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1. INTR INTROD ODUC UCTI TION ON
The Runge-Kutta method is popular method for solving initial value problem. t is most a!!urate and stable method. The Runge-Kutta method of order p"# 2 are nonlinear one step methods. The$ are nonlin sin!e the fun!tion f&'%$( is not involved in the Runge-Kutta formulas as a !ombination of values% but it emerges as itself argument. )o*ever% the e'pli!it +uler method is the one st linear method% and it is also the Runge-Kutta method of order p#1. ,e shall !onsider Run Kutta e'pli!it methods in the follo*ing general form $n1 # $n hn Φ&'n % $n % hn (%
n#0%1%//.% -1%
*here determines the number of mesh points et s be an integer &the number of stages( and a 21% a31% a32%/.%as1%as2%/... as%s-1%b1%/bs%!2%/..%!s be real !oeffi!ients. Then the method 1 # f&'0%$0( 2 # f&'0 !2h%$0 ha21 1( //// s # f&'0 !sh% $0 h &a s1 1 /. a s%s-1 s-1 s-1(( $1 # $0 h&b1 1 /.. b s s( is !alled an s-stage e'pli!it Runge-Kutta method.
The Runge5Kutta methods are iterative *a$s to !al!ulate the solution of a differential e6uation. tarting from an initial !ondition% the$ !al!ulate the solution for*ard step b$ ste The most !ommon method is the fourth-order Runge5Kutta method% often simpl$ referred as the Runge5Kutta method. Sign up to vote on this title Runge- Kutta method is a more general and improvised method than that of the +uler8s Useful Not useful
method. Methods based on higher order appro'imations are !alled Runge-Kutta methods. Runge-Kutta methods are a !lass of methods *hi!h 9udi!iousl$ uses the information on th
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n this se!tion% *e dis!uss the formulation of method% !on!ept of !onvergen!e% stabilit$% !onsisten!$ for RK4 method. n spite of the fa!t that Runge Kutta methods are all e'pli!it impli!it Runge Kutta method is also observed. t has the same idea of +uler method. +ule method is the first order a!!urate< in addition it re6uires onl$ a single evaluation of f &' to obtain $ n1 from $n. n !ontrast% Runge Kutta method has higher a!!ura!$. t re-evaluate the fun!tion f at t*o !onse!utive points &' n% $n( and &'n1% $n1(. t re6uires four evaluation step. =ue to this% Runge-Kutta method is 6uite a!!urate% and it has faster rates of !onvergen!e.
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n this se!tion% *e dis!uss the formulation of method% !on!ept of !onvergen!e% stabilit$% !onsisten!$ for RK4 method. n spite of the fa!t that Runge Kutta methods are all e'pli!it impli!it Runge Kutta method is also observed. t has the same idea of +uler method. +ule method is the first order a!!urate< in addition it re6uires onl$ a single evaluation of f &' to obtain $ n1 from $n. n !ontrast% Runge Kutta method has higher a!!ura!$. t re-evaluate the fun!tion f at t*o !onse!utive points &' n% $n( and &'n1% $n1(. t re6uires four evaluation step. =ue to this% Runge-Kutta method is 6uite a!!urate% and it has faster rates of !onvergen!e.
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2. HISTORY
t arise *hen eonhard +uler have made improvements on +uler method to produ!e mproved +uler method. Then% Runge is reali>ed this method *hi!h is similar method *it the se!ond order Runge Kutta method. ? fe* $ears later late r in 1@A@ Runge a!6uired :ourth ;rder of Runge Kutta method and after*ards% it is developed b$ )eun&1@00( and Kutta&1@01(.
The idea of generali>ing the +uler method% b$ allo*ing for a number of evaluations of the derivative to tae pla!e in a step% is generall$ attributed to Runge &1A@B(. :urther !ontributions *ere made b$ )eun &1@00( and Kutta &1@01(. The latter !ompletel$ !hara!teri>ed the set of Runge-Kutta methods of order 4% and proposed the first methods o order B. pe!ial methods for se!ond order differential e6uations *ere proposed b$ $stro &1@2B(% *ho also !ontributed to the development of methods for first order e6uations. t * not until the *or of )uta &1@BC%1@B7( that si'th order methods *ere introdu!ed.
in!e the advent of digital !omputers% fresh interest has been fo!used on Runge-Kutta methods% and a large number of resear!h *orers have !ontributed to re!ent e'tensions to theor$% and to the development of parti!ular methods. ?lthough earl$ studies *ere devote entirel$ to e'pli!it Runge-Kutta methods% interest has no* moved to in!lude impli!it methods% *hi!h have be!ome re!ogni>ed as appropriate for the solution of stiff differentia e6uations.
? number number of different approa!hes have been used in the anal$sis of Runge-Kutta methods The methods most !ommonl$ emplo$ed b$ s!ientists to integrate o.d.e.s *ere first develo b$ the Derman mathemati!ians E.=.T. E.=.T. Runge and M.,. Kutta in the latter half of the nineteenth !entur$.
n 1@@B% regarded as the !entenar$ of the Runge-Kutta method% a !onferen!e *as held at t E, in ?msterdam to !elebrate this o!!asion and% subse6uentl$ D. ,arner ,arner and the autho a utho *rote an a!!ount of the of Runge- Kutta methods as *e understood it at the time. ? more general histor$ of numeri!al methods of ordinar$ differential e6uations appeared a fe* $e Sign up to vote this title later as a millennium revie*. n the light of these e'isting surve$s% iton seems appropriate to Useful Not useful aim for something on a smaller s!ale in the present paper.
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3. TECHNICALITIES OR DERIVATIONS
Runge-Kutta methods are single step methods and are reall$ a!!urate than the Ta$lor
series method. Runga- Kutta method is a more general and improvised method than t of the +uler8s method. Methods based on higher order appro'imations are !alled Run Kutta methods. Runge-Kutta Metho o! O"e" T#o$ The Runge-Kutta 2nd order method is a numeri!al te!hni6ue used to solve an ordinar$ differential e6uation of the form
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The flo* !hart asso!iated *ith the R-K method of order 2 is :igure 1
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dy = y = f ( x , y ) dx
*ith $&'0( # $0
Taing h as the length of the e6uispa!ed interval% *e get $ &'0 h( # $&' 0( *1 1 *2 2 *here 1 # h f&' 0%$0( 2 # h f&'0 ah% $ 0 b 1( and # *1 1 *2 2
///.1
///.2
The parameters *1% *2% a% and b are !hosen in order to mae &1( as a!!urate as possibl G$ Ta$lor eries +'pansion% *e have
h
2
$ &'0 h( # $&' 0( h$0′ 2 !
y0
h
''
3
3!
' ' '
y0
/.
///
in!e $′ # f&'% $( $′′ # f ′&'%$( # f ' f $ f $′′′ # f '' 2f '$ f f $$ f 2 f ' f $ f $2 f You're Reading a Preview
Thus% from &3( *e obtain
Unlock full access with a free trial.
Download With Free Trial 2 h $ &'0 h( # $&' 0( h f&' 0 % $0( 2 ! [ f x + f x f y ]( x 0, y 0)
h
3
[ f xx + 2 f xy f + f yy f 2 + f x f y + f 2 y f ]( x 0, y 0 ) ;&h4( 3! ////4 Sign up to vote on this title
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*here the subs!ripts on f denote partial derivatives *ith respe!t to the indi!ated variables.
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o* b$ Ta$lorFs e'pansion for fun!tion of t*o variables% *e have 2 # h f&'0 ah % $ 0 b 1( # h H f&'0% $0( h&af ' bf $f(&'0%$0(
h
2
( a2 f xx +2 ab f xy f + b2 f yy f 2)( x 0, y 0) /////I 2
ubstituting this e'pression for 2 in &1( % *e get $ &'0 h( # $&' 0( h H * 1 f&'0% $0( *2 f&'0 ah% $0 b 1(I # $&'0( h &* 1 *2( f&'0 % $0( h2*2 Haf ' bff $I &'0% $0( 3
h
2
2
2
w 2 [ a f xx + 2 ab f f xy + b f f yy ]( x 0, y 0) ;&h4( 2
//..B
Eomparing e6uations &4( and &B(% to mae the !orresponding po*ers of h and h 2 agree% *e You're Reading a Preview must have Unlock full access with a free trial.
*1 *2 # 1
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1
a# b #
2 w2
This is a s$stem of nonlinear e6uations in the four unno*ns * 1on %* 2. There Sign upa%tob% vote this title are man$ solutions of this depending on the !hoi!e of * 2. These !hoi!es Useful!orresponding Not useful to some of t
1
standard numeri!al integration formulas. f *e !hoose * 2 #
2
then
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h )en!e *e get $&' 0 h ( # $&' 0(
2
[ f ( x0 , y 0 ) + f ( x 0+ h , y 0 + h f ( x 0 , y 0 ))]
,ith 1 # h f&'0%$0( 2 # h f&' 0 h% $ 0 h f&' 0% $0(( This method is !alled the Runge-Kutta %etho !o" o"e" t#o. The steps for the !al!ulation involved here are 1. 2. 3.
dentif$ f&'%$(% '0 % $0% and tae the value of h. :ind 1 # h f&'0%$0(. :ind 2 # h f&'0 h% $ 0 1( 1
4.
:ind #
2
& 1 2(
B. ;btain $1 # $0 % *here $ 1 is at '1 # '0 h
?lternative form for Runge-Kutta method of order t*o You're Reading a Preview
Alternativ eUnlock Form full access with a free trial. k1 f (xi , yi )
Trial h, yi With h kFree k2 f (xi Download 1) yi 1 yi h w1 k1 w2 k2 1 :or e'ample% !hoosing α = 1% then β = 1% w1 = w2 =
2
e!ond ;rder Runge - Kutta method be!omes K 1
=
h f & xi % yi (
K 2
=
h f & xi 1
+
h% yi + K 1 ( Sign 0 up to vote on this title
h
( K 1 + K 2 ) = yi + ( f & xi % yi ( + f & xi 1 % yi 1 ( ) Useful 2 2 This is Heun8 s Method *ith a ingle Eorre!tor
yi
1 =
+
yi
+
+
+
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1
Ehoosing α =
then β =
1
% w1 = 0% w2 = 1 2 2 7e!ond ;rder Runge - Kutta method be!omes K 1
=
h f & xi % yi (
K 2
=
h f & xi
yi
= +1
yi
+
K 2
+
h 2
=
% yi
yi
+
+
K 1 2
(
h f & xi
+
h 2
% yi
+
K 1 2
(
This is the Midpoint Method
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Th&" o"e" Runge-Kutta Metho$
Third order Runge-Kutta method is more a!!urate than the se!ond order Runge-Kutta method. :ollo*ing are the steps for third order method. 1. dentif$ f&'% $(% '0% $0% and tae the value of h 2. :ind 1 # h f&'0% $0( k 1 h 3. :ind 2 # h f&'0 2 % $0 2 ( 4. :ind 3 # h f&'0 h% $ 0 5 1 2 2( 1
B. :ind # 6 H 1 4 2 3I C. ;btain $1 # $0 % *here $ 1 is at '1 # '0 h
'ou"th o"e" Runge-Kutta Metho$
This is most !ommonl$ used for all engineering appli!ations. This method is a!!urate the fourth order term of the Ta$lor e'pansion so the lo!al error is proportional to h follo*ing steps are used for !omputation. 1. dentif$ f&'% $(% '0% $0% and tae the value of h 2. :ind 1 # h f&'0% $0( k 1 h You're a Preview 3. :ind 2 # h f&'0 2 % $0 2 Reading (
h
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k 2
4. :ind 3 # h f&'0 2 % $0 2 ( Download With Free Trial B. :ind 4 # h f&'0 h% $0 3( 1
C. :ind #
6
H 1 2 2 2 3 4I
7. ;btain $1 # $0 % *here $ 1 is at '1 # '0 h
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k 1
=
f & xi % yi (
k 2
=
f & xi
+
k 3
=
f & xi
+
k 4
=
f & xi
+
1 4 1 4 1
h% yi
+
h% yi
+
h% yi
−
1 4 1 A 1
k 1h( k 1h +
1 A
k 2 h(
k 2 h + k 3h( 2 3 @ k B = f & xi + h% yi + k 1h + k 4 h( 4 1C 1C 3 2 12 12 A k C = f & xi + h% yi − k 1h + k 2 h + k 3h − k 4 h + k B h( 7 7 7 7 7 h ( 7k 1 + 32k 3 + 12k 4 + 32k B + 7k C ) yi 1 = yi + @0 2 3
+
H&ghe" o"e" Runge-Kut
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Figure2: Error grahp size-step h function
+rrorJsi>e-step Draph in logarithmi! s!ale of the tree methods seen here -
The graph !ontaining red line is for +uler method The graph !ontaining green line is for the middle point method *ith order 2 ?nd the bla! line is for the Runge fourth order Kutta !lassi! method.
ote the differen!e in slope% *hi!h in!reases *ith the order of the method. You're Reading a Preview Unlock full access with a free trial.
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E(a%)*e+$ olve the follo*ing s$stem to find x&1.02( using RK2 x" &t ( = 1 + x 2 &t ( + t 3 %
x&1( = −4% h = 0.01
So*ut&on$ olve the follo*ing s$stem to find x&1.02( using RK2 2
3
x" &t ( = 1 + x &t ( + t %
x&1( = −4% h = 0.01
T+1 2
3
K1
=
h f &t % x( = 0.01&1 + x
K2
=
h f &t + h% x + K1 ( = 0.01&1 + & x + 0.1A(
x&1 + 0.01( = x&1( +
1 2
+
t ( = 0.1A
( K1 + K 2 )
4+
=−
1 2
2
+
3
&t + .01( ( = 0.1CC2
&0.1A + 0.1CC2( = −3.A2C@
You're Reading a Preview Unlock full access with a free trial.
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3
K 1
=
h f &t % x( = 0.01&1 + x
K 2
=
h f &t + h% x + K 1 ( = 0.01&1 + & x + 0.1CCA(
x&1.01+ 0.01( = x&1.01( + 3.A2C@ +
=−
1 2
1 2
+
t ( = 0.1CCA 2
+
3
&t + .01( ( = 0.1B4C
( K 1 + K 2 ) Sign up to vote on this title
&0.1CCA + 0.1B4C( = −3.CCC2 Useful
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So*ut&on$ Diven $8 # '-$$% $H0.00I # 1.0 &Lsing RK method of order 4( *ith step-length # 0.1
K1 # -0.1000000014@011C12 K2 # -0.0AB2B00010B42B71B K3 # -0.0ACCBCC@017A347B4 K4 # -0.07341@C01104C227A $H0.10I # 0.@137@4B024@000AC
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,. ALICATIONS O' RUNE-KUTTA METHOD
Runge Kutta methods are *idel$ used methods for the integration of initial value problems for ordinar$ differential e6uations. The$ !an also be used for the time integration of initial value problems for time dependent partial differential e6uations appl$ing the so !alled methods of line.
Man$ numeri!al methods have been developed for =?+s &=ifferential algebrai!
e6uations( for me!hani!s. )istori!all$% =?+s *ere redu!ed to underl$ing ;=+s and solved *ith a standard Runge-Kutta &RK( or multistep method. Methods *here the !onstraints are differentiated to obtain lo*er inde' !onstraints are !alled inde' redu!t methods.
To demonstrate the use of the Runge-Kutta method% t*o spreadsheets are in!luded th
solve the !lassi! !hemi!al engineering tan problem. These problems involve !on!entration andJor volume var$ing *ith time in a tan and re6uire a material balan to derive a differential e6uation.
The pra!ti!al appli!ation of the Runge-Kutta method is to !al!ulate the profiles of ba!*ater !urves-for spatiall$ and graduall$-varied flo*.
? preadsheet olution of a $stem ;rdinar$ =ifferential +6uations Lsing the :ou You'reof Reading a Preview
;rder Runge-Kutta Method. olving s$stems of ordinar$ differential e6uations &;=+ Unlock full access with a free trial. b$ using the fourth-order Runge-Kutta &RK4( method in !lassroom or in e'amination 6uite tedious% tiring and boring sin!e it involves man$ iterative !al!ulations. )en!e% t Download With Free Trial is a need to design a suitable tool in tea!hing and learning the numeri!al methods involved% espe!iall$ those for solving s$stems of ;=+s. )ere% *e present a ne* approa!h to solving s$stems of ;=+s b$ the RK4 method through the use of an +NE spreadsheet to ta!le these dra*ba!s. n doing so% *e emplo$ the !on!ept of relative ro*% relative !olumn and fi'ed !olumn in the spreadsheet to obtain the solution of s$stems of ;=+s b$ the RK4 method.
Signb$ upRunge to vote on this title olution for the first order fu>>$ differential e6uation 5Kutta method of ord
Useful Not useful t*o *ith ne* parameters and )armoni! mean of iFs *hi!h are used in the main form in order to in!rease the order of a!!ura!$ of the solution.
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The s!heme of Runge-Kutta fourth order Method for the numeri!al solution of the s$ of si' !oupled nonlinear ;=+s.
The e6uation of transient flo* of an ideal gas through porous media% as an e'ample o
!lass of non-linear =+. has been !onsidered. The solution of the problem has been -obtained b$ using t*o distin!t approa!hes% vi>.% the Runge-Kutta methods and the fin differen!e t$pe methods. ?s observed b$ the authors the Runge-Kutta approa!h $ield better results and to over!ome the limitations of this method the area of irregular grid net*or should be investigated.
RK method is ver$ effi!ient in solving se!ond order differential e6uations. Thus% *e
!on!lude that b$ !arr$ing out the transient anal$sis of a s$stem% *e !an find out the response of the s$stem b$ !hanging the !onditions from one stead$ state value to ano This response helps in designing a s$stem *hi!h meets our re6uirements% and *e !an further optimi>e the time domain parameters of the s$stem.
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/. ALICATION DESCRI0ED IN THIS REORT
A))*&at&on$ T"an+&ent Ana*+&+ o! E*et"&a* C&"u&t+ U+&ng Runge-Kutta Metho an &t+ A))*&at&on1. hat &+ an e*et"&a* &"u&t4
?n RE !ir!uit &or ER !ir!uit( is an ele!tri!al !ir!uit !onsisting of a resistor% an indu!tor% and a !apa!itor% !onne!ted in series or in parallel. The RE part of the name due to those letters being the usual ele!tri!al s$mbols for resistan!e% indu!tan!e and !apa!itan!e respe!tivel$. The !ir!uit forms a harmoni! os!illator for !urrent and *ill resonate in a similar *a$ as an E !ir!uit *ill. hat &+ t"an+&ent an t"an+&ent ana*+&+4
The time var$ing !urrents and voltages resulting from the sudden appli!ation of sour! usuall$ due to s*it!hing are !alled TR?+T. n transient anal$sis *e start b$ *riting the !ir!uit e6uations using basi! !on!epts of KE% KO% node-voltage anal$s and mesh-!urrent anal$sis. =ue to the involvement of integrals and derivatives in !urrent-voltage relationships for indu!tan!es !apa!itan!es% *e obtain integroYou're Reading and a Preview differential e6uations *hi!h are !onverted to pure differential e6uations b$ Unlock full access with a free trial. differentiating *ith respe!t to time. Thus the stud$ of transients re6uires solving of differential e6uations. The order of the differential e6uation depends on the number o Download With Free Trial energ$ storage elements present in the !ir!uit. ome Gasi! Eon!epts1. Kir!hhoffFs Eurrent a* 5 The sum of !urrents flo*ing in and out of a node is >ero :or a node%P #0 . 2. Kir!hhoffFs Ooltage a* 5The sum of voltages a!ross various !ir!uit elements in a mesh is >ero. n a mesh%P v #0. 3. Ooltage a!ross a !apa!itor5 O # & ʃ i dt( JE i.e integral of !urrent flo*ing through Sign up to vote on this title over a period of time divided b$ !apa!itan!e Usefulof !apa!itan!e Not usefuland derivati 4. Eurrent supplied b$ !apa!itor- # E &dvJdt( i.e produ!t of voltage a!ross !apa!itor
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ANALYSIS O' RLC CIRCUIT ?n RE !ir!uit &or ER !ir!uit( is an ele!tri!al !ir!uit !onsisting of a resistor% an indu!tor% and a !apa!itor% !onne!ted in series or in parallel. The RE part of the name due to those letters being the usual ele!tri!al s$mbols for resistan!e% indu!tan!e and !apa!itan!e respe!tivel$. The !ir!uit forms a harmoni! os!illator for !urrent and *ill resonate in a similar *a$ as an E !ir!uit *ill. The main differen!e that the presen!e the resistor maes is that an$ os!illation indu!ed in the !ir!uit *ill die a*a$ over tim it is not ept going b$ a sour!e. This effe!t of the resistor is !alled damping. The presen!e of the resistan!e also redu!es the pea resonant fre6uen!$ some*hat. ome resistan!e is unavoidable in real !ir!uits% even if a resistor is not spe!ifi!all$ in!luded a !omponent.
:igure3 RE Eir!uit =iagram The RE filter is des!ribed as a se!ond-order !ir!uit% meaning that an$ voltage or !ur in the !ir!uit !an be des!ribed b$ a se!ond-order differential e6uation in !ir!uit anal$ &t(#!.QO!&t(J Qt ,here% You're Reading a Preview E#!apa!itan!e O!&t(#voltage a!ross !apa!itan!e Unlock full access with a free trial. Then *e *rite KO e6uation for the !ir!uit asFree Trial Download With Q&t(J QtR&t(O!&t(#Oin ubstituting for &t(%*e get E Q2 O! &t(J Q2&t(RE QO!&t(J Q&t(O!&t(#Oin :or the !ase of the series RE !ir!uit these t*o parameters are given b$ Sign up to vote on this title
,here ω0 # natural fre6uen!$.
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? useful parameter is the damping fa!tor% % *hi!h is defined as the ratio of these t*o
n the !ase of the series RE !ir!uit% the damping fa!tor is given b
%$The value of the damping fa!tor determines the t$pe of transient that the !ir!uit *ill e'hibit. ome authors do not use and !all S the damping fa!tor. =ifferent !onditions for damping fa!tors% f% " 1% the s$stem is !alled over damped. # 1% the s$stem is !alled !riti!all$ damped. 1% the s$stem is !alled under damped.
'ORMULATION O' RK METHOD
You're method Readingof a Preview Runge5Kutta method is an effe!tive solving ordinar$ differential e6uation 1storder. f the given ordinar$Unlock differential e6uation of higher order sa$ UnF then it ! full access with a free is trial. be !onverted to a set of n 1storder differential e6uations b$ substitution. The Runge-Kutta method uses the formulas Download With Free Trial
t1#th V 91#V 9& 12 22 3 4(JC ,here 1#h f&t 9% V 9( 2#h f&t 9hJ2%V 9 1J2 ( 3#h f&t 9hJ2%V 9 2J2 ( 4#h f&t 9h%V 9 3(
*here K#0%1%2/.m-1
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Not useful Useful 1 is the in!rement based on the slope at the beginning of the interval% using $n< 2 is the in!rement based on the slope at the midpoint of the interval% using $ n 1J2 < is again the in!rement based on the slope at the midpoint% but no* using $ J2 <
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o%
o* let% &t(#'1 O!&t(#'2
et% N&i(#'1 V&i(#'2 You're a Preview olving the above e6uation using 4thReading order R-K method f1#hf&t%'1%'2( Unlock full access with a free trial.
g1#hg&t%'1%'2(
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f2#hf&&thJ2(%&'1f1J2(%&'2g1J2(( g2#hg&&thJ2(%&'1f1J2(%&'2g1J2(( f3#hf&&thJ2(%&'2f2J2(%&'2g2J2(( g3#hg&&thJ2(%&'1f2J2(%&'2g2J2((
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Useful Not useful f4#hf&&th(%&'1f3(%&'2g3( g4#hg&&th(%&'1f3(%&'2g3((
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SOLUTION O' THE RLC CIRCUIT$
'1#'1&&f1f4(2&f2f3((JC.0 '2#'2&&g1g4(2&g2g3((JC.0 ,here% h#&Tf -T0 (Jn here h#step si>e Tf #final time To#initial time
+'ample etFs tae an e'ample to get the transient anal$sis of !ir!uit for an over damped s$stem. et R#300% # 10m)%E#1u: and Oin#10O )ere% &0(#0%O!&0(#0 You're Reading a Preview Unlock full access with a free trial.
o% ,0 #1000
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This gives a result *ith magnitude 1B000 o*% after dividing the t*o values *e get 1.B% therefore the !ondition is over damped. Sign up to vote on this title
RAHS 'OR DI''ERENT CONDITIONS O' DAMIN$ Useful Not useful A. O5e" Da%)e Con&t&on ? plot for voltage developed a!ross !apa!itor and time for a time interval of 1ms.
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:igure4 Ooltage Os time graph for over damped !onditions 0. C"&t&a** Da%)e Con&t&on+ let us tae R#200W then% S#10000 and #1.B ? plot for voltage developed a!ross !apa!itor and time for a time interval of 1ms You're Reading a Preview Unlock full access with a free trial.
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C. Une" Da%)e Con&t&on+ et us tae R#100W Then% S#B000 and% #0.B ? plot for voltage developed a!ross !apa!itor and time for a time interval of 1ms
:igure C. Ooltage Os time for under damped !onditions You're Reading a Preview Unlock full access with a free trial.
A))*&at&on$ A S)"ea+heet So*ut&on o! a S+te% o! O"&na" D&!!e"ent&a*
E6uat&on+ U+&ng the 'ou"th-O"e" Runge-Kutta Metho 2. Download With Free Trial olving s$stems of ordinar$ differential e6uations &;=+s( b$ using the fourth-ord Runge-Kutta &RK4( method in !lassroom or in e'aminations is 6uite tedious% tirin and boring sin!e it involves man$ iterative !al!ulations. )en!e% there is a need to design a suitable tool in tea!hing and learning the numeri!al methods involved% espe!iall$ those for solving s$stems of ;=+s. )ere% *e present a ne* approa!h to solving s$stems of ;=+s b$ the RK4 method through the use of an +NE+ spreadsheet to ta!le these dra*ba!s. n doing so% *e emplo$ the !on!ept of rela ro*% relative !olumn and fi'ed !olumn in the spreadsheet to obtain the solution of Sign up to vote on this title s$stems of ;=+s b$ the RK4 method.
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E(a%)*e$ :or illustration% a simple R-!ir!uit is !onsidered% *here *e tae a !onstant voltage i
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:igure 7 ? simple R !ir!uit ,e assume that R and are !onstants. ?lso% the &defining( relation bet*een the !harg &in !oulombs( and the !urrent is
///.. &1( ?!!ording to Kir!hhoffFs e!ond a*% the !harge 6 satisfies the differential e6uation
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?s an e'ample% *e assume that R #1B ohms% L # 3 henr$s% volts and the initial !ondit Download With Free Trial and *hen t # 0. ,e then *ant to find and for 0 X t X B *ith b$ using the RK4 method The e'a!t solutions for the !harge 6 and the !urrent i are given respe!tivel$% b$
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G$ !omparing the solution obtained b$ the RK4 method *ith the e'a!t solution obtai
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tep 1 ,rite the se!ond-order differential e6uation &e6uation &2(( as a s$stem of first order differential e6uations b$ using e6uation &1(.
tep 2 ,rite the RK4 formula given belo*
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Table1 olution of a $stem of ;=+s &+'ample 1( b$ the RK4 Method
:igure A sho*s the !urrent i at a !onstant voltage of 120 volts for a R series !ir!uit a sho*n in :igure B. oti!e that% the !urrent i rea!hes the stead$ state solution at i # A You're Reading a Preview amperes as time t goes to infinit$. Unlock full access with a free trial.
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:igure A Eurrent i of a Eonstant Ooltage for a R eries Eir!uit
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7. ADVANTAES AND DISADVANTAES
ADVANTAES O' RUNE-KUTTA METHOD
The$ are eas$ to implement. The$ are also stable and thus are suitable for solving stiff ;=+s. ?s it is eas$ to !hange step si>e of this method during integration% therefore this
method is eas$ to program. There is no need to no* the derivatives of the fun!tion f. The idea of families of Runge Kutta method is too !ompli!ated% but higher order
provides mu!h better appro'imated solutions than +uler method. The most popula Runge-Kutta method is the method of order four. t is good !hoi!e to get more a!!urate and more effi!ient solutions for solving the spe!ified ordinar$ differentia e6uations. The appro'imated solution !onverges faster to e'a!t solution and the or of RK4 is 4 and the trun!ation error is ; &h B(. t is a one step method. Thus the global error is of the same order as lo!al error. t is eas$ for ?utomati! +rror Eontrol ?utomati! +rror Eontrol Lniform grid spa!ing 5 in this !ase% time steps 5 are good some !ases but not al*a$s. ometimes *e deal *ith problems *here var$ing the g You're Reading Preview si>e maes sense. )o* do $ou no* *henato !hange the step si>eY f *e have an r order s!heme and r 1th order *e a!an Unlock s!heme% full access with free tae trial. the differen!e bet*een these to be the error in the s!heme% and mae the step si>e smaller if *e prefer a smaller Download Witherror. Free :or Trial?utomati! error !ontrol $ou ar error% or larger if *e !an tolerate a larger !omputing a useless &r1(th order s!heme . . . *hat a *asteZ Gut *ith Runge Ku *e !an tae a fifth order method and a fourth order method% using the same s. ;n little e'tra *or at ea!h step
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Method is re-evaluating the fun!tion f at ea!h time to obtain the predi!table solutio
t re6uires four evaluations per step. o% the !omputation of fun!tion ma$ tae lon time. The derivation of Runge-Kutta method is obtained from Ta$lor series% but tedious to !al!ulate higher derivative. To avoid this% the fun!tion f is evaluated at m points. The$ re6uire relativel$ large !omputer time. The error estimation is not eas$ to be done for base !hoi!e of step si>e. The above simple RK methods do not *or *ell for stiff differential e6uations &eg
linear differential e6uations *ith *idel$ spread eigenvalues( or *hen ver$ high a!!ura!$ is re6uired. n parti!ular% the$ are not good for s$stems of differential e6uations *ith a mi' of and slo* state d$nami!s.
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8. RESEARCH AERS
The resear!h papers *hi!h are referred here are more or less !on!erned *ith the appli!ati and introdu!tion of Runge-Kutta method. )ere in the resear!h paper mostl$ it is given abo appli!ation that uses this method. The first resea r!h paper is about Transient ?nal$sis of +le!tri!al Eir!uits Lsing Runge-Kutta Method and its ?ppli!ation1. The RK method is ve effi!ient in solving se!ond order differential e6uations. Thus% *e !an !on!lude that b$ !arr$ing out the transient anal$sis of a s$stem% *e !an find out the response of the s$stem !hanging the !onditions from one stead$ state value to another.
?nother resear!h is about ? preadsheet olution of a $stem of ;rdinar$ =ifferential +6uations Lsing the :ourth-;rder Runge-Kutta Method in *hi!h it is designed for a s$ste of ;=+s *ith t*o e6uations% the +NE+ !ommands involved are not ver$ diffi!ult if !ompared to the M?T)E?=% M?+% M?T?G% or E rogramming. )en!e% the users ! e'tend the !on!ept presented here to solve a s$stem of ;=+s up to n e6uations or solve ;=+s b$ other numeri!al methods.
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9. CONCLUSION
n this paper% *e have dis!ussed numeri!al methods for solving s$stems of ordinar$ differential e6uations. ome ne!essar$ !onditions and definitions are given to e'amine th numeri!al methods. )ere *e have dis!ussed about Runge- Kutta method. Runge- Kutta method is a more general and improvised method than that of the +uler8s method. Method based on higher order appro'imations are !alled Runge-Kutta methods. Runge Kutta met gives better results and it !onverge faster to anal$ti!al solution and has less iteration to ge a!!ura!$ solution. ?lso its appli!ation is being dis!ussed. Runge Kutta methods are *idel used methods for the integration of initial value problems for ordinar$ differential e6uatio T*o resear!h papers have been dis!ussed based on its appli!ation. ? preadsheet olutio a $stem of ;rdinar$ =ifferential +6uations Lsing the :ourth-;rder Runge-Kutta Metho ?nd for transient anal$sis of the s$stem.
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:. RE'ERENCES
1. Anu; Suhag -Transient ?nal$sis of +le!tri!al Eir!uits Lsing Runge-Kutta Method and it ?ppli!ation. & nternational [ournal of !ientifi! and Resear!h ubli!ations% Oolume 3% s 11% ovember 2013(
2. Kim Dai Ta$% ie ong Ke% Rosmila ?bdul-Kahar- ? preadsheet olution of a $stem ;rdinar$ =ifferential +6uations Lsing the :ourth-;rder Runge-Kutta Method 3. httpsJJ***.tu-ilmenau.deJfileadminJmediaJsimulationJehreJdivJe!\lides3.pdf 4. httpJJ***.!fm.bro*n.eduJpeopleJsgJ?M3Bodes.pdf B. httpJJnm.mathfor!ollege.!omJtopi!sJrunge\utta\4th\method.html
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1<.LAIARISM REORT
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