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8- 1

Chapter 8 Differential Equations

• An equation that defines a relationship between anunknown function and one or more of its derivatives

is referred to as a differential equation.

• A first order differential equation:

• Example:

)! y x f dx

dy=

"#$"#%obtain we&and%n'(ubstituti

%

"'etweit(olvin'

#&at%condition boundarwith"

%

%

−===

+=

===

x y x y

c x y

x y xdx

dy



8- 2

• Example:

• A second*order differential equation:

• Example:

)!%

%

dx

dy y x f

dx

yd =

)! x ycdx

dy−=

+%++ y xy x y ++=



8- 3

,alor (eries Expansion

• -undamental case the first*order ordinar differential

equation:

.nte'rate both sides

• ,he solution based on ,alor series expansion:

$$ atsub/ect to )! x x y y x f

dx

dy===

∫ ∫ = x

x

y

ydx x f dy

$$

)! ∫ +== x

xdx x f y x g y

$

)!)!or $

)!)!+ and )!where

###)!++0%

)!)!+)!)!)!

$$$$

$

%

$$$

x f x g x g y

x g x x

x g x x x g x g y

==

+−

+−+==



8- 4

Example : First-order

Differential Equation

1iven the followin' differential equation:

,he hi'her*order derivatives:

&at&such that2 % === x y xdx

dy

3nfor$

4

4

2

2

%

%

≥=

=

=

n

n

dx

yd

dx yd

xdx

yd



8- 5

,he final solution:

& where

)&!)&!2)&!2&

)4!02

)&!)4!

0%

)&!)2)!&!&

02

)&!

0%

)&!)&!&)!

$

2%

2

$

%%

$

2

22

%

%%

=

−+−+−+=

−+

−+−+=

−

+

−

+−+=

x

x x x

x x

x x x

dx

yd x

dx

yd x

dx

dy x x g



8- 6

x 5ne ,erm ,wo ,erms ,hree ,erms -our ,erms

& & & & &

&#& & &#2 &#22 &#22&

&#% & &#4 $#6% &#6%8

&#2 & &#7 %#&6 %#&76

&#3 & %#% %#48 %#633

&#" & %#" 2#%" 2#26"

&#4 & %#8 2#88 3#$74

&#6 & 2#& 3#"6 3#7&2

&#8 & 2#3 "#2% "#82%

&#7 & 2#6 4#&2 4#8"7

% & 3 6 8

,able: ,alor (eries (olution



8- 7



8- 8

1eneral Case

• ,he 'eneral form of the first*order ordinar

differential equation:

• ,he solution based on ,alor series expansion:

$$

atsub/ect to )! x x y y y x f dx

dy===

###)!++0%

)!

)!+)!)!)! $$

$

$$$$$ +

−

+−+== y x g

x x

y x g x x y x g x g y



8- 9

Eulers 9ethod

• 5nl the term with the first derivative is used:

• ,his method is sometimes referred to as the one-step

Euler’s method since it is performed one step at a

time#

edx

dy x x x g x g +−+= )!)!)! $$



8- 10

Example: 5ne*step Eulers 9ethod

• Consider the differential equation:

• -or x &#&

,herefore at x&#& y&#33&22 !true value)#

&at&such that3 % === x y xdx

dy

∫ ∫ =&#&

&

%

&3 dx xdy

y

33&22#$2

3&

&#&

&2

==− x y



8- 11

$#$$8262#iserror,he

&#32%74'!&#&$)

valueestimatedthestepsfiveafter$#$%of si;estepa-or

$#$%$822#toreducediserror,he

3%$"#&<)$"#&!3)=$"#&&$#&!)$"#&!)&$#&!

%#&%#$&<)&!3)=$$#&$"#&!)&!)$"#&!

:)$"#& and &!at

twiceequationsr+appl Euleand$#$"of si;estepa>se

value)#absolute!in $#$3&22error ,he

3#&<)&!3=&#$&)&#&!

'et we&#$)!of si;estepa?ith

%

%

%

$

=

=−+=

=+=−+=

==

=

=+=

=−=∆

g g

g g

x x

g

x x x



8- 12

Errors with Euler@s 9ethod

• Local error : over one step si;e#

Global error : cumulative over the ran'e of the solution#

• ,he error ε usin' Euler@s method can be approximated usin' the

second term of the ,alor series expansion as

• .f the ran'e is divided into n increments then the error at the end

of ran'e for x would be nε .

<#=inmaximumtheis where

0%

)!

$%

%

%

%%

$

x xdx

yd

dx

yd x x −=ε



8- 13

Example: Analsis of Errors

$$88#$)$%#$)!&#&)!3!"

$%%#$)$"#$)!&#&)!3!%

$33#$)&#$)!&#&!3 :&#&at

error theonlimitsupperthe$#$%#and$#$"$#&of si;esstep-or

)!3)8!0%

)! b boundediserrorthe,hus

8

&at&thatsuch 3

%

$%#$

%

$"#$

%

&#$

%

$

%

$

%

%

%

==

==

===

−=−

=

=

===

ε

ε

ε

ε

x

x x x x x x

xdx

yd

x y xdx

dy



8- 14

,able: ocal and 1lobal Errors with a (tep (i;e of $#&#

x Exactsolution

Bumerical(olution

ocalError!)

1lobalError!)

& & & $ $

&#& &#33&2222 &#3 *%#8466&"& *%#8466&"&

&#% &#76$4446 &#883 *%#2$$3$4 *3#2768237

&#2 %#"74 %#34 *&#7$$2"7" *"#%288%7

&#3 2#2%"2222 2#&24 *&#4$2837% *"#4724438

&#" 3#&444446 2#7% *&#274 *"*7%

&#4 "#&%8 3#8% *&#&74$368 *4#$$4%3$%

&#6 4#%&62222 "#833 *&#$"$8%"4 *4#$$36&8

&#8 6#33%4446 6 *$#72&"4"6 *"#736487

&#7 8#8&% 8#%74 *$#82%&78" *"#8""4"&3

% &$#222222 7#63 *$#638286& *"#63&72""



8- 15

x Exactsolution

Bumerical(olution

ocalError!)

1lobalError!)

& & & $ $

&#$" &#%&$&446 &#% *$#83$&$36 *$#83$&$36

&#& &#33&2222 &#3%$" *$#63$$""" *&#33"3%$7

&#&" &#473" &#44%" *$#4"87738 *&#88834%6

&#% &#76$4446 &#7%6 *$#"7%$&4% *%#%&"82%%

&#%" %#%6$8222 %#%&" *$#"2"6678 *%#3"86&"4

&#2 %#"74 %#"%6" *$#38672$& *%#4284637

&#2" %#736&446 %#84"" *$#3346"48 *%#66&$%2

&#3 2#2%"2222 2#%2 *$#3&$7843 *%#84488$"

&#3" 2#62&" 2#4%% *$#2674"$6 *%#7233648

&#" 3#&444446 3#3$%" *$#2"% *%#78

,able: ocal and 1lobal Errors with a (tep (i;e of $#$"#



8- 17

x Exact solution Bumerical (olution ocal Error!) 1lobal Error!)

& & & $ $

&#$% &#$8&4&$6 &#$8 *$#&387&26 *$#&387&26

&#$3 &#&4438"2 &#&42%2% *$#&3$8%&7 *$#%687$$"

&#$4 &#%"3488 &#%3764 *$#&2236%8 *$#27%646

&#$8 &#234%8%6 &#227438 *$#&%46488 *$#37%8&28

&#& &#33&2222 &#32%74 *$#&%$4%7 *$#"8$7324&#% &#76$4446 &#7"2&% *$#$742343 *$#87$27%3

&#2 %#"74 %#"4838 *$#$672$&" *&#$4$$7%3

&#3 2#2%"2222 2#%86$3 *$#$446%$& *&#&"&"428

&#" 3#&444446 3#&&48 *$#$"6$88 *&#&748

&#4 "#&%8 "#$4864 *$#$37"$4 *&#%&26%8"

&#6 4#%&62222 4#&3&7% *$#$323$"" *&#%&%7"2

&#8 6#33%4446 6#2"2%8 *$#$283$7% *&#%$&$$2%

&#7 8#8&% 8#6$683 *$#$23%"42 *&#&8%$%3"

% &$#222222 &$#%&24 *$#$2$64&2 *&#&"86$76

,able: ocal and 1lobal Errors with a (tep (i;e of $#$%#



8- 21

(econd*order un'e*utta 9ethods

• ,he modified Eulers method is a case of the second*

order un'e*utta methods# .t can be expressed as

xh x x x

x x g y x g y

h y xhf yh x f y x f y y

ii

iiii

iiiiiiii

∆=∆+=∆+==

++++=

+

+

+

)! )! where

))<!!)!="#$

&

&

&



8- 22

• ,he computations accordin' to Eulers method:

&# Evaluate the slope at the start of an interval that is

at ! xi yi) #

%# Evaluate the slope at the end of the interval

! xi+&

yi+&

) :

2# Evaluate yi+& usin' the avera'e slope S & of and S % :

)!& ii y x f S =

)! &% hS yh x f S ii ++=

hS S y y ii )!"#$ %&& ++=+



8- 24

• ,he computational steps for the third*order method:

&# Evaluate the slope at ! xi yi)#

%# Evaluate a second slope S % estimate at the mid*point

in of the step as

2# Evaluate a third slope S 2 as

3# Estimate the quantit of interest yi+& as

)"#$"#$! &% hS yh x f S ii ++=

)%! %&2 hS hS yh x f S ii +−+=

hS S S y y ii <3=4

&2%&& +++=+

)!& ii y x f S =



8- 25

-ourth*order un'e*utta 9ethods

&# Compute the slope S & at ! xi yi)#

%# Estimate y at the mid*point of the interval#

2# Estimate the slope S % at mid*interval#

3# evise the estimate of y at mid*interval

# atthatsuch )! $$ h x x x y y y x f dx

dy

=∆===

)!& ii y x f S =

)!%

%F& iiii y x f h

y y +=+

)"#$"#$! &% hS yh x f S ii ++=

%%F&

%

S h

y y ii +=+



8- 26

"# Compute a revised estimate of the slope S 2 at mid*

interval#

4# Estimate y at the end of the interval#

6# Estimate the slope S 3 at the end of the interval

8# Estimate yi+& a'ain#

)"#$"#$! %2 hS yh x f S ii ++=

2& hS y y ii +=+

)! 23 hS yh x f S ii ++=

)%%!4

32%&& S S S S h y y ii ++++=+



8- 27

Gredictor*Corrector 9ethods

• >nless the step si;es are small Eulers method

and un'e*utta ma not ield precise

solutions#

• ,he Gredictor*Corrector 9ethods iterate

several times over the same interval until the

solution conver'es to within an acceptable

tolerance#

• ,wo parts: predictor part and corrector part #



8- 28

Euler*trape;oidal 9ethod

• Eulers method is the predictor al'orithm#

• ,he trape;oidal rule is the corrector equation#

• Eluer formula !predictor):

• ,rape;oidal rule !corrector):

,he corrector equation can be applied as man times as

necessar to 'et conver'ence#

H

H&

i

i jidxdyh y y +=+

<=

% &&H

H&

−+

+ ++= jii

i ji

dx

dy

dx

dyh y y



8- 29

Example 8*4: Euler*trape;oidal 9ehtod

&at&thatsuch :Groblem === x y y xdx

dy

&#&)&!&#$&

&#$

&&&

is &#&atforestimate)!predictor initial,he

$&

$$

H$$&

$$

=+=

+=

==

=

y

dx

dy y y

dx

dy

x y

&"247#&&#&&#&

:estimatetheimprovetousedisequationcorrector,he

$&

==dx

dy



8- 30

[ ]

[ ]

[ ] &$687#&&"68%#&&

%

&#$&

%

&"68%#&&$687#&&#&

&$687#&&"66&#&&%

&#$&

%

&"66&#&&$648#&&#&

&$648#&&"247#&&%

&#$&

%

%&$$

H$2&

%&

&&$$

H$%&

&&

$&$$

H$&&

=++=

++=

==

=++=

++=

==

=++=

++=

dx

dy

dx

dyh y y

dx

dy

dx

dy

dx

dyh y y

dxdy

dx

dy

dx

dyh y y

#haveweAnd

#&#&at&#&$687toconver'es (ince

2&H&

%&2&

y y

x y y y

=

==



8- 31

%%246#&)&"68%#&!&#$&$687#&

:equation predictorthe%#&atof estimate -or the

H&H&$% =+=+=

=

dx

dyh y y

x y

[ ]

22%$2#&%2%&"#&%#&

%2%&"#&2%633#&&"68%#&%&#$&$687#&

%

2%633#&%%246#&%#&

:equationcorrector,he

%%

&%H&

H&&%

&%

==

=++=

++=

==

dx

dy

dx

dy

dx

dyh y y

dx

dy



8- 32

[ ]

[ ]

&#%2%27#is %#&atof estimate,he

#iterationsthreeinconver'esal'orithmcorrectortheA'ain

%2%27#&22%&"#&&"68%#&%

&#$

&$687#&

%

22%&"#&%2%28#&%#&

%2%28#&22%$2#&&"68%#&%

&#$&$687#&

%

2%H&

H&2%

2%

%%H&

H&%%

=

=++=

++=

==

=++=

++=

x y

dx

dy

dx

dyh y y

dx

dy

dx

dy

dx

dyh y y



8- 33

9ilne*(impson 9ethod

• 9ilnes equation is the predictor euqation#

• ,he (impsons rule is the corrector formula#

• 9ilnes equation !predictor):

-or the two initial samplin' points a one*step

method such as Eulers equation can be used#• (impsoss rule !corrector):

<%%=2

3

H%H&H

H2$&

−−

−+ +−+=iii

iidx

dy

dx

dy

dx

dyh y y

<3=2 H&H&

H&&

−+

−+ +++=ii ji

i jidx

dy

dx

dy

dx

dyh y y



8- 34

Example 8*6: 9ilne*(impson 9ehtod

#3#& and 2#&atestimatewant to?e

&at&thatsuch :Groblem

==

===

x x y

x y y xdx

dy

& & &

&#& &#&$687 &#&"68%

&#% &#%2%27 &#22%&"

Assume that we have the followin' values

obtained from the Euler*trape;oidal methodin Example 8*4#

x ydx

dy



8- 36

[ ]

[ ]

2637%#&

&"68%#&)22%&"#&!3"%323#&2

&#$&$687#&

"%323#&2637&#&2#&

2637&#&

&"68%#&)22%&"#&!3"%3%3#&2

&#$&$687#&

"%3%3#&26363#&2#&

22

%2

%2

&2

=

+++=

==

=

+++=

==

y

dxdy

y

dx

dy

,he computations for x&#2 are complete#



8- 37

,he 9ilne predictor equation for estimatin' y at x&#3:

( )( ) ( )[ ]

"264%#&

&"68%#&%22%&"#&"%323#&%2

&#$3&

%%2

3

H&H%H2

H$$3

=

+−+=

+−+= dx

dy

dx

dy

dx

dyh

y y

624&$#&"264%#&3#&&3

==dx

dy

,he corrector formular:



8- 38

( )[ ]

( )[ ]

complete#isit,hen

"267&#&22%&"#&"%323#&3624&6#&2

&#$%2%27#&

624&6#&"267&#&3#&

"267&#&

22%&"#&"%323#&3624$&#&2

&#$%2%27#&

3

2

%3

%3

H%H2$3

H%&3

=+++=

==

=

+++=

+++=

y

dxdy

dx

dy

dx

dy

dx

dyh y y



8- 39

east*(quares 9ethod

• ,he procedure for derivin' the least*squares

function:

&# Assume the solution is an nth*order polnomial:

%# >se the boundar condition of the ordinar

differential equation to evaluate one of !bob&b%

Ibn)#2# Define the ob/ective function:

n

n x xb xbbb y ++++=

%

%&$J

dxe x∫ = %

dx

dy

dx

yd e −=

J where



8- 40

3# -ind the minimum of - with respect to the unknowns

!b&b% b2Ibn) that is

"# ,he inte'rals in (tep 3 are called the normal

equationsK the solution of the normal equations ields

value of the unknowns !b&b% b2Ibn)#

∫ =

∂

∂=

∂

∂

xall ii

dx

b

ee

b

$%



8- 41

Example 8*8: east*squares 9ethod

%F%

:solutionAnaltical

&#x$interval for theit(olve

$at&thatsuch :Groblem

xe y

x y xydx

dy

=

≤≤

===

&

&

$

&$

&$

J

&J

is modellinearthe,hus #& ields

)$!&J

condition boundarthe>sin'

J

bdx

yd

xb y

b

bb y

xbb y

=

+=

=

+==

+=• -irst assume a linear model is used:



8- 42

$)"32

%

%!

$)&)<!&!=%%

&

)&!

:functionerror,he

$

"

&

32

&

%

&

$ $

%

&&

&

%

&

&&&

=++−−

=−+−=

−=

+−=−=

∫ ∫ x

x x

xb x xb x xb

dx x xb xbdxdb

dee

xdb

de

xb xb xybe

x y

x

x

2%&"

2%

&"

&

&J

,hus # b'etwe&withinte'ralabove

thesolve &$ran'etheininterestedarewe(ince

+=

==≤≤



8- 45

( ) ( )[ ]( )

3

&

&%

"

&"

8

:limituppertheas & >sin'

$

34"%3

2

2

%

$&%&

&

%&

$

34

%

"

&

%3

%%

%

2

&&

$

%2

%

%

&

%

&

=+

=

=

+++−−+−

=−−−+−

−=∂

∂

∫

bb

x

x xb xb x xb xb

xb xb

dx x x x xb xb

xb

e

x

x



8- 47

x ,rue y Lalue Bumerical y Lalue Error !)

$ &# &# *

$#% &#$%$% &#$$%% *&#8

$#3 &#$822 &#$463 $#$

$#4 &#&76% &#&7"4 $#$

$#8 &#266& &#28448 $#$

&#$ &#4386 &#43&& $#$

,able: A quadratic model for the least*squares method

%F% xe y = %68664#$&3447#$&J x x y +−=



8- 48

1alerkin 9ethod

• Example: 1alerkin 9ethod

,he same problem as Example 8*8#

>se the quadratic approximatin' equation#

i

i

i

x i

b

e!

!

niedx!

∂

∂=

==∫

method squaresleast-or the

factor# n'a wei'hti is where

###%& $

# and et %

%& x! x! ==



8- 49

%

%&

%&

&

$

%2

%

%

&

&

$

2

%

%

&

8""%4#$%42&4#$&J

:result final ,he

3

&

2

&

&"

%

2

&

&"

6

&%

&

:equations normal followin' 'et the ?e

$<)%!)&!=

$<)%!)&!=

x x y

bb

bb

dx x x x xb xb

xdx x x xb xb

+−=

=+

=+

=−−+−

=−−+−

∫

∫



8- 50

,able: Example for the 1alerkin method

x ,rue y value Bumerical y value Error!)

$ &# &# **

$#% &#$%$% $#78&4 $#$

$#3 &#$822 &#$2&4 $#$

$#4 &#&76% &#&"$$ $#$

$#8 &#266& &#2248 $#$

&#$ &#4386 &#"7%& $#$

%F% xe y = %8""%4#$%42&4#$&J x x y +−=



8- 51

Mi'her*5rder Differential Equations

• (econd order differential equation:

,ransform it into a sstem of first*order differential

equations#

dx

dy

dx

dy y y y

ydx

dy

y y x f dx

dy

===

=

=

&%&

%&

%&%

and where

)!

=

dx

dy y x f

dx

yd

%

%



8- 52

• .n 'eneral an sstem of n equations of the followin'

tpe can be solved usin' an of the previousl

discussed methods:

)###!

)###!

)###!

)###!

%&

%&22

%&%%

%&&&

nnn

n

n

n

y y y x f dx

dy

y y y x f dx

dy

y y y x f dx

dy

y y y x f dx

dy

=

=

=

=



8- 53

Example: (econd*order Differential Equation

E" # #

E" $

d# % d

%

%

%

&$ :Groblem −==

& d#

d%

E" # #

E" $

d# d&

=

−==

&$

:into med transfor be can.t

%

h & % # f % %

h & % # f & &

& % # E"

iiiii

iiiii

)!

)!

:equations followin' thesolve tomethod s Euler+>se

$%2&3#$ and $$at24$$ Assume

&&

%&

+=+=

−====

+

+

bl d d iff i l i



8- 54

,able: (econd*order Differential Equation

>sin' a (tep (i;e of $#& -t

X

(ft)

Y

(ft)

Exact Z Exact Y

(ft)

0 0 -0.0231481 0 -0.0231481 0

0.1 0.000275 -0.0231481 -0.0023148 -0.0231344 -0.0023144

0.2 0.0005444 -0.0231206 -0.0046296 -0.0230933 -0.004626

0.3 0.0008083 -0.0230662 -0.0069417 -0.0230256 -0.0069321

0.4 0.0010667 -0.0229854 -0.0092483 -0.0229319 -0.0092302

0.5 0.0013194 -0.0228787 -0.0115469 -0.0228125 -0.0115177

0.6 0.0015667 -0.0227468 -0.0138347 -0.0226681 -0.0137919

0.7 0.0018083 -0.0225901 -0.0161094 -0.0224994 -0.0160505

0.8 0.0020444 -0.0224093 -0.0183684 -0.0223067 -0.018291

0.9 0.002275 -0.0222048 -0.0206093 -0.0220906 -0.020511

d#

d&

d#

d% & =

, bl ( d d Diff i l E i



,able: (econd*order Differential Equation

>sin' a (tep (i;e of $#& -t !continued)

d#

d&

d#

d% & =

X

(ft)

Y

(ft)

Exact Z Exact Y

(ft)

1 0.0025 -0.0219773 -0.0228298 -0.0218519 -0.0227083

2 0.0044444 -0.0185565 -0.0434305 -0.0183333 -0.04296663

3 0.0058333 -0.0134412 -0.0298019 -0.0131481 -0.0588194

4 0.0066667 -0.007187 -0.0704998 -0.0068519 -0.0688889

5 0.0069444 -0.0003495 -0.0746352 0.00000000 -0.071228

6 0.0066667 0.0065157 -0.0718747 0.0068519 -0.0688889

7 0.0058333 0.0128532 -0.06244066 0.0131481 -0.0588194

8 0.0044444 0.0181074 -0.0471107 0.0183333 -0.042963

9 0.0025 0.0217227 -0.0272183 0.0278519 -0.0227083

10 0.000000 0.0231435 -0.00466523 0.0231481 0.000000

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