One Dimensional Equations1

8/13/2019 One Dimensional Equations1

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College of Engineering and Computer ScienceMechanical Engineering Department

Engineering Analysis Notes

Last updated: July 7, 2004 Larry Caretto

Introduction to Orthogonal Functions and Eigenfunction Expansions

Goal of these notes

Function sets can form vector spaces and the notions of vectors and matrix operations orthoonality, !asis sets, eienvalues, can !e carried over into analysis of functions that areimportant in enineerin applications" #$n dealin %ith functions %e have eienfunctions in placeof eienvectors"& 'hese notes lin( the previous notes on vector spaces to the application tofunctions"

'he eienfunctions %ith %hich %e %ill !e dealin are solutions to differential e)uations"*ifferential e)uations, !oth ordinary and partial differential e)uations, are an important part of

enineerin analysis and play a ma+or role in the 0-./ course" tudents are expected to havea !ac(round in ordinary differential e)uations so that %e %ill !ein %ith a !rief revie% of thistopic" 1e %ill then discuss po%er series solutions to differential e)uations and apply thistechni)ue to /essels differential e)uation" 'he series solutions to this e)uation, (no%n as/essel functions, usually occur in cylindrical eometries in the solution to the same pro!lems thatproduce sines and cosines in rectanular eometries"

1e %ill see that /essel functions, li(e sines and cosines, form a complete set so that any functioncan !e represented as an infinite series of these functions" 1e %ill discuss the turm3Loiuvillee)uation, %hich is a eneral approach to eienfunction expansions, and sho% that sines, cosines,and /essel functions are special examples of functions that satisfy the turm3Liouville e)uation"

'he /essel functions are +ust one example of special functions that arise as solutions to ordinary

differential e)uations" .lthouh these special functions are less %ell (no%n than sines andcosines, the idea that these special functions !ehave in a similar eneral manner to sines andcosines in the solution of enineerin analysis pro!lems, is a useful concept in applyin thesefunctions %hen the pro!lem you are solvin re)uires their use"

'hese notes !ein !y revie%in some concepts of differential e)uations !efore discussin po%erseries solutions and Fro!enius method for po%er series solutions of differential e)uations" 1e%ill then discuss the solution of /essels e)uation as an example of Fro!enius method" Finally%e %ill discuss the turm3Liouville pro!lem and a eneral approach to special functions that formcomplete sets"

What is a differential equation?

. differential e)uation is an e)uation, %hich contains a derivative" 'he simplest (ind of adifferential e)uation is sho%n !elo%:

00)( xxatyywithxfdx

dy=== -5

$n eneral, differential e)uations have an infinite num!er of solutions" $n order to o!tain a uni)uesolution one or more initial conditions #or !oundary conditions& must !e specified" $n the a!oveexample, the statement that y 6 y0at x 6 x0is an initial condition" #'he difference !et%een initial

nineerin /uildin 8oom -999 ail Code ;hone:


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and !oundary conditions, %hich is really one of namin, is discussed !elo%"& 'he differentiale)uation in -5 can !e ?solved@ as a definite interal"

=x

x

dxxfyy

0

)(- 0 25

'he definite interal can !e either found from a ta!le of interals or solved numerically,dependin on f#x&" 'he initial #or !oundary& condition #y 6 y0at x 6 x0& enters the solution directly"Chanes in the values of y0or x0 affect the ultimate solution for y"

. simple chane ma(in the riht hand side a function of x and y, f#x,y&, instead of a function ofx alone ives a much more complicated pro!lem"

00),( xxatyywithyxfdx

dy === 95

1e can formally %rite the solution to this e)uation +ust as %e %rote e)uation 25 for the solution toe)uation -5"

=x

x

dxyxfyy

0

),(- 0 45

Aere the definite interal can no loner !e evaluated simply" 'hus, alternative approaches areneeded" )uation 45 is used in the derivation of some numerical alorithms" 'he #un(no%n&exact value of f#x,y& is replaced !y an interpolation polynomial %hich is a function of x only"

$n the theory of differential e)uations, several approaches are used to provide analytical solutionsto the differential e)uations" 8eardless of the approach used, one can al%ays chec( to see aproposed solution is correct !y su!stitutin a proposed solution into the oriinal differentiale)uation and determinin if the solution satisfies the initial or !oundary conditions"

Ordinary differential equationsinvolve functions, %hich have only one independent varia!le"'hus they contain only ordinary derivatives" Partial differential equationsinvolve functions %ithmore than one independent varia!le" 'hus, they contain partial derivatives" 'he a!!reviationsB* and ;* are used for ordinary and partial differential e)uations, respectively"

$n an ordinary differential e)uation %e are usually tryin to solve for a function, y#x&, %here thee)uation involves derivatives of y %ith respect to x" 1e call y the dependent varialeand x theindependent variale" #Bther varia!les are conventionally used as the independent anddependent varia!les often t is used as the independent varia!le and x is used as the dependentvaria!le" $n any physical pro!lem, the appropriate varia!le for the physical )uantities underconsideration are used"&

'he order of the differential equationis the order of the hihest derivative in the e)uation")uations -5 and 95 are first3order differential e)uations" . differential e)uation %ith first, secondand third order derivatives only %ould !e a third order differential e)uation"

$n a linear differential equation, the terms involvin the dependent varia!le and its derivativesare all linear terms" 'he independent varia!le may have nonlinear terms" 'hus x9d2yDdx2E y 6 0is a linear, second3order differential e)uation" ydyDdx E sin#y& 6 0 is a nonlinear first3orderdifferential e)uation" #ither term in this e)uation ydyDdx or sin#y& %ould ma(e the differentiale)uation nonlinear"&



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*ifferential e)uations need to !e accompanied !y initial or oundary conditions" .n nthorderdifferential e)uation must have n initial #or !oundary& conditions in order to have a uni)uesolution" .lthouh initial and !oundary conditions !oth mean the similar thins, the term ?initialconditions@ is usually used %hen all the conditions are specified at one initial point" 'he term?!oundary conditions@ is used %hen the conditions are specified at t%o different values of theindependent varia!le" For example, in a second order differential e)uation for y#x&, thespecification that y#0& 6 a and y#0& 6 !, %ould !e called t%o initial conditions" 'he specificationthat y#0& 6 c and y#L& 6 d, %ould !e called !oundary conditions" 'he initial or !oundary conditionscan involve a value of the varia!le itself, lo%er3order derivatives of the varia!le, or e)uationscontainin !oth values of the dependent varia!le and its lo%er3order derivatives"

Some simple ordinary differential equations

From previous courses, you should !e familiar %ith the follo%in differential e)uations and theirsolutions" $f you are not sure a!out the solutions, +ust su!stitute them into the oriinal differentiale)uation"

)(000

0ttkeyyttatyywithkydt

dy ==== 5

)cos()sin(22

2

kxBkxAyykdx

yd +== =5

kxkx eBeAkxBkxAyykdx

yd +=+== '')cosh()sinh(22

2

75

$n e)uations =5 and 75 the constants . and / #or . and /& are determined !y the initial or!oundary conditions" ote that %e have used t as the independent varia!le in e)uation 5 and xas the independent varia!le in e)uations =5 and 75"

'here are four possi!le functions that can !e a solution to e)uation =5: sin#(x&, cos#(x&, ei(x

, ande3i(x, %here i26 3-" imilarly there are four possi!le functions that can !e a solution to e)uation75: sinh#(x&, cosh#(x&, e(x, and e3(x" $n each of these cases the four possi!le solutions are notlinearly independent"- 'he minimum num!er of functions %ith %hich all solutions to thedifferential e)uation can !e expressed is called a !asis set for the solutions" 'he solutions sho%na!ove for e)uations =5 and 75 are !asis sets for the solutions to those e)uations"

Bne final solution that is useful is the solution to eneral linear first3order differential e)uation"'his e)uation can !e %ritten as follo%s"

)()( xgyxfdx

dy=+


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Po!er series solutions of ordinary differential equations

'he solution to e)uation =5 is composed of a sine and a cosine term" $f %e consider the po%erseries for each of these, %e see that the solution is e)uivalent to the follo%in po%er series"

=

=

+

++

=

+++

++= 0

2

0

12642753

)!2(

)1(

)!12(

)1(

!6!4!21!7!5!3 n

nn

n

nn

n

x

Bn

x

A

xxx

B

xxx

xAy

-05

1e are interested in seein if %e can o!tain such a solution directly from the differential e)uation". proof !eyond the level of this course can !e used to sho% that the follo%in differential

e)uation )()()(

)()(

2

2

xryxqdx

xdyxp

dx

xyd=++ has po%er series solutions, y#x& in a reion, 8,

around x 6 a, provided that p#x&, )#x& and r#x& can !e expressed in a po%er series in some reiona!out x 6 a" Functions that can !e represented as a po%er series are called analytic functions"2

'he po%er series solution of )()()(2

2

xryxq

dx

dyxp

dx

yd=++ re)uires that the three functions

p#x&, )#x& and r#x& !e represented as po%er series" 'hen %e assume a solution of the follo%inform"

==

00 )-()(

n

nn xxaxy --5

Aere the anare un(no%n coefficients" 1e can differentiate this series t%ice to o!tain"

=

=

==0

202

2

0

10 )-()1()-(

n

nn

n

nn xxann

dx

ydandxxna

dx

dy-25

u!stitutin e)uations --5 and -25 into our oriinal differential e)uation ives the follo%inresult"

)()-()()-()()-()1(0

00

10

0

20 xrxxaxqxxnaxpxxann

n

nn

n

nn

n

nn =++

=

=

=

-95

1e then set the coefficients of each po%er of x on !oth sides of the e)uation to !e e)ual to eachother" 'his ives an e)uation that %e can used to solve for the un(no%n ancoefficients in termsof one or more coefficients li(e a0and a-, %hich are used to determine the initial conditions" 'his

is !est illustrated !y usin e)uation =5, 022

2

=+ ykdx

yd, as an example" Aere %e have p#x& 6 0,

)#x& 6 (

2

, and r#x& 6 0, so for this example, e)uation -95 !ecomes"

0)-()-()1(0

02

0

20 =+

=

=

n

nn

n

nn xxakxxann -45

'he only %ay to assure that e)uation -45 is satisfied is to have the coefficient of each po%er of #x x0& vanish" o %e et the po%er series solution !y settin the coefficients of each po%er of #x

2ee the !rief discussion of po%er series in .ppendix . for more !asic information on po%erseries"



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x0& e)ual to Hero" 'his can !e done if the po%er of #x x0& is the same in !oth sums so that %ecan %rite the t%o suns as a sinle sum" 'o do this %e use the follo%in process:

-" u!stitute the index m 6 n 2 in the first sum" 'his %ill eventually ive the same po%er of #x x0& in !oth sums" $n order to do this %e su!stitute n 6 m E 2 in the sum terms and replacethe lo%er limit of n 6 0 !y m 6 0 2 6 2" 'hese t%o steps ive the follo%in result"

0)-()-()1)(2(0

02

202 =+++

=

=

n

nn

m

mm xxakxxamm -4a5

2" 'o have the same sums, %e have to have the same lo%er limit" 1e can evaluate the sumsfor m 6 32 and m 6 3- separately and then have the sum over m runnin from Hero to infinity"'his tas( is easy here since !oth the m 6 32 and m 6 3- term are Hero"

0)-()-()1)(2(000

02

002 =+++++

=

=

n

nn

m

mm xxakxxamm -4!5

9" ince !oth m and n are dummy indices, %e can replace either of these !y any other letter and

have the same result for the sum" .lthouh %e used m 6 n 2 to rearrane the first sum, %ecan no% replace the m index in the first sum !y n to match the second sum" 'his ives thefollo%in result"

0)-()-()1)(2(0

02

002 =+++

=

=

n

nn

n

nn xxakxxann -4c5

4" 1e can no% com!ine the t%o sums !ecause !oth have the same upper and lo%er limits and!oth have the same po%er of x x0" 'he result of these four steps allo%s us to re%ritee)uation -45 as sho%n !elo%"

[ ] 0)-()1)(2(

)-()1()-()-()1(

0

0

22

2

20

0

02

0

20

=+++=

=+

=+

=

=

=

n

nnn

n

nn

n

nn

n

nn

xxakann

xxannxxakxxann

-5

'he last sum in e)uation -5 e)uals Hero only if the coefficient of #x x 0&nvanishes for each n"

'his ives the follo%in relationship amon the un(no%n coefficients"

)1)(2(0)1)(2(

2

22

2 ++==+++ ++

nn

akaorakann nnnn -=5

'his ives us an e)uation for anE2in terms of the coefficient previously found for an" 1e cannotuse this e)uation to find a0or a-, so %e assume that these coefficients %ill !e determined !y the

initial conditions" Ao%ever, once %e (no% a0, %e see that %e can find all the even num!eredcoefficients as follo%s



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234)12)(22(

2

)12)(22(

2)10)(20(

04

02

2

22

4

02

02

2

=

++

=++

=

=++

=

ak

akk

aka

akaka

-75

Continuin in this fashion %e see that the eneral pattern for a su!script %hose coefficient is aneven num!er is the follo%in"

evennn

aka

nn

n)!(

)1( 02

= -


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.s !efore, %e can chec( this eneral relationship !y o!tainin an expression for anE2and sho%inthat the ratio anE2Danas computed from e)uation 2-5 satisfies e)uation -=5" 'his chec( is left asan exercise for the reader"

o% that %e have expressions for anin terms of the initial values a0and a-%e can su!stitutethese expressions #in e)uations -


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'he Fro!enius method is used to solve the follo%in differential e)uation"

0)()()()(

22

2

=++ yx

xc

dx

xdy

x

xb

dx

xyd295

$n this e)uation !#x& and c#x& are analytic at x 6 0" ote that the conventional po%er seriesmethod cannot !e used for this e)uation !ecause the coefficients of dyDdx and y are not analyticat x 6 0"

$n the Fro!enius method the solution is %ritten as follo%s"

=

+

=

==00

)(n

rnn

n

nn

r xaxaxxy 245

Aere the anare un(no%n coefficients and the value of r is also un(no%n" 'he value of r is

determined durin the solution procedure so that a00" 1e can differentiate the series for y#x& in

e)uation 245 t%o times to o!tain"

=

+

=

+ ++=+=0

2

2

2

0

1 )1)(()(n

rnn

n

rnn xarnrn

dx

ydandxarn

dx

dy25

1e assume that !#x& and c#x& are constants, simple polynomials or series expressions that havethe follo%in eneral forms"

=

=

==00

)()(n

nn

n

nn xcxcxbxb 2=5

1e can su!stitute e)uations 245, 25, and 2=5, into e)uation 295 and multiply the result !y x2too!tain the follo%in e)uation"

0)(

)1)((

000

1

0

0

22

=

+

+

+++

=

+

=

=

+

=

=

+

n

rnn

n

nn

n

rnn

n

nn

n

rnn

xaxcxarnxbx

xarnrnx

275

1e can com!ine the x2and x terms outside the sums %ith the x terms inside the sums as follo%s"

0)(

)1)((

0000

0

=

+

+

+++

=

+

=

=

+

=

=

+

n

rnn

n

nn

n

rnn

n

nn

n

rnn

xaxcxarnxb

xarnrn

2


9/35

.s in the conventional po%er series solution, %e re)uire the coefficient of each term in the po%erseries to vanish to satisfy e)uation 2


10/35

0)(1)(

2

22

2

2

=

++ yx

x

dx

xdy

xdx

xyd9=5

1e see that this is the eneral form iven in e)uation 295 %ith !#x& 6 - and c#x& 6 x22" 8ecall

the !asic solution format for Fro!enius method from e)uation 245, copied !elo%"

=

+

=

==00

)(n

rnn

n

nn

r xaxaxxy 245

'he derivatives of this solution %ere iven in e)uation 25, also copied !elo%"

=

+

=

+ ++=+=0

2

2

2

0

1 )1)(()(n

rnn

n

rnn xarnrn

dx

ydandxarn

dx

dy25

$f %e multiply e)uation 9=5 !y x2and su!stitute e)uations 245 and 25 %e o!tain the follo%inresult"

( ) 0)()1)((0

22

00

=+++++ =

+

=

+

=

+

n

rnn

n

rnn

n

rnn xaxxarnxarnrn 975

xcept for the final sum that is multiplied !y x2, %e can com!ine all three summation operatorsinto a sinle sum"

[ ] 0)()1)((0

2

0

2 =+++++

=

++

=

+

n

rnn

n

rnn xaxarnrnrn 9


11/35

'he indicial e)uation is the e)uation that sets the coefficient of a0e)ual to Hero" For /essels

e)uation the indicial e)uation is r226 0, %hose solution is r 6 " 'he first Fro!enius method

solution %ill !e the same reardless of the value of" Ao%ever, if = 0, this is a dou!le root and

ifis an inteer, the t%o roots of the indicial e)uation %ill differ !y an inteer"4 'he first solution

%ill !e the same in all three cases" 1e %ill fist o!tain this solution and defer the consideration of

the second solution" 1e %ill ta(e the root r 6, that ives a positive value for r" 1e %ill assume

thatis positive" u!stitutin r 6, into the po%er series solution in e)uation 4-5 ives

( )[ ] [ ]{ } 0)(12

2221

122 =++++

=

+

+

n

nnn xaanxa

425

ach coefficient in this e)uation #for each po%er of x& must e)ual Hero for series sum on the left3

hand side of e)uation 425 to !e Hero" 'he coefficient of the x-Eterm %ill not vanish unless a-6 0"

'hus %e conclude that a-is Hero" 'he remainin po%ers of x in e)uation 425 have a commone)uation for the coefficient" 1hen %e set this e)uation to Hero, %e o!tain the follo%in result"

[ ] [ ] ( ) 022)( 22222222 =++=+++=++

nnnnnn aannaannaan 495

1e can solve this e)uation to o!tain a recurrence relationship that ives anin terms of an32"

)2(2

+= nn

aa nn 445

.ccordin to e)uation 445, if an6 0, then anE26 0" ince %e have sho%n that a-must !e Hero, %econclude that all values of an%ith an odd su!script must !e Hero" ince the further application ofe)uation 445 %ill !e for even su!scripts only, %e chane the index from n to m %here n 6 2m"'his allo%s us to re%rite e)uation 445 as follo%s"

)(2)22(2 2 22222 +

=+

= mm

a

mm

aa mmm 45

1e %ill ta(e a0as an un(no%n coefficient that is determined !y the initial conditions on thedifferential e)uation" 1e can then use e)uation 45 to %rite the first fe% #even3num!ered&coefficients in terms of a0" ettin m 6 - in e)uation 45 ives us a2"

)1(2)1)(1(2 20

2

2)1(22 +

=

+

=

aaa 4=5

ext %e set m 6 2 in e)uation 45 to o!tain a4in terms of a2, and then use e)uation 4=5 to et a4in terms of a0"

)1)(2)(2(2)2)(2(2

)1(2

)2)(2(2)2)(2(2 40

2

20

22

2

2)2(2

4 ++=

+

+

=

+=

+

= aa

aaa

475

48emem!er thatis (no%n from the oriinal differential e)uation" $t is typically a mathematical or physical

parameter from the oriinal pro!lem statement"



12/35

ext %e use m 6 9 in e)uation 45 to compute a=!y %hich point %e should !e a!le to see thepattern in the a2mcoefficients"

)1)(2)(3)(2)(3(2)3)(3(2

)1)(2)(2(2

)3)(3(2)3)(3(2 6

0

2

40

2

4

2

2)3(26

+++

=+

++

=

+

=+

= a

a

aaa

4


13/35

'he $essel function of the first &ind of integer order n' (n)*+is defined !y this e)uation, %iththe ar!itrary constant, ., omitted" 'hus, %e %rite

=+ +

=0

2

2

)!(!2

)1()(

mnm

mmn

nnmm

xxxJ 25

;lots of these /essel functions for some lo% values of n are sho%n !elo%" ote that J0#0& 6 -%hile Jn#0& 6 0 for all n 0"

$essel "unctions of the "irst ,ind for -nteger Orders

./01

./02

./03

./04

/

/04

/03

/02

/01

/05

/06

/07

/08

/09

4

/ 4 3 2 1 5 6 7 8 9 4/ 44 43 42 41 45

*

(n

)*+ n : /

n : 4

n : 1

First Frobenius method solution for noninteger 'he only chane necessary %hen %e

consider noninteer, is that %e do not have the factorials used a!ove in the case of inteer 6

n" $nstead, %e use the amma function in the definition of a0" #ee appendix C for !ac(roundon the amma function and its a!ility to eneraliHe factorial relationships to noninteer values"&

'hat is %e replace e)uation 4G5 !y the follo%in e)uation for noninteer"

)1(2

10 +

= Aa 95

u!stitutin this expression for a0into e)uation 4G5, and usin e)uation C395 from .ppendix Con amma functions ives the follo%in result"

)1(!2

)1(

)1()1)(2()1)((!2

)1(222 ++

=

+++++

= ++ mmA

mmm

Aa

m

m

m

m

m

45

1e su!stitute this e)uation for a2m#%ithout the ar!itrary constant, .& into the eneral po%er seriessolution from e)uation 245 to define the $essel function of the first &ind of )noninteger+ order

' ()*+"



14/35

=+

++

=

02

2

)1(!2

)1()(

mm

mm

mm

xxxJ 5

ote that this definition for noninteeris the same as e)uation 25 for inteer 6 n, since

e)uation C3=5 sho%s that #m E n&K 6 #m E n E -&"

Second Frobenius-method solution o% that %e have the first solution for inteer and

noninteer, includin6 0, %e have to f ind another linearly independent solution to /essels

e)uation to form a !asis for all possi!le solutions to this e)uation" 'his re)uires us to consider

each case separately" 'he simplest case is %henis not an inteer" Aere the values of J#x&

and J3#x& are linearly independent and %e can %rite a solution to /essels e)uation as follo%s"

)()()( xBJxAJxy += =5

Aere . and / are ar!itrary constants determined !y the initial conditions on the oriinal

differential e)uation" 'he values of J3#x& are found from e)uation 5"

1henis an inteer #%ith the usual notation that inteer 6 n&, %e can use e)uation 25 to sho%that the /essel functions for order n and order n are linearly dependent" 'hese t%o orders of/essel functions are simply related as sho%n !elo%" 'his result confirms that J3nis not a linearlyindependent solution to /essels e)uation"

)()1()( xJxJ nn

n = 75

Second Frobenius method solution for integer = n 'o find a second solution for inteer

%e have to separately consider the special case %here6 n 6 0" $n this case, %here the indicial

e)uation had a dou!le root, the second solution is sho%n in e)uation 945" 'o et the secondsolution for /essels e)uation %ith n 6 0, %e modify e)uation 945 !y settin the first solution toJ0#x& and the value of r in e)uation 945 to the value of the dou!le root, r 6 0" 'his ives thee)uation sho%n !elo% for the second solution

=+=

102 )ln()()(

m

mmxAxxJxy


15/35

( ) 0)()( 222

22 =++ ynx

dx

xdyx

dx

xydx =05

1e %ill also use e)uation G5 as the second solution" 1hen %e do this, %e have to remem!er toset ( 6 - and to set the lo%er limit of the summation e)ual to - to ma(e the e)uation for eneral nin G5 correspond to e)uation


16/35

( )

[ ]

=

+

=

=

=

+

=

=

=

+

=++

=++

0

2

0

0 0

22

0

22

0 0

)2(

)()1)((

)()1)((

m

nm

mm

nm

m

m m

nmm

nmm

m

nmm

m m

nmm

nmm

xAxAnmm

xAxAnnmnmnm

xAnxxAnmxAnmnm

=5

1e can use e)uation 25 for Jnto et a series for the derivative term, dJnDdx, in e)uation =45"

=

=+

+

+

+

=+

+

++

=+

+

=

+

=

0 02

2

2

12

02

2

)!(!2

)2()1(2

)!(!2

)2()1(2

)!(!2

)1(2

)(2

m mnm

nmm

nm

nmm

mnm

nmmn

nmm

xnmk

nmm

xnmkx

nmm

x

dx

dkx

dx

xdJkx

=5

1e can no% su!stitute e)uations =5 and ==5 into e)uation =45, remem!erin that %e havepreviously sho%n that the first ro% of =45 is e)ual to Hero"

0)2()!(!2

)2()1(2

0 0

2

02

2

=+++

+

=

=

+

=+

+

m m

nmm

nmm

mnm

nmm

xAxAnmmnmm

xnmk =75

Second Frobenius method solution for n = 0 .t this point %e %ant to return to the separateconsideration of n 6 0" 8ecall that %e had to set n 6 0 and ( 6 - in this case" 1e also had toincrease the lo%er limit on the summation of the .mx

mterms from Hero to one" a(in thesechanes in e)uation =75 ives the follo%in result for n 6 0"

( ) 0

!2

)2()1(2

1 1

22

022

2

=++

=

=

+

= m m

mm

mm

mm

mm

xAxAmm

xm=


17/35

[ ] 04)!1(!2

)1(

32

2221

122

2

=++++

=

=

m

mmm

mm

mm

xAAmxAxAmm

x7-5

ince e)uation 7-5 is a po%er series %hose sum is e)ual to Hero, the coefficient of every po%erin this series must vanish to for the sum to e)ual Hero for any value of x" 1e see that the lo%estpo%er of x occurs in the term .-x" 1e must have .-6 0 for the x coefficient to vanish"

1e next consider the x2term" For the coefficient of this term to vanish %e must satisfy thefollo%in e)uation"

4

10414

)!11(!12

)1(2222)1(2

1

==+=+

AAA 725

ince the first sum in e)uation 7-5, %hich comes from the first derivative of Jn#x&, has only evenpo%ers of x, the coefficients for odd po%ers of x are iven !y the second sum only" ettin thecoefficient of odd po%ers of x e)ual to Hero ives the follo%in result"

moddm

AA mm 2

2= 795

ince %e have previously sho%n that .-6 0, e)uation 795 tells us that .96 0" $n fact %e canapply it se)uentially to all values of .m%ith an odd su!script to sho% that all values of .mfor%hich m is odd are Hero" 'his leaves us %ith only even values of m to consider" For convenience%e can re%rite the second sum in e)uation 7-5 to have only even po%ers" 'o do this %etemporarily define the index ( 6 mD2 and replace m !y 2( in the second sum in e)uation 7-5"/efore doin this %e set .26 0 and start the sum at m 6 4 since the m 6 9 term is Hero"

[ ] 0)2(4)!1(!2

)1(

2

2222

222

122

2

=+++

=

=

k

kkk

mm

mm

xAAkxAmm

x745

From e)uation 745, %e can see that the coefficients of even po%ers, x2m, for m -, %ill vanish ifthe follo%in e)uation is satisfied"

10)2()!1(!2

)1(222

2

22 >=++

mAAmmm mmm

m

75

'his allo%s us to define ne% coefficients in terms of old ones !y the follo%in e)uation"

2

22

2222)2()!1(!)2(2

)1(

m

A

mmmA m

m

m

m

= 7=5

For m 6 2, %e can compute .4in terms of .26 M"

128

3

16

41

)1)(2)(16)(4(

1

)22()!12)(!2()22(2

)1(2

222)2(2

2

4 ==

=

AA 775

For m 6 9, %e can compute .=in terms of .4and use the result that .46 39D-2< to et a value for.="



18/35

284,13

11

284,13

9

284,13

2

608,4

3

912,6

1

36128

3

)2)(6)(36)(16(

1

)32()!13)(!3()32(2

)1(2

422)3(2

3

6

=+=+=

=

= A

A

7


19/35

ince the last expression in e)uation


20/35

)2(0)2(

2

2njj

AAAAnjj

j

jjj ==+


21/35

1hen %e apply this e)uation for m 6 -, usin the result that ( 6 ( .0, %e o!tain .2nE2,

)44()!1()1(2

)2(

)2)(22()!1(!1)2)(22(2

)2()1(2 2

302

2

1

022n

A

nn

nA

n

A

nn

nAA n

nn

nn +

+++

=+

++

+= +++ G95

1e can continue to apply e)uation G45 to et additional values of .m" 'he eneral result for thiscoefficient is tedious to o!tain and %e %ill s(ip the steps that lead to the follo%in result"

=+

= =

+

=++

++

+

+

+=1

022

2

0 1 122

212

)!(!2

)!1(11

)!(!2

)1(

)ln()()(n

mnm

mn

m

m

k

nm

knm

mmn

n

nmm

xmnx

kknmm

xx

xxJxy

G45

.lthouh this solution provides a second linearly independent solution to /essels e)uation for6

n 0, it is conventional to ta!ulate the function Nn#x& sho%n !elo% as the second solution" 'he

definition of this second solution is similar to the one used in o!tainin e)uation


22/35

sho%n a!ove the the plots of Nn#x& sho%n !elo%" 'he .'L./ functions !essel+#,x& and

!essely#,x& evaluate J#x& and N#x& for !oth inteer and noninteer"

$essel "unctions of the Second ,ind of -nteger Order

.4

./09

./08

./07

./06

./05

./01

./02

./03

./04

/

/04

/03

/02

/01

/05

/06

/ 4 3 2 1 5 6 7 8 9 4/ 44 43 42 41 45

*

;n)*+ n : /

n : 4

n:1

'his plot sho%s that the values of Nn#x& approach minus infinity as x approaches Hero due to theln#x& term in Nn#x&" /ecause the common application of /essel functions is to pro!lems %ith

radial eometries, %e usually have to ta(e C26 0 in the eneral solution, y 6 C-J#x& E C2N#x&, to

have a solution that remains finite at x 6 0"

Summary of $essel functions

'he differential e)uation of the form x2d2yDdx2E xdyDdx E #x232&y 6 0 are (no%n as

/essels e)uation" $ts main enineerin application is to pro!lems in radial eometries"

'he eneral solution to /essels e)uation is y 6 C-J#x& E C2N#x& %here C-and C2are

constants that are determined !y the !oundary conditions on the differential e)uation"

'he functions J#x& and N#x& are (no%n as the /essel functions of orderof the first and

second (ind, respectively" Ralues of these functions, %hich have oscillatory !ehavior,

may !e found in various ta!les and computer li!rary solutions"

For inteer n, at x 6 0, J0#x& 6 - and Jn#x& 6 0" .s x approaches Hero, Nn#x& approaches

minus infinity"

.lthouh not discussed in these notes, $t is possi!le to transform some e)uations into the

form of /essels e)uation"

1e %ill consider /essel functions in su!se)uent portions of the 0-./ course"



23/35

Summary of "roenius method

Fro!enius method is used to determine series solutions to the differential e)uation 'he differentiale)uation of the form x2d2yDdx2E x!#x&dyDdx E c#x&y 6 0" $n these notes %e have applied thismethod to the solution of /essels e)uation"

'he eneral form of the Fro!enius method solution is the infinite series y#x& 6 xr

#a0E a-xE a2x

2E O"&

'he eneral solution is differentiated and su!stituted into the oriinal differential e)uation"

ettin the coefficients of each po%er of xne)ual to Hero ives e)uations that can !esolved for r and the a icoefficients"

ettin the coefficient of xr6 0 ives a )uadratic e)uation for r (no%n as the indicial

e)uation"

'here are three possi!le cases for the t%o roots of the indicial e)uation: #-& the t%o roots

are the same #2& the t%o roots differ !y an inteer #other than Hero& #9& the t%o roots aredifferent and their difference is not an inteer"

$n all three cases, the first solution is y-#x& 6 xr#a0E a-x E a2x

2E O"&, %here r is one root to

the indicial e)uation" 'his must !e ta(en as the reater of the t%o roots if the roots differ!y an inteer"

'he values of the aicoefficients in the solution are determined in the same %ay as in the

eneral po%er series solution the coefficients of each po%er of xnin the solution must !eHero"

$f the t%o roots of the indicial e)uation are different, and the difference is not an inteer,

the second solution is is y2#x& 6 x8#.0E .-x E .2x

2E O"&, %here 8 is the second root tothe indicial e)uation"

$f the t%o roots are the same, the second solution is is y2#x& 6 y-#x& ln#x& E #.-x E .2x2E

.9x9E O"&"

$f the t%o roots differ !y an inteer, the second solution is is y2#x& 6 ( y-#x& ln#x& E #.0E

.-x E .2x2E .9x

9E O"&, %here ( may !e Hero"

'he coefficients .i#and the value of (& are found !y ma(in the coefficients of all po%ers

of x e)ual to Hero"

Nou should understand ho% Fro!enius method %or(s, and !e a!le to apply this method"Ao%ever, it does not play a role in the solution of most practical enineerin pro!lems"


24/35

[ ] 0)()()( =++

yxpxqdx

dyxr

dx

dG75

%ith the follo%in !oundary conditions at x 6 a and x 6 !" $n these !oundary conditions, at leastone of the t%o constants (-and (2is not Hero" imilarly, at least one of the t%o constants S-and S2is not Hero"

0)(0)( 2121 =+=+== bxax dx

dybyand

dx

dykayk G


25/35

)*,(),( jiji fLffLf = -045

$t is possi!le to have an operator that is self3ad+oint or Aermetian operator that is an operator for%hich LT 6 L" 1e can sho% that the operator in the turm3Liouville e)uation, as defined ine)uation -025, satisfies the definition of a self3ad+oint usin the inner3product definition ine)uation -095" For a self3ad+oint operator, e)uation -045 tells us that #Lfi,f+& 6 #fi,Lf+&"

dxxfpxqdx

dfxr

dx

fdfdxxpfxq

dx

dfxr

dx

fd b

a

jj

ij

b

a

ii )()()()()()(2

2

2

2

=

-05

1e no% state t%o important results for Aermetian #or self3ad+oint& operators in eneral and for theturm3Liouville operator in particular:

-" 'he eienvalues of any self3ad+oint or Aermetian operator are real"

2. 'he eienfunctions of any self3ad+oint or Aermetian operator defined over a reion a x

! form an orthoonal set over that reion"

9" 'he eienfunctions form a complete set if the vector space has a finite num!er ofdimensions as in an n x n matrix"

4" 'he turm3Liouville operator has a complete set of einefunctions over an infinite3dimensional vector space"

'he notion of orthoonal functions is an extension of the notions of inner products andorthoonality for vectors" For functions, the inner product is defined in terms of the interal in

e)uation -095" $f %e have a set of functions, fi#x& defined on an interval a x !, are orthoonal

over that interval if the follo%in relationship holds"

( ) ijib

a

jiji adxxpxfxfff == )()()(, * -0=5

1e can define a set of orthonormalfunctions !y the follo%in e)uation"

( ) ijb

a

jiji dxxpxfxfff == )()()(, * -075

.ny set of orthoonal functions i#x& can !e converted to a set of orthonormal functions fi#x& !ydividin !y the s)uare root of the inner product, # i,i&, %hich %e can also %rite as the t%o norm, UUiUU2" $f %e understand that %e are al%ays usin the t%o3norm, %e can drop the su!script and%rite this more simply as UUiUU"

( )

===b

a

ji

i

ii

i

i

ii

dxxpxgxg

xg

gg

xg

g

xgxf

)()()(

)(

,

)()()(

* -0


26/35

$f %e have a complete set of eienfunctions in a reion a x ! %e can expand any other

function , f#x&, in that reion !y the follo%in e)uation"

=

=0

)()(m

mm xyaxf -0G5

1e can derive a simple expression for the coefficients in this e)uation if the eienfunctions areorthoonal" $f %e multiply !oth sides of this e)uation !y #x&yn#x& and interate !et%een x 6 a and

x 6 !, %e o!tain the follo%in result"

=

===

00

)()()()()()()()()(m

b

a

mnm

b

a mmmn

b

a

n dxxyxyxpadxxyaxyxpdxxfxyxp

--05

1e can re%rite these interals usin the inner product notation"

( ) ( )

