onlne Scence Lees A: Mhemcs Physcs nd Mechncs Edos-n-chef Syed Tuseef Mohyud-Dn HITEC Unvesy Tl Cn Pksn syeduseefs@homl.com J-Hun He onl Engneeng Looy of Moden Slk Soochow Unvesy Suzhou Chn hejhun@sud.edu.cn Assoce Edos Pof. Xu Chen College of Mhemcs nd Comuonl Scence Shenzhen Unvesy Shenzhen Eml: uchen@szu.edu.cn K-Teng Wu College of Mhemcs nd Infomon Scences ejng oml Unvesy ejng 6 Schun Chn Xn-We Zhou Demen of Mhemcs Kunmng Unvesy Chn km_wzhou@63.com Edol Bod P. Donld Ael Tny Wesen Unv De Mh Sc Lngley BC VY Y Cnd Eml: DAel@wu.c Eedun Buhe Demen of Mhemcs Hohho Unvesy fo onles 56 Tongdo Rod Hohho 5 Chn. Eml: eedunuhe@63.com Mehd Dehghn Demen of Aled Mhemcs Amk Unvesy of Technology Tehn IRA E-ml : mdehghn@u.c. Engu Fn School of Mhemcs Fudn Unvesy Shngh 33 Chn E-ml: fneg@fudn.edu.cn Jn-Wen Feng College of Mhemcs & Comuonl Scence Shenzhen Unvesy Shenzhen 586 Chn Eml: fengjw@szu.edu.cn eem Fz Moden Tele Insue Donghu Unvesy Shngh Chn Hn L School of MechnclCollege of Scence Inne Mongol Unvesy of Technology Chn lhnm@63.com Jun Lu College of Mhemcs nd Infomon Scence Qujng oml Unvesy Yunnn 655 Chn Yong Lu School of Teles Tnjn Polyechnc Unvesy Tnjn 36 Chn Eml: luyong@ju.edu.cn Hong-C M College of Scence Donghu Unvesy Shngh Chn Eml: hongcm@dhu.edu.cn Jn-Guo ng Se Key Looy of Eloson Scence nd Technology Bejng Insue of Technology Bejng 8 Chn We-Dong Song Se Key Looy of Eloson Scence nd Technology Bejng Insue of Technology Bejng 8 Chn Eml: swdgh@.edu.cn Ahme Yldm Ege Unv De Mh TR-35 Bonov Tukey hme.yldm@ege.edu. Elcn Yusufoglu Dumlun Unvesy Demen of Mhemcs Kuhy Tukey E-ml ddess: eyusufoglu@dumlun.edu. Sheng Zhng Demen of Mhemcs Boh Unvesy Jnzhou Chn zhsheng@yhoo.com.cn Tng Zhong Jshou Unv De Mh Zhngjje 7 Hunn Peoles R Chn zhongng_5@6.com Xo-ng Wng Demen of Conol Scence nd Engneeng Huzhong Unvesy of Scence nd Technology WuhnHue37 Chn E-ml:38868888@39.com Cheng-Bo Zheng Ynshn Unvesy Eml: czheng@ysu.edu.cn
Insucons fo Auhos Puose nd Scoe onlne Scence Lees AISS 76-75 offes fs ulcon fo novel nd fone nonlne scences. I encouges he sumsson of new esech on:. ew nonlne henomen no mhemcl nlyss s needed.. ew mhemcl models dffeenl model fcl model fconl dffeenl model dffeenl-dffeence model fuzzy model sochsc model nd ohes fo vous nonlne olems. 3. ew mehods nlycl mehod numecl mehod omzon mehod sscl mehod llomec mehod nd ohes fo nonlne equons. Genelly one emle should e gven somemes only he soluon ocedue s enough.. ew neeon of nonlne henomenon o new soluon of nonlne equon. 5. ew heoes fo elnon of ny nonlne olems Alhough he jounl concenes mnly on Lees whn 6 ges mn-evew cles whn ges e nved y he Edo nd wll e ulshed fom me o me. Mnusc Sumsson Mnuscs mus no hve een coyghed clssfed o sumed fo ulcon elsewhee. Mnusc should e sen o he edol offce v eml nlsedo@yhoo.cn n oh MS-Wod fle *.doc nsed of *.doc nd df fle. Only ognl conuons whch e no ulshed n ny lnguge nd e no sumed fo ulcon elsewhee e cceed fo ee evew. Mnuscs should e cefully eed fo ccodng o he jounl s emle lese vs www.nonlnescence.com fo deled nfomon Illusons All llusons nd hooghs should e cle sh lck nd whe ns wh good cons on glossy e. Colo llusons e encouged u uhos wll e sked y he ulshe o cove he full coss ncued n colo nng. Coygh Pes e consdeed fo ulcon on he undesndng h hey hve no een sumed o ny ohe ulshe. The sumed woks mus e ognl nd no evously ulshed elsewhee. The coygh of es cceed fo ulcon ecomes he oey of he ulshe nd eoducon of ny of he e elsewhee s no llowed whou he emsson of he ulshe. Auhos who wsh o eoduce llusons whch hve ledy een ulshed elsewhee mus on he wen emsson of he coygh holde. Pge Chges Thee s no ge chge. Bu he uho hs o y colo llusons $ e ge he uho s lso encouged o y voluny ge chge of $3 e e. The coesondng uho wll eceve sof coy of hs e n PDF fle. o sngle off-n of he e wll e ovded he uho cn ook n ssue of he jounl o o nng. Suscon Infomon onlne Scence Lees A ISS 76-75 www.onlnescence.com s ee-evewed jounl ulshed y Asn Acdemc Pulshe Lmed Room 38 Cenl Plz 8 Hou Rod Wnch Hongkong Chn. Suscon Re fo volume ssues: $. Suscon odes should e sen v eml AsnAcdemcPulshe@gml.com.
onlne Scence Lees A Mhemcs Physcs nd Mechncs Vol. 3 os.- COTETS He nsfe nlyss n dvegng nd convegng chnnels S. T. Mohyud-Dn U. Khn. Ahmed Z. A. Zd S.I. U. Khn X.-J. Yng-----------------------6-79 Collz olem nd he homooy euon mehod P. D. Ael--------------------------------------------------------------------------------------------------------7-77 Anlyc soluons fo nonlne Schödnge-ye sysem M.M. K-------------------------------------------------------------------------------------------------------78-9 An dve secl vonl eon mehod fo solvng nonlne nl vlue olems A. Ghon J. Se-djf M. Gchzn------------------------------------------------------------95- Soluon of me-fconl econ dffuson equon y usng homooy nlyss mehod A. A. Rg K. M. Hemd M. S. Mohmed M. A. Ad El Slm--------------------------------5
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 He nsfe nlyss n dvegng nd convegng chnnels Syed Tuseef Mohyud-Dn * Um Khn veed Ahmed Z. A. Zd S.I. U. Khn nd Xo-Jun Yng 3 Demen of Mhemcs Fculy of Scences HITEC Unvesy Tl Cn Pksn COMSATS Insue of Infomon Technology Unvesy Rod Aod Pksn 3 College of Scence Chn Unvesy of Mnng nd Technology Xuzhou Jngsu 8 Chn Asc He nsfe nlyss of Jeffey-Hmel flow s esened; consevon lws long wh smly nsfoms hs een used o deemne he dffeenl equons whch goven he flow. Vonl Ieon Mehod VIM s hen emloyed o solve he e-fomuled sysem of equons. Fo he ske of comson numecl soluon s lso esened. A concee nlyss of he mees ffecng he flow s gven nd ghcl demonson s lso ovded. Jeffey Hmel Flows e ge monce due o he ccl lcon n ndusl nd ologcl scences. Keywods: Jeffey-Hmel flows; He nsfe; convegng nd dvegng chnnels; Lgnge mulle nonlne olems.. Inoducon Afe he foundng nvesgons done y Jeffey nd Hmel [-] ove he flow eween wo non-llel wlls; hs olem hs go consdele enon fom he scenss due o s wde nge of lcons h nclude eosce chemcl cvl envonmenl mechncl o-mechncl engneeng. Rves nd chnnels e lso dsclly smlfed models desced y hs wok. Von n ngle eween he wlls nd ohe flow mees wh he effecs on flud flow hs een dscussed eensvely n [3-6]. To nvesge hese yes of flows n ee wy mny sudes hve een ced ou nd we cn fnd moe flele nd elned mel n leue now dys howeve mny es e sll oen nd eseche e conung o nsec he olem fuhe; he nsfe nlyss s one of hem emeue dsuon effecs he flow ehvo n mny cses so o undesnd s effecs on he flow he nsfe nlyss s essenl. Ths cle sudes he he nsfe effecs on Jeffey Hmel flows. Lws of consevon of momenum nd enegy led us o flow model descng he flow chcescs. Pl dffeenl equons oned e hen conveed n o nonlne odny dffeenl equons usng smly nlyss. The Effecs of dffeen mees on velocy nd emeue ofles e dscussed nd he effecs on oh dvegng nd convegng chnnels e esened. Due o nonlney of he equons ec soluons e unlkely; so elle nd effcen echnque clled Vonl Ieon Mehod VIM [7-] s used o solve he e-fomuled sysem of equons. VIM s song nlycl * Coesondng uho. Eml: syeduseefs@homl.com Syed Tuseef Mohyud-Dn Coygh Asn Acdemc Pulshe Ld. Jounl Homege: www.onlnescence.com
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 6 echnque nd hs een emloyed y sevel eseches n ecen mes o sudy dffeen ye of olems [-6]. The ogno of hs hghly elle scheme s Chnese Mhemcn J- Hun He [8-7] who elzed he el oenl of hs echnque nd conveed no n oe eve scheme. The mn osve feues of hs echnque [8-8] s s smlcy selecon of nl omon comly wh he nonlney of hyscl olems of dvesfed comle nue mnml lcon of negl oeo nd d convegence [8].. Govenng Equons In hs olem we hve consdeed n ncomessle vscous flud flow due o souce o snk he nesecon of wo gd lne wlls ngle eween wlls s. Flow s ssumed o e symmec nd uely dl. Unde hese ssumons velocy feld ke he fom V [ u ] whee u s funcon of oh nd θ. The equon of connuy moon nd enegy n ol coodnes n he sence of ody foces unde mosed ssumons ecome u u u u u u u θ υ ρ θ υ ρ u 3. θ θ ρ u u T T T k T u c Suong oundy condons e w T T u T u U u θ θ θ θ 5 whee w T s he emeue he wll γ nd δ s he velocy sl nd heml sl mees esecvely. Fom he connuy equon we cn we θ θ u f. 6 Usng he dmensonless mees f m f F θ θ T w T β 7
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 63 Elmnng fom Eqs. nd 3 nd usng Eqs.6 nd 7 we ge sysem of nonlne odny dffