The Variational Iteration Method Which Should Be Followed

Transcription

1 From he SelecedWork of J-Ha He The Varaoal Ierao Mehod Whch Shold Be Followed J-Ha He, ogha Uvery Go-Cheg W, ogha Uvery F. A, Hog Kog Polyechc Uvery Avalable a: hp://work.bepre.com/j_ha_he/49/

2 J.H. He, G. C. W, F. A, ol. Sc. Le. A, Vol., o., -, Bref Revew Arcle The Varaoal Ierao Mehod Whch Shold Be Followed J-Ha He, Go-Cheg W, F. A. ogha Uvery, 88 Ya a Xl Road, Shagha 5, Cha Emal: jhhe@dh.ed.c.eparme of Appled Mahemac, Hog Kog Polyechc Uvery, Hghom, Kowloo, Hog Kog, Cha Emal:maa@e.poly.ed.hk Abrac Th paper propoe hree adard varaoal erao algorhm for olvg dffereal eqao, egro-dffereal eqao, fracoal dffereal eqao, fracal dffereal eqao, dffereal-dfferece eqao ad fracoal/fracal dffereal-dfferece eqao. The phycal erpreao of he fracoal calcl ad he fracal dervave are gve ad a applcao o dcree lace eqao dced. The paper he exame he accelerao of ome erao formlae wh parclar empha beg placed o he expoeal Padé approxma ha ggeed for olary olo ad he odal Padé approxma ha ally ed for perodc ad compaco olo. The paper po o ha here may o be ay phycal meag o he exac olo of may olear eqao ad ree he mporace of earchg for approxmae olo ha afy boh he eqao ad he approprae al/bodary codo. The varaoal erao mehod parclarly able for olvg h kd of problem. Approxmae al/bodary codo ad po bodary al/codo are alo dced, wh he varaoal erao mehod beg capable of recoverg he correc al/bodary codo ad fdg he olo mlaeoly. Keyword: Varaoal erao mehod; olear eqao; fracoal dffereal eqao; fracal dffereal eqao; dffereal-dfferece eqao; fracal dffereal-dfferece eqao; fracal paceme; poro flow; Loka Volerra eqao; predaor-prey model; olary olo; expoeal Padé approxma; odal Padé approxma; approxmae al/bodary codo; po bodary /al codo. Bographcal oe: J. H. He a Profeor a ogha Uvery. He ha pblhed more ha arcle ISI-led joral, he oal cao more ha 6,8 ad H-dex 9. He he edor--chef of Ieraoal Joral of olear Scece ad mercal Smlao. H crre reearch ere maly cover olear cece, exle egeerg, aoechology, ad bology. hp://work.bepre.com/j_ha_he/ Copyrgh Aa Academc Pblher Ld. Joral Homepage:

3 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - Coe. Irodco. Varaoal Ierao Algorhm-I. Varaoal Ierao Algorhm-II 4. Varaoal Ierao Algorhm-III 5. Varaoal Ierao Algorhm for Ordary ffereal Eqao ad Paral ffereal Eqao 6. Varaoal Ierao Algorhm for Fracoal ffereal Eqao 7. Phycal Uderadg of Fracoal Calcl 8. Varaoal Ierao Algorhm for Fracal ffereal Eqao 9. Phycal Uderadg of Fracal ffereal Eqao. Varaoal Ierao Algorhm for ffereal-dfferece Eqao. Phycal Uderadg of ffereal-dfferece Eqao. Varaoal Ierao Algorhm for Fracal-dfferece Eqao ad Fracoal-dfferece Eqao. Sere Solo, Expoeal Padé Approxma ad Sodal Padé Approxma 4. Approxmae Solo v Exac Solo 5. Approxmae Ial/Bodary Codo ad Po Bodary Codo 6. Coclo. Irodco The varaoal erao mehod wa fr propoed a he ed of he la cery[,] ad flly developed 6 ad 7[-5], ad ha bee exevely worked o by mero ahor who cceflly appled he mehod o varo kd of olear problem (ee mmary Ref.[6,7]). I rece year, accordg o he Web of Scece, he mber of pblcao o he mehod ha creaed coderably (ee Table ) becae a wde rage of olear eqao are ow olvable boh accraely ad coveely by h mehod depe he drawback ha repeaed ad eceary erao volved each ep. I h paper, we how ome adard applcao of he mehod o varo problem. Table. The mber of pblcao o VIM. Year Oc.7,9 mber of pblcao o VIM 6 5. Varaoal Ierao Algorhm-I The varaoal erao mehod[-7] ha bee how o olve a large cla of olear problem effecvely, ealy, ad accraely wh he approxmao covergg rapdly o accrae olo. To llrae he bac dea of he echqe, we coder followg geeral olear yem: L[ ( )] + [ ( )] =, () where L a lear operaor ad a olear operaor.

4 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac The bac cocep of he mehod o corc a correco fcoal for he yem (), whch read { } ( ) ( ) ( ) ( ) d + = + λ L + %, () where λ a geeral Lagrage mlpler ha ca be defed opmally va varaoal heory, he -h approxmae olo, ad ~ deoe a rerced varao,.e. δ ~ =. To llrae how rerced varao work he varaoal erao mehod, we coder a mple algebrac eqao: We re-wre Eq.() he form x x+ =. () ~ x x x+ =, (4) where x ~ called a rerced varable whoe vale amed o be kow (he al ge). Solvg x from (4) lead o he rel or erao form, x = ~, (5) x x + =. (6) x Afer defyg he mlpler Eq.(), we have he followg erao algorhm: Eq.(7) called Varaoal Ierao Algorhm-I. Example { } ( ) ( ) ( ) ( ) d. + = + λ L + (7) Coder he followg olear eqao of k-h order ( k ) ( k ) + f (,,, L, ) =. (8) The varaoal erao formlao corced a follow ( ) ( ) ( )d, f% (9) ( k ) + = + λ + where δ % =, f = f (,,, L ). Afer defyg he mlpler, we have f

5 4 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - ( k ) + ( ) = ( ) + ( ) ( ) ( ) + f d. ( )! () The ma mer of h erao formla le he fac ha ( ), he al olo, ca be freely choe, wh eve kow parameer coaed. However, ome repeaed ad eceary erao are volved h erao algorhm a each ep. For al vale problem, we ca beg wh k ( k ) ( ) = () + () + () +L (). ()! k! Th lead o a ere olo covergg o he exac olo. For bodary vale problem, he al ge ca be expreed he form ( ) = a g ( ) + a g ( ) + L + a g ( ), () where gk ( ) are kow fco, a k are kow o be frher deermed afer a few erao by he bodary codo. k k. Varaoal Ierao Algorhm-II Afer defyg he Lagrage mlpler λ Eq.(), we ca corc he erao formla + ( ) = ( ) + λ( )d, () ead of he erao algorhm Eq.(7). The proof of he above erao formlao wa gve Ref.[4]. We call Eq.() Varaoal Ierao Algorhm-II. For Example gve above, Varaoal Ierao Algorhm-II gve + ( ) = ( ) + ( ) ( ) f d. (4) ( )! oe: m afy he al/bodary codo. Th he ma horcomg of he algorhm. Example Coder he followg mple eqao We fr corc a erao formlao g a Lagrage mlpler + =, () =. (5) + = + λ( ( ) ( )). + % d (6)

6 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 5 The mlpler ca be ealy defed a λ= ad Varaoal Ierao Algorhm-I read ( ( ) ( )) + = + d (7) whle Varaoal Ierao Algorhm-II read = + ( ) d. (8) If we beg wh ( ) () = =, we oba a coverge ere whch coverge o he exac olo ( ) = /( + ). ( ) = = + O( ) (9) = + + O( ), 4. Varaoal Ierao Algorhm-III From Varaoal Ierao Algorhm-II, Eq.(), we have ( ) = ( ) + λ ( )d. () + + Sbracg Eq.() from Eq.(), we oba he followg Varaoal Ierao Algorhm-III { } ( ) = ( ) + λ ( ) ( ) d. () Oe commo propery of boh Varaoal Ierao Algorhm-I ad Varaoal Ierao Algorhm-III he allowed depedece of he al ge o kow parameer whoe vale cold be defed afer a few erao by g he al/bodary codo. Varaoal Ierao Algorhm-III, parclar, hghly able for bodary vale problem of hgh order. Example Coder he followg 4 h -order dffereal eqao (4) (4) + f (,,,, ) =. () We have he followg algorhm: Varaoal Ierao Algorhm-I:

7 6 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - Varaoal Ierao Algorhm-II: Varaoal Ierao Algorhm-III: + ( ) = ( ) + ( ) { ( ) } d. + f () 6 + ( ) = ( ) + ( ) f d. (4) 6 + ( ) = ( ) + ( ) ( f f )d. (5) 6 The al gee for Algorhm-I ad Algorhm-III ca be choe o coa kow parameer ch a = A + B + C + (6) ( ), where A, B, C, ad are kow o be deermed. Algorhm-II able for al vale problem, ad he al olo alway choe o be ( ) = () + () + () + (). (7) 6 Example 4 We coder a forh-order egro-dffereal eqao[4] wh he followg bodary ad al codo We rodce f ( x ) a x (4) x x y ( x) x( e ) e y( x) y( x) dx = + + +, (8) y () =, y() = + e, y () =, y () = e. (9) x x f ( x) = x( + e ) + e + y( x) y( x) dx. () Varaoal Ierao Algorhm-III for Eq.(8) y+ ( ) = y+ ( ) ( ) ( f + f )d, () 6 x

8 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 7 where x x x f ( x) x( e ) e y ( x) y ( x) dx = We beg wh y x = + e a+ bx+ cx + dx () ( ) x ( ) ad y ( ) = y( ) ( ) f d. () 6 The kow a,b, c ad d ca be defed afer a few erao by corporag he al/bodary codo. If erao are ffce, for ace, he from Eq.(9), we have The 4 kow ca be ealy deermed from Eq.(4) (7). y () =, (4) y (), = + e (5) y () =, (6) y (). = e (7) 5. Varaoal Ierao Algorhm for Ordary ffereal Eqao ad Paral ffereal Eqao I h eco, we mmarze he varaoal erao algorhm for ome freqely ed dffereal eqao. We have + f (, ) = + ( ) = ( ) ( )d + f + ( ) = ( ) f d + ( ) = ( ) ( f f )d + + f (, ) = ( ) + ( ) = ( ) e ( )d + + f ( ) + ( ) = ( ) e f d ( ) + ( ) = ( ) e ( f f )d (8) (9)

9 8 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - + f (,, ) = + ( ) = ( ) + ( )( )d + f + ( ) = ( ) + ( ) f d + ( ) = ( ) + ( )( f f )d (4) + ω + f (,, ) = + ( ) = ( ) + ( + ω + f ) ω( )d ω + ( ) = ( ) + f ( )d ω ω + ( ) = ( ) + ( f f ) ω( )d ω (4) + f (,, ) = ( ) ( ) + ( ) = ( ) + ( e e )( + f )d ( ) ( ) + ( ) = ( ) + ( e e ) f d ( ) ( ) + ( ) = ( ) + ( e e )( f f )d + f (,,, ) = + ( ) = ( ) ( ) ( )d + f + ( ) = ( ) ( ) f d + ( ) = ( ) ( ) ( f f )d (4) (4) (4) (4) + f (,,,, ) = (4) + ( ) = ( ) + ( ) ( + f )d 6 + ( ) = ( ) + ( ) f d 6 + ( ) = ( ) + ( ) ( f f )d 6 (44)

10 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 9 ( k ) ( k ) + f = (,,, L, ) k k ( k ) + ( ) = ( ) + ( ) ( ) ( )d + f ( k )!. k k + ( ) = ( ) + ( ) ( ) f d ( k )! k k + ( ) = ( ) + ( ) ( ) ( f f )d ( k )! (45) The algorhm Eq.(4) are oly ed for olear ocllaor, ad he al ge ca be choe o be ( ) = ()co ω + ()ω ω, (46) where ω he freqecy of he ocllaor. For example, we coder a ocllaor of he form whch ca be rewre he form + =, (47) + ω + =, (48) f where f = ω +. Smlar erao algorhm ca be corced for paral dffereal eqao. Coder, for example, he paral dffereal eqao L + L+ L+ =, (49) where L, L ad L are he lear operaor wh repec o x, y ad z repecvely ad a olear operaor. Correco fcoal ca be corced a follow x { } + = + λ L (, y, z) + L % (, y, z) + L % (, y, z) + % (, y. z) d, / y { } = + λ L % ( x,, z) + L ( x,, z) + L % ( x,, z) + % ( x,, z) d, (5) + / + / + / + / + / + / z { } = + λ L % ( x, y, ) + L % ( x, y, ) + L ( x, y, ) + % ( x, y, ) d. + + / + / + / + / + / The varaoal erao algorhm (I, II ad III) ca be ealy corced afer he mlpler λ, λ, ad λ are defed. Example 5 Coder he followg paral dffereal eqao

11 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - We ca beg wh (, ) ( x, y) x y + ( x y) = x y,. (5) ( x, y) = (,) + x (, y) + y ( x,), (5a) x y ad he varaoal erao algorhm a follow (, ) (, ) x y y + ( x, y) = ( x, y) + ( x) + (, y) d, y x (, y) + ( x, y) = ( x, y) + ( ) (, ) d, y y x (, y) (, y) + ( x, y) = ( x, y) + ( ) (, y) + (, y) d. y y (5b) We ca alo cover he paral dffereal eqao o ordary dffereal eqao f oly ravelg wave olo are ogh. Applyg he raformao η = x+ ω y, we ge he OE or d ( + ω ) =, (5) dη + ω d = dη, (54) wh al codo ( ) η = A ad ( η) = B. We ca he corc he followg erao algorhm η d ( ) + ( η) = ( η) + ( η)( ( ))d, d + ω η + ( η) = ( η) ( η) ( )d, + ω η + ( η) = ( η) ( η)( ( ) ( ))d, + ω (55) o ha, begg wh ( ) η = () + η () = A+ Bη, (56) a ere olo ealy obaed.

12 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 6. Varaoal Ierao Algorhm for Fracoal ffereal Eqao The rece decade have weed a rapd developme he heory of he fracoal calcl ad applcao, wh fracoal ychrozao [8,9,] aracg parclar ere. Coder he fracoal dffereal eqao where a f, + = (57) p ( ) p he Capo fracoal dervave ha wdely ed he lerare ad defed p ( m+ ) ( ) ( τ ) = dτ p a p m Γ ( m +, m<p<m+, (58) p) ( τ ) for ome real mber p called he order of he fracoal dervave wh Γ deog he gamma fco. Fracoal dffereal eqao have receved mch aeo recely becae of her exac decrpo of may olear pheomea. The varaoal erao mehod a grealy ccefl echqe ha wa fr propoed 998 o olve fracoal dffereal eqao (ee Ref.[]) whoe effecvee ad accracy wa clearly demoraed by ragaec[], Odba ad Moma[] by applyg o ome complex dffereal eqao of fracoal order. I 7, Moma ad Odba[] howed ha he homoopy perrbao mehod alo capable of olvg fracoal dffereal eqao aalycally. A pree, he varaoal erao ad he homoopy perrbao mehod are he ma mahemacal ool for acklg fracoal dffereal eqao, ee Ref.[4-]. I he cae < <, we re-wre Eq.(57) he form ad he varaoal erao algorhm are gve a follow d d + + f = (59) d d + ( ) = ( ) ( + f )d d + ( ) = ( ) ( + f)d d d d + ( ) = ( ) ( + f ) ( + f ) d. d d (6) I he cae < <, he above erao formla are alo vald. We ca re-wre Eq.(57) he form

13 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - ad he followg erao formlae are ggeed d d f + + = (6) d d + ( ) = ( ) + ( )( + f )d d + ( ) = ( ) + ( )( + f )d (6) d d d + ( ) = ( ) + ( ) ( + f ) ( + f ) d. d d Whe cloe o, Eq.(6) beer whle Eq.(6) recommeded for approachg. For he cae < < +, where a aral mber, he erao formla + ( ) = ( ) + ( ) ( ) ( + f )d ( )! d + ( ) = ( ) + ( ) ( ) ( + f)d (6) ( )! d d d + ( ) = ( ) + ( ) ( ) ( + f ) ( + f ) d ( )! d d or + + ( ) = ( ) + ( ) ( ) ( + f )d! + + d + ( ) = ( ) + ( ) ( ) ( + f )d +! d d d + ( ) = ( ) + ( ) ( ) ( + f ) ( + f ) d + +! d d (64) ca be ed. Whe cloe o, Eq.(6) recommeded ad Eq.(64) work more effecvely for approachg Phycal Uderadg of he Fracoal Calcl Alhogh he fracoal calcl wa veed by ewo ad Lebz over hree cere ago, oly became a ho opc recely owg o he developme of he comper ad exac decrpo of may real-lfe problem. To gve a phycal erpreao of he fracoal calcl, we beg wh a mple fco

14 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac Whe x= x ad x= x, we have y= x. (65) y = x ad y x. = (66) We herefore have he dfferece y= y y = x x = ( x x )( x + x ) = ( x + x ) x. (67) I cae x x or x, we have he dffereal dy= xdx. (68) The above dervao, however, oly vald for coo fco. To how applcao, we coder he aco Srg Theory[] S = mc d, (69) where m he ma of a parcle, c he peed of lgh, ad d he relavc merc ha ca be expreed he form Sbo of Eq.(7) o Eq.(69) lead o = =. (7) c d c d dx dy dz c d S mc d from whch he eqao of free moo obaed he al way = (7) c d m ( ) =. d c (7) The above dervao ame ha pace ad me are boh coo. The dace bewee wo po, for example, cao be expreed he form of Eq.(7) a dcoo world (e.g., he Jla e). ow coder a plae wh fracal rcre (ee Fg.). The hore pah bewee wo po o a le ad we have[] de = kd, (7) where d E he acal dace bewee wo po ( dcoo le Fg.), d he le dace bewee wo po (coo le Fg.), he fracal dmeo ad k a coa.

15 4 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - Fg. The dace bewee wo po a dcoo paceme. The aco a dcoo paceme ca herefore be wre he followg form g fracoal calcl S = mc d E + / = mc k( ) d. (74) c Fracoal calcl herefore vald for dcoo problem. ow we coder a well-kow predaor-prey model (he Loka Volerra eqao)[] dx x( a by), d = (75) dy = y( c dy), (76) d where y he mber of predaor (for example, wolve), x he mber of prey (for example, rabb) ad a,b,c, ad d are parameer repreeg he eraco of he wo pece. I geeral, he growh of he wo poplao dcoo ad a mple modfcao of he predaor-prey model o replace dy/d ad dx/d by fracoal dervave x x( a by), = (77) β y = y( c dy), (78) β where he poplao of he predaor ad prey may be grealy affeced by he fracoal order, ad β.

16 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 5 8. Varaoal Ierao Algorhm for Fracal ffereal Eqao The fracal dervave defed by raformg he adard eger dmeoal pace-me (x, ) o a fracal pace-me [4,5] d( ) ( ) ( ) = lm, (79) d where he order of he fracal dervave. A a example, coder he fracal dervave relaxao eqao[4] d d + B =, < <, ()=, (8) whoe aalycal olo [4] ( ) = exp( B ). (8) We wre dow a geeral fracal dffereal eqao of he form d d + f = (8) ad e he raformao = x o cover Eq.(8) o a ordary dffereal eqao d f dx + = (8) o ha he erao algorhm above ca be drecly appled: x d + ( x) = ( x) ( + f )d x dx x + ( x) = ( x) f d x x + ( x) = ( x) { f ) } d f x (84) or

17 6 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - d + ( ) = ( ) ( + f ) d d + ( ) = ( ) f d = { f f } x + ( ) ( ) ) d. (85) A aoher example, coder he -h order fracal dffereal eqao d + f = (86) d whch ca be covered by he raformao = x (87) o he ordary dffereal eqao d + f =, (88) dx ad he varaoal erao algorhm are x ( ) + ( x) = ( x) + ( ) ( x) ( + f )d ( )! + ( x) = ( x) + ( ) ( x) f d ( )! + ( x) = ( x) + ( ) ( ) ( f f )d. ( )! (89) 9. Phycal Uderadg of Fracal ffereal Eqao The fracal dervave mpler ha fracoal coerpar may applcao ad alo vald for dcoo cae. We re-wre Eq.(79) he form d( x) ( A) ( B) = lm, dx k A B x% x% A B (9) where k a coa ad A ad B are arbrary po dcoo pace or paceme (a how Fg.). ( x% A, x% B ) are called he fracal coordae ad are defed by

18 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 7 x% = k( x ) = k( x ) (9) A A A x% = k( x ) = k( x ), (9) B B B where ( x A, x B ) are he coordae ad he fracoal dmeo x-dreco. Sbg Eq.(9) ad (9) o (9), we oba d( x) ( A) ( B) = lm. dx A B ( x ) ( x ) A B (9) Fg. A chemac dagram of dace bewee O ad A (.e. he fracal coordae of A) a fracal pace, =l/l, whle he fracoal dmeo for he plae are l8/l. The fracal dffereal model parclarly able for decrbg dcoo maer ad he preferred model for decrbg flow or hea codco hrogh poro meda. For ace, he prcple of ma coervao ca be wre he form ρ ( ) ( v) ( w) k ρ k ρ k ρ =, x y z (94) where,, ad are he fracal dmeo of poroy he x, y, ad z dreco repecvely ad k ( =,,) are coa ha are relaed o he fracal dmeo. I parclar, we have k =, whe =. Smlarly, he momem eqao for oe-dmeoal poro flow ca be wre he form

19 8 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - k P + k = + k ( µ k ), x ρ x x x (95) whe = ad k =, Eq.(95) r o o be he clacal oe. The oe-dmeoal hea codco eqao poro meda ca be expreed a T T + k ( µ k ) =, x x (96) where µ he codco coeffce ad k = whe =. A ocllaor wgg a poro medm ca alo be decrbed by fracal dffereal eqao. For example, he ffg eqao wh fracal damp ca be expreed a d d kµ ε. d = (97) d I eay o eablh fracal dffereal eqao for dcoo meda by replacg clacal approach by k / x. / x he. Varaoal Ierao Algorhm for ffereal-dfferece Eqao The dffereal-dfferece model aoher approach o he formlao of dcoo problem ha ca be wre he geeral form k d k d + f ( L,,,, L) =. + + (98) The varaoal erao algorhm are gve a follow k k k d,, + ( ) =, ( ) + ( ) ( ) ( + f )d k ( k )! d k k, + ( ) =, ( ) + ( ) ( ) f d ( k )! k k, + ( ) =, ( ) + ( ) ( ) ( f, f, )d. ( k )! (99). Phycal Uderadg of Fracal ffereal Eqao The dffereal-dfferece model ha recely araced mch aeo becae of ably o exacly

20 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 9 decrbe may real-lfe problem exle egeerg [6], aoechology [7], ad rafed hydroac flow [7]. Aalycal mehod for olvg dffereal-dfferece eqao have herefore bee evely vegaed, wh he exp-fco mehod [8-], he homoopy perrbao mehod [], ad he varaoal erao mehod [-] beg he mo wdely ed. A beer phycal deradg of dffereal-dfferece eqao ca be obaed by coderg he flow hrogh a lace where he coervao of ma reqre[6] d d ρ +ρ ρ + + =, () wh ρ ad beg repecvely he ga dey ad velocy a he -h lace po (ee Fg.). ρ Ga flow dρ d ρ + + Ga flow (-)-h lace po -h lace po (+)-h lace po Fg. Coervao of ma.. Varaoal Ierao Algorhm for Fracal-dfferece eqao ad Fracoal-dfferece eqao We fr wre dow ome fracal-dfferece eqao ad ome fracoal-dfferece eqao. ) Volerra-ype eqao d = ( + ) (a) d = ( ). + (b) ) cree mkdv lace eqao = ( )( ) + (a) = ( )( ). + (b) ) The Hybrd-Lace yem read

21 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - = ( + + β )( ) + (a) = ( + + β )( ) + (b) The fracal-dfferece eqao ca be wre he form d d + f ( L,,,, L) =. + + (4a) Ug he raformao = x, we oba he dffereal-dfferece eqao d + f = (4b) dx ad he varaoal erao algorhm are mlar o Eq.(89). The erao formla for a geeral fracoal-dfferece eqao + f ( L,,,, ) + + L = d (5) are mlar o Eq.(6) or Eq.(64).. Sere Solo, Expoeal Padé Approxma ad Sodal Padé Approxma If we beg wh k ( k ) ( ) = () + () + () + () + L + (), (6)!! k! a ere olo obaed. There, however, a mple way of accelerag he covergece. Coder Eq.(5) of Example aga. ffereag boh de of Eq.(5) wh repec o rel + = (7) o ha, by Eq.(5), he fr-order dffereal eqao red o a ecod-order dffereal eqao whch read =, () =, () =. (8) The erao formla ca be readly obaed, whch read + ( ) = ( ) ( ). d (9)

22 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac Begg wh ( ) = () + () =, erao formla (9) gve 4 ( ) = ( )( ) d = + + ( ) O () whch he ame a ha eeded for 4 erao by he erao formla (8). Padé approxmae echology alo wdely ed o accelerae covergece. Here, we gge he Expoeal Padé Approxma whch wa fr propoed he moograph o-perrbave mehod for rogly olear problem [5]. The Expoeal Padé Approxma ha he form pq ( ) = p = q = c exp d, () a expb where he coeffce a, b,c,d, are deermed from he followg codo: he fr (p +q+) compoe of he expao of he raoal fco pq () a Maclar ere cocde wh he fr (p +q+) compoe of he ere (). Coder a example: I clear ha + =, ( ) =. () ad ( ) =, () () =, (4) ad ha he lm a, we have =±. Accordgly, we chooe a mple Expoeal Padé Approxma of he form Machg he codo e = + e a b. (5) ( ) = b=, (6) ( ) = b(a b) =, (7) we have a = / ad b = whch lead o he rel.5 e = + e. (8)

23 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - The Expoeal Padé Approxma very efl for olary olo. For Eq.(54) of Example 5, we have he -h order approxmae olo ( η) ad he Expoeal Padé Approxma read M = Aη, (9) = p = pq ( η) =. q = c exp dη a expbη () We e ( k ) d dη d ( η) =, k =,,, L, () dη ( k ) ( η) pq ( k ) ( k ) o defy he kow a, b,c,d.. For perodc olo or compaco-lke olo, we gge he Sodal Padé Approxma Coodal Padé Approxma of he form pq p + c ( )( d+ φ ) = ( ) = q. () β + a ( )( b + ϕ ) = pq p + c co( d+ φ ) = ( ) = q β+ a co( b + ϕ ) = () ca alo be corced a mlar maer. A a example, coder he ffg eqao + + =, () A, (). ε = = (4) I erao formla ca be readly obaed, whch read { } + ( ) = ( ) + ( ) ( ) +ε ( ) d. (5) Begg wh ( ) = () + () = A, erao formla (5) yeld

24 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac ( ) = A ( A+ ε A ) = A B, (6) { ε } { ε 4 6 } { ε ε ε 4 ε 6} ( ) = A+ ( ) A B + ( A B ) d = A+ ( ) A B + ( A A B + AB B ) d = A+ ( ) A+ A B(+ A ) + AB B d = A A+ ε A + B + ε A ε AB + εb 4* 6*5 8* ( ) ( ), (7) where B= ( A+ ε A ) /. We e he mple Sodal Padé Approxma he form ( ) = Aco ω. (8) A mple calclao how ha (4) () = B(+ ε A ) = A( + ε A )(+ ε A ) (9) ad () = Aω. () (4) 4 Therefore, by eg (4) (4) () = (), we mmedaely oba / 4 ( A )( A ) () ω= + ε + ε or 4 /4 ( A A ). ω= + ε + ε () To ee he effecvee of he Sodal Padé Approxma, coder fr he cae whe ε. Uder h codo, Eq.() ca be approxmaed a ω ε A 4 = +, ε () whch very cloe o he perrbao rel have ω = + ε A. ow coder he cae whe ε. We 8 ω / 4 = ε A,. ε (4) The approxmae perod ca be wre a

25 4 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - π T = =, ε, (5) / 4 ε A ε A ad he exac oe T= 4 π / d x + εa k x, (6) εa where k =. Sce ( + εa ) 6.74 lm T ex =, ε ε A (7) we have T lm = =.7. ε T 6.74 ex (8) Alhogh he error of h approxmao cold be a large a % whe ε ed o fy, he approxmae rel vald for he whole olo doma < ε <. Obvoly, he accracy ca be mproved by g hgher order Sodal Padé Approxma. 4. Approxmae Solo v Exac Solo There are may rel o he exac olo of olear eqao where he al or bodary codo are o codered. Thee olo are called mahemacal olo becae he phycal cora o he real-world problem ha beg modeled o accoed for. Or ma am, however, o fd olo of he derlyg problem ha afy all he al/bodary codo ha ex. Thee olo, arally, are called he phycal olo of he problem. Coder, for example, he well-kow KdV eqao 6 + =, x x (9) wh a olary olo[4,5] c c = ech ( x c ξ). (4) 4 May mahemacal olo for Eq.(9) cold be fod ha carry o phycal meag ( =, for ace, a exac olo of Eq.(9) ha ha o phycal meag a all). Oher reearcher, o he oher had, beg wh ome very good al codo, ay

26 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 5 c c ( x,) = ech ( x ξ), (4) 4 ad fd ha he codo fac oo good o olve he eqao. For a ravellg olo, for example, we mgh ge a olo of he form c c ( x, ) = ech ( x+ a ξ) (4) 4 where he kow coa a ca be deermed by bg (4) o (9). The al codo may praccal ao, however, may be gve he form of a ere x ( x,) = a x. (4) = Geerally we eek ravelg wave olo he frame ( x, ) = ( ξ ), ξ = x c, (44) o ha po bg (44) o (9), we have c 6 + =, (45) where prme deoe he dffereal wh repec o ξ. Iegrag (45) yeld c + =, (46) where a egral coa ad he varaoal erao algorhm-ii for (46) Begg wh ξ { } + ( ξ ) = ( ξ ) ( ξ ) ( ) + ( ) + d. (47) herefore, whch afe Eq.(4), he -h order approxmae olo ( ξ ) = aξ, (48) = ( ξ) = bξ (49) = ealy obaed. Ug he Expoeal Padé Approxma ow, we ame ha he olary olo ca be expreed he form

27 6 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - o ha by eg q ( ξ ) = exp( aξ ) + exp( aξ ) + p (5) k d d q ( ) ( ) k, k = ~ (5) dξ dξ aξ aξ p k bξ = k = exp( ) + exp( ) + ξ = ξ = a, p ad q ca be defed wh eae. 5. Approxmae Ial/Bodary Codo ad Po Bodary Codo The al or bodary codo are omeme gve oly approxmaely becae he meared daa ally fxed g a ere ch a Eq.(4). Sppoe ha a approxmae al codo ( ) = + ε (5) gve Example above o ha he varaoal erao formla Eq.(8) gve ( ) = + ε (5) ε ( ε ). = + + (54) Ug he Padé Approxma, we ame ha he approxmae olo ca be expreed a b ( ) = + a (55) ad defy a ad b a a= ( + ε ) ad b= + ε. (56) We herefore oba he approxmae olo ( + ε ) ( ) = + ( + ε ) (57) ad he redal (by bg Eq.(57) o Eq.(5)) R( ) = ( ) + ( ). (58) We ow locae R () = (59)

28 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 7 o recover he al codo ad fd, from (59), ha ε =. (6) The more reaoable al codo ad approxmae olo are herefore ad () = (6) ( ) = +, (6) repecvely ad we have recovered he correc al codo ad exac olo by chace. For a geeral bodary vale problem wh approxmae bodary codo ( a) f f L+ = (6) = + ε, ( b) g ε g = + (64) for ome kow f, f, g, g, he varaoal erao mehod ca be ed o oba he -h order approxmae olo ( ) ad he redal R( ) = L + (65) wh b R d a m. (66) Th ca be ed o opmally deerme ε ad ε. I may praccal applcao, however, oly po bodary codo are gve becae daa vale ca oly be meared a he bodary po. Coder, for example, he eqao L( x, ) + ( x, ) =, x [ a, b] (67) where he vale of x ca be meared a dffere po a me =, ay, ( b a) ( x,) = f, x =. (68) M Ug erao formla algorhm-i, we have + ( x, ) = ( x, ) + λ( L ( x, ) + ( x, )) d (69) o ha he -h order approxmae olo ( ) ca be calclaed from he al approxmao r ( x, ) = p + p x+ p x + p x + L + p x. (7) r

29 8 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - The redal alo follow by bg ( ) o Eq.(67) o ha by eg R( x, ) = L + (65) ( x,) = f, = ~ M (66) we have R( x,) =. (67) Remark. We have oly ed r M eqao Eq.(67) ad M + eqao Eq.(66) o olve for p ( = ~ r). 6. Coclo Th paper a elemeary rodco o he correc applcao of he varaoal erao mehod ha reqre oly a bac kowledge of Advaced Calcl o derad. ew cocep ch a he fracoal calcl ad he fracal dervave are explaed hercally ad he dcree lace eqao alo dced. The olo procedre yemacally addreed ad parclar aeo pad hrogho he paper o gvg a ve grap of he mehod ad correc applcao. Approxmae al/bodary codo ad po bodary codo are fr propoed ad dced h paper. Referece [] He JH. Approxmae aalycal olo for eepage flow wh fracoal dervave poro meda, Comper Mehod Appled Mechac ad Egeerg, 67 (-)(998), [] He JH. Varaoal erao mehod - a kd of o-lear aalycal echqe: ome example, Ieraoal Joral of o-lear Mechac, 4 (4)(999) [] He JH. Varaoal erao mehod - ome rece rel ad ew erpreao, Joral of Compaoal ad Appled Mahemac, 7()(7): -7 [4] He JH, W XH. Varaoal erao mehod: ew developme ad applcao, Comper & Mahemac wh Applcao, 54(7): [5] He JH. o-perrbave mehod for rogly olear problem, Berl: derao.de-verlag m Iere GmbH, 6 [6] He JH. Some aympoc mehod for rogly olear eqao, Ieraoal Joral of Moder Phyc B, ()(6) 4-99 [7] He JH. A elemeary rodco o recely developed aympoc mehod ad aomechac exle egeerg, Ieraoal Joral of Moder Phyc B, ()(8): [8] Y YG, We GG, L HX, e al. The ychrozao of hree fracoal-order Lorez chaoc yem, Ieraoal Joral of olear Scece ad mercal Smlao, ()(9): 79-86

30 ISS 76-75: olear Scece Leer A- Mahemac, Phyc ad Mechac 9 [9] She LJ, Tam LM, Lao SK, e al. Paramerc aaly ad mplve ychrozao of fracoal-order ewo-lepk yem, Ieraoal Joral of olear Scece ad mercal Smlao, ()(9) -44 [] X C, W G, Feg JW, e al. Sychrozao bewee wo dffere fracoal-order chaoc yem, Ieraoal Joral of olear Scece ad mercal Smlao, 9()(8): [] ragaec GE. Applcao of a varaoal erao mehod o lear ad olear vcoelac model wh fracoal dervave, Joral of Mahemacal Phyc, 47 (8)(6): Ar. o. 89 [] Odba ZM, Moma S. Applcao of he varaoal erao mehod o olear dffereal eqao of fracoal order, Ieraoal Joral of olear Scece ad mercal Smlao, 7 (): [] Moma S, Odba Z, Homoopy perrbao mehod for olear paral dffereal eqao of fracoal order, Phyc Leer A, 65(7): 45-5 [4] Y YG, L HX. Applcao of he mlage homoopy-perrbao mehod o olve a cla of hyperchaoc yem, Chao Solo ad Fracal, 4(4)(9): -7 [5] Odba ZM. Exac olary olo for vara of he KdV eqao wh fracoal me dervave, Chao Solo ad Fracal, 4()(9): 64-7 [6] Moma S, Odba Z, Hahm I. Algorhm for olear fracoal paral dffereal eqao: a eleco of mercal mehod, Topologcal Mehod olear Aaly, ()(8): -6 [7] Odba Z, Moma S. Applcao of varaoal erao ad homoopy perrbao mehod o fracoal evolo eqao, Topologcal Mehod olear Aaly, ()(8): 7-4 [8] Gaj ZZ, Gaj, Jafar H, e al. Applcao of he homoopy perrbao mehod o copled yem of paral dffereal eqao wh me fracoal dervave, Topologcal Mehod olear Aaly, ()(8): 4-48 [9] Ae I, Yldrm A, Applcao of he varaoal erao mehod o fracoal al-vale problem, Ieraoal Joral of olear Scece ad mercal Smlao, (7)(9): [] Yldrm A, A algorhm for olvg he fracoal olear Schrodger eqao by mea of he homoopy perrbao mehod, Ieraoal Joral of olear Scece ad mercal Smlao, (4)(9): [] a S. Solo of fracoal vbrao eqao by he varaoal erao mehod ad he modfed decompoo mehod, Ieraoal Joral of olear Scece ad mercal Smlao, 9(4)(8): 6-66 [] a S, Gpa PK, Rajeev, A fracoal predaor-prey model ad olo, Ieraoal Joral of olear Scece ad mercal Smlao, (7)(9): [] He JH. Srg heory a cale depede dcoo pace-me, Chao, Solo ad Fracal, 6()(8): [4] Che W, Zhag X, Korošak. Ivegao o fracoal relaxao-ocllao model, Ieraoal Joral of olear Scece ad mercal Smlao, pre [5] Che W. Tme-pace fabrc derlyg aomalo dffo, Chao, Solo ad Fracal, 8(4)(6) [6] W GC, Zhao L, He JH, ffereal-dfferece model for exle egeerg, Chao, Solo & Fracal, 4 (9), [7] He JH, Lee EWM, Varaoal prcple for he dffereal-dfferece yem arg rafed hydroac flow, Phyc Leer A, 7 (9), [8] Zh S, Exp-fco mehod for he hybrd-lace yem, Ieraoal Joral of olear Scece ad mercal Smlao, 8() (7), [9] Zh S, Exp-fco mehod for he dcree mkdv lace, Ieraoal Joral of olear

31 J.H. He, G. C. W, F. A, ol. Sc. Le. A, (): - Scece ad mercal Smlao, 8() (7), [] a CQ, Wag YY, Exac ravellg wave olo of Toda lace eqao obaed va he Exp-fco mehod, Zechrf fr arforchg A, 6(8): [] Yldrm A, Exac olo of olear dffereal-dfferece eqao by He homoopy perrbao mehod, Ieraoal Joral of olear Scece ad mercal Smlao, 9() (8), -4. [] Yldrm A, Applyg He varaoal erao mehod for olvg dffereal-dfferece eqao, Mah. Prob. Eg. 8 (8), Arcle I [] Mokhar R., Varaoal erao mehod for olvg olear dffereal-dfferece eqao, Ieraoal Joral of olear Scece ad mercal Smlao, 9() (8), 9-. [4] He JH., Solo Perrbao. I Meyer, Rober (Ed.) Ecyclopeda of Complexy ad Syem Scece, Vol 9, pp Sprger ew York, 9 [5] He JH., Zh S., Solo ad Compaco. I Meyer, Rober (Ed.) Ecyclopeda of Complexy ad Syem Scece, Vol 9, pp Sprger ew York, 9