
-1.164938344
1.657764312
-1.179344367
3.67492E-13
-1.044356655
-0.003178281
1.070119773
-1.520695408
1.080275913
0.003254214
-1.105428917
1.569582752
-1.114100668
-0.003321966
1.138545217
-1.615324513
1.145675353
0.003378976
-1.169334777
1.657757855
-1.174903203
-0.003423445
67
0.003655078
-0.003178281
0.003794588
2.71691E-06
-0.003862395
0.003254214
-0.004003186
-3.89276E-06
0.004065018
-0.003321966
0.004205132
5.15366E-06
-0.004259188
0.003378976
-0.004396433
-6.45626E-06
0.004441164
-0.003423445
0.004169645
-0.000678281
-0.003651958
0.009448352
-0.009902415
0.000754214
0.009994841
-0.019465871
0.017729686
-0.000821966
-0.018530569
0.032628972
-0.027807029
0.000878976
0.029393037
-0.049100492
0.040231207
-0.000923445
Table 7: Perturbation and Numerical Values Determined for the Van Der Pol
Equation (ε=.05)
X
0
3.926991
7.853982
11.78097
15.70796
19.63495
23.56194
27.48894
31.41593
35.34292
39.26991
43.1969
47.12389
51.05088
54.97787
58.90486
62.83185
66.75884
70.68583
74.61283
78.53982
82.46681
86.3938
90.32079
94.24778
98.17477
102.1018
106.0288
109.9557
113.8827
117.8097
121.7367
125.6637
129.5907
Y Perturbation
1
-0.748123272
-0.0125
0.874350428
-1.294524311
0.95638341
0.0125
-1.082610566
1.589048623
-1.164643548
-0.0125
1.290870703
-1.883572934
1.372903685
0.0125
-1.499130841
2.178097245
-1.581163823
-0.0125
1.707390979
-2.472621556
1.789423961
0.0125
-1.915651117
2.767145868
-1.997684099
-0.0125
2.123911254
-3.061670179
2.205944236
0.0125
-2.332171392
3.35619449
-2.414204374
Y Multiple
Scales
1
-0.759718006
9.54311E-15
0.866366918
-1.299739054
0.970378338
-3.46252E-14
-1.066325186
1.569586644
-1.149934542
7.11143E-14
1.218955321
-1.764777172
1.273265719
-1.11311E-13
-1.314329515
1.881740046
-1.344415243
1.38149E-13
1.365939003
-1.943389019
1.381071207
-1.63382E-13
-1.391578068
1.973582897
-1.398809787
2.15621E-13
1.403757128
-1.987824615
1.407127574
-2.39125E-13
-1.409417186
1.994421102
-1.410969541
Y Numerical
1
-0.747966033
-0.013403983
0.880398134
-1.299528534
0.954246161
0.015498069
-1.085391266
1.56907201
-1.128744127
-0.017147211
1.242782828
-1.764004837
1.24846984
0.017093211
-1.340725129
1.880847285
-1.318537991
-0.015052885
1.39240169
-1.942499258
1.35620113
0.011594317
-1.416362673
1.972767831
-1.376194256
-0.007354759
1.425900721
-1.987115154
1.387399208
0.002739235
-1.428444397
1.993825342
-1.394422714
68
Absolute Error
(Multiple
Scales vs
Numerical)
0
0.011751972
-0.013403983
0.014031216
0.00021052
-0.016132177
0.015498069
-0.01906608
-0.000514635
0.021190415
-0.017147211
0.023827506
0.000772334
-0.024795879
0.017093211
-0.026395613
-0.00089276
0.025877251
-0.015052885
0.026462687
0.000889761
-0.024870076
0.011594317
-0.024784605
-0.000815066
0.022615531
-0.007354759
0.022143593
0.000709461
-0.019728366
0.002739235
-0.019027211
-0.00059576
0.016546827
Absolute Error
(Perturbation vs
Numerical)
0
0.000157239
-0.000903983
0.006047706
-0.005004223
-0.002137249
0.002998069
-0.0027807
-0.019976613
0.03589942
-0.004647211
-0.048087875
0.119568096
-0.124433846
0.004593211
0.158405712
-0.29724996
0.262625832
-0.002552885
-0.314989289
0.530122298
-0.43322283
-0.000905683
0.499288444
-0.794378037
0.621489843
0.005145241
-0.698010534
1.074555025
-0.818545028
-0.009760765
0.903726995
-1.362369148
1.01978166
133.5177
137.4447
141.3717
145.2987
149.2257
153.1526
157.0796
161.0066
164.9336
168.8606
172.7876
176.7146
180.6416
184.5686
188.4956
192.4226
196.3495
-0.0125
2.54043153
-3.650718801
2.622464512
0.0125
-2.748691667
3.945243113
-2.830724649
-0.0125
2.956951805
-4.239767424
3.038984787
0.0125
-3.165211943
4.534291735
-3.247244925
-0.0125
2.90529E-13
1.412020647
-1.997450568
1.412731717
-3.41747E-13
-1.413212462
1.998836407
-1.413537354
3.9285E-13
1.413756859
-1.999469221
1.413905133
-3.87055E-13
-1.414005279
1.999757945
-1.414072913
4.38051E-13
0.002044676
1.427706849
-1.996965526
1.399495979
-0.006901328
-1.425451914
1.998454034
-1.403666262
0.011788486
1.422497587
-1.999179232
1.407417697
-0.016688209
-1.419219222
1.999549071
-1.410973045
0.021592419
69
0.002044676
0.015686202
0.000485042
-0.013235738
-0.006901328
-0.012239452
-0.000382373
0.009871092
0.011788486
0.008740728
0.000289989
-0.006487436
-0.016688209
-0.005213943
-0.000208874
0.003099868
0.021592419
0.014544676
-1.112724681
1.653753275
-1.222968532
-0.019401328
1.323239753
-1.946789079
1.427058387
0.024288486
-1.534454218
2.240588192
-1.63156709
-0.029188209
1.745992721
-2.534742664
1.83627188
0.034092419