=

=0

,,

m

mnmn yyafy ---5

ince the ymform an orthoonal set, the only nonHero term in the summation on the riht is theone for %hich n 6 m" .ll other inner products are Hero !ecause of orthonality" 'hus %e have asimple e)uation to solve for amafter settin n 6 m every%here in the e)uation"

( )

( )

( )

===

b

a

mm

b

a

m

m

m

mm

mm

dxxyxyxp

dxxfxyxp

y

fy

yy

fya

)()()(

)()()(,

,

,2

--25

$f %e have an orthonormal set, the denominator of the ame)uation is one"

'he most common eienfunction expansions are the Fourier series of sines and cosines" 'hesefunctions are solutions of a turm Liouville differential e)uation defined !y e)uations G75 and

G


27/35

( ) ( )=

==1

0

)()sin(2

2

1

,,dxxfxm

fy

y

fya m

m

mm

--45

For example, %e can expand the simple function f#x& 6 c, a constant, usin the follo%in

coefficients"

[ ] [ ])cos(12

)cos(2

)sin(2 1

0 =

== mm

cxm

m

ccdxxma

b

a

m --5

'he term - cos#m& 6 0 if m is even and 2 if m is odd" 1e can thus %rite the amcoefficients for

this expansion as follo%s"

=oddm

m

c

evenm

am

4

0

--=5

$n this case, our eienfunction expansion in e)uation -0G5 !ecomes

=

=

===5,3,10

)sin(4

)()(mm

mm xmm

cxyacxf --75

1e see that %e can divide !y c and o!tain the follo%in expression"

=

=,5,3,1

)sin(41

m m

xm--


28/35

Orthogonal Sine Series for f)*+ : 4

/

/03

/01

/06

/08

4

403

401

/ /03 /01 /06 /08 4

*

Seriessum

4 term

3 terms

2 terms

5 terms

4/ terms

1e see that, as %e ta(e additional terms, the series converes to the value of f#x& 6 -, !ut thereare continued oscillations a!out this point" $n addition, at the !oundaries of x 6 0 and x 6 -, theseries sum is Hero"

$essel%s equation as a Sturm.=iouville prolem and its eigenfunction e*pansions

1e can sho% that /essels e)uation 9=5 is a turm3Liouville e)uation" 1e multiply e)uation 9=5!y x2and define a ne% varia!le, H 6 xD( such that dyDdH 6 #-D(&dyDdH and #-D(2&d2yDdH2" 1e canthe re%rite e)uation 9=5 in terms of H" $f %e divide the result !y H, %e o!tain"

( ) 011)(1 22

2

2222

2

2

2

2 =

+

++=

++ yzk

zdz

dy

dz

ydzyzk

dz

dy

kkz

dz

yd

kkz

z--G5

Ao%ever, since %e started %ith an e)uation for y as a function of x in e)uation 9=5 and %e havenot chaned this, %e could formally %rite y 6 y#x& 6 y#(H& in e)uation --G5 and !elo% in e)uation-205" ince Hd2yDdH2E dyDdH can !e %rittenas dH dyDdH5DdH, %e can reduce e)uation --G5 tothe turm Liouville form of e)uation G75 as follo%s"

( ) 022

22

2

2

=++

=

+

+

=

+

++ yzq

dz

dyr

dz

dyzk

zdz

dyz

dz

dyzk

zdz

dy

dz

ydz -205

1e see that /essels e)uation has the follo%in definitions of the functions r, ), and p, and the

eienvalue , in the turm3Liouville e)uation: r 6 H, ) 6 3n2DH, p 6 H, and 6 (2"

5Nou can sho% that this is correct !y applyin the rule for differentiation of products to dH dyDdH5DdH"



29/35

$f %e consider the case of inteer6 n over a reion 0 H 8 the eneral solution of /essels

e)uation, remem!erin that %e no% have y#(H& as our dependent varia!le, is a linear com!inationof Jn#(H& and Nn#(H&" Ao%ever, Nn#(H& approaches infinity as H approaches Hero" 'hus, Nn#(H&cannot !e part of our eneral solution %hen %e include H 6 0" $f the !oundary condition at H 6 8is y#(8& 6 0, %e must select ( to satisfy this e)uation" $t turns out that there are an infinite num!erof values that %e can select for %hich (8 6 0" 'here are (no%n as the Heros of the /essel

function" #ee the plots of the /essel function a!ove that sho% the initial locations at %hich thefirst fe% /essel functions are Hero"&

For convenience %e define mnand (mnas follo%s"

RkRkJJ mnmnmnnmnn

=== 0)()( -2-5

'he values of mn, called the Heros of Jn, are the values of the arument of Jnfor %hich Jnis Hero"

'here are an infinite of values, mn, for %hich Jn#mn& 6 0" ote the t%o su!scripts for mn m is an

eienfunction3countin index that ranes from one to infinity" *o not confuse this eienfunctionindex %ith the index, n, for Jn" 'he index for Jnis set !y the appearance of n in the oriinal

differential e)uation" 'hus our set of eienfunctions, for fixed n, %ill !e Jn#(-nH&, Jn#(2nH&, Jn#(9nH&,Jn#(4nH&, etc"

'his set of functions, Jn#(mnH& is orthoonal, since it is a solution to a turm3Liouville pro!lem"'he orthoonality condition satisfied !y these functions is defined from e)uation -0=5 %ith a%eihtin function, p#H& 6 H, as discussed in the pararaph follo%in e)uation -205" .pplyine)uation -0=5 to this pro!lem %e have the interval from 0 to 8 #in place of a to !& and %e have areal function so %e do not have to consider the complex con+uate" 'his ives the follo%inorthoonality condition"

( ) ( ) mom

R

onznmnznonznmnznom azdzzkJzkJrkJzkJff === 0

)()()(),(, -225

1e can apply the usual e)uation for an eienfunction expansion from -0G5 to this set of /esselfunctions"

=

===

11

)()()(m

mnnmm

mm zkJazyazf -295

$f %e multiply !oth sides of this e)uation !y HJn#(onH&dH and interate from 0 to 8 %e o!tain thefollo%in result"

==

=

R

onoom

R

onnmnnm

R

onn zdzzkJazdzzkJzkJazdzzkJzf0

2

1 00

)()()()()( -295

$n the final step %e use the orthoonality relationship %hich ma(es all terms in the sum, exceptfor the term in %hich m 6 o, Hero" 1e can use an interal ta!le to evaluate the normaliHationinteral"=

6'he eneral form of this interal, %hich is found !y interation !y parts follo%ed !y su!stitution of /esselse)uation, is sho%n !elo%"



30/35

2

)()(

21

2

0

2 zkJRzdzzkJ onR

ono = -245

Com!inin e)uations -295 and -245 and usin m in place of o as the coefficient su!script ivesthe follo%in result for the eienfunction expansion coefficients"

=R

mnn

mn

m zdzzkJzfzkJR

a0

21

2 )()(

)(

2-295

.lthouh this seems li(e an unli(ely choice of eienfunctions %ith %hich to expand an ar!itraryf#x&, %e %ill see that such expansions !ecome important in the consideration of partial differentiale)uations in cylindrical eometries"

Summary of these notes

1e have developed the !ac(round to solve the turm3Liouville pro!lem form several eneralcases !y developin the tools of po%er series solutions and Fro!enius method" 1e sho%ed thedetails of ho% one applies Fro!enius method to the solution of /essels e)uation" 'he textcovers other applications of this method"

1e discussed the turm3Liouville pro!lem" 1e noted that this pro!lem %as defined !y thedifferential e)uation and !oundary conditions in e)uations G75 and G


31/35


32/35

$ntroduction to orthoonal eienfunction expansions July 7, 2004, L" " Caretto ;ae 92

#ppendi*

'he various sections in this appendix contain !ac(round material on the topics covered in thesenotes"

#ppendi* # > Po!er Series )


33/35


the price %e pay for reducin the infinite series for the truncation error to a sinle term is that %elose the certainty a!out the point %here the derivative is evaluated" $n principle, this %ould allo%us to compute a !ound on the error !y findin the value of X, !et%een x and a, that made theerror computed !y e)uation .35 a maximum" $n practice, %e do not usually (no% the exactfunctional form, f#x&, let alone its #mE-& thderivative" 'he main result provided !y e)uation .35for use in numerical analysis is that the remainder #or truncation error& depends on the step siHe,x a, raised to the po%er mE-, %here m is the num!er of terms in the partial sum"

$n the a!ove discussion %e have not considered %hether the series %ill actually represent the

function" . series is said to !e convergentin a certain reion, a 8, called the convergence

interval, if %e can find a certain num!er of terms, m, such that the remainder is less than any

positive value of a small )uantity " 'he half3%idth of the reion, 8, is called the radius of

convergence" . real function, f#x&, is called analytic, at a point x 6 a, if it can !e represented !ya po%er series in x a, %ith a radius of converence, 8 0"

For simplicity %e can represent the coefficient in a 'aylor series as a sinle coefficient, !n" 'hisdefinition and the resultin form for the po%er series are sho%n !elo%"

==

==0

)-()(!

1

n

n

nax

n

n

n

axbxfthat!odx

fd

nb .3=5

1e have the follo%in results for operations on po%er series" Consider t%o series,

=

=0

)-()(n

nn axbxf and

=

=0

)-()(n

nn axcxg " 'he sum of the t%o series is

=

+=+0

)-)(()()(n

nnn axcbxgxf .375

'he product of the t%o series re)uires a dou!le summation"

++++++=

=

=

=

=

=

=

2021120011000

0 000

)-)(()-)((

)-()-()-()-()()(

axcbcbcbaxcbcbcb

axcaxbaxcaxbxgxfn m

mm

nn

m

mm

n

nn

.3


34/35


n

nnornnn

!)!1()!1(! ==

From this definition, %e can see %hat the values for -K and 0K are"

11

!1

)!11(!012

!2

)!12(!1

!

)!1( ======= andnn

n

1e also notice that %hen %e have ratios of similar factorials %e can reduce them as follo%s"

)1)(2(!

!)1)(2(

!

)!1)(2(

!

)!2(++=

++=

++=

+nn

n

nnn

n

nn

n

n

#ppendi* C > Gamma "unctions

'he amma function, #x& is defined !y the follo%in interal"

=0

1)( dttex xt C3-5

$n this definition, t is a dummy varia!le of interation and the arument of the amma function, x,appears only in the term tx3-" 'he amma function can !e sho%n to !e a eneraliHation of

factorials" 'o do this %e use e)uation C3-5 to evaluate #xE-&" 'o do this, %e simply replace x

!y x E - every%here in that e)uation and interate !y parts to o!tain

( ) ( )

=

===+

+

vdvdv

tdeteedtdttex xtxttxxt

00

00

11 )()1(C325

$f %e restrict the value of x to !e reater than Hero, the term e 3t txvanishes at !oth the upper andlo%er limit" #.t the upper limit, %e have to apply lAopitals rule to et this result"& 'he d#tx& termin the interal is simply xtx3-dt and %hen %e perform this differentiation %e see that the resultin

interal is simply x#x&"

( ) ( ) )(0)1(0

1

00

xxdttextdetex xtxtxt ===+

C395

'he amma function for an arument of - is particularly simple to compute"

1)10()1(000

11 =====

ttt edtedtte C345

1e can apply e)uation C395 to this result to o!tain the value of the amma function for 2, 9, and4 as follo%s"

6)3(3)13()4(2)2(2)12()3(1)1(1)11()2( ==+===+===+= C35

1e see that the eneral result of e)uation C35 is


35/35


!)()1( nnnn ==+ C3=5

1e can apply e)uation C395 and the result that #-& 6 - to compute #0&"

+==+ )0()1()()()1( that!ox

xxxxx C375

xtendin this relationship to neative inteers tells us that the value of the amma function forneative inteers is infinite" Ao%ever, the value of the amma function for and noninteernum!er, includin neative ones, may !e found from e)uation C395, once the values of theamma function are (no%n in the interval !et%een Hero and one" umerical methods areenerally re)uired to find these noninteer values of the amma function" Bne exception to thisis the value of the amma function of one3half that can !e found !y contour interation in thecomplex plane" 'he result of such interation is

=

2

1C3

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One Dimensional Equations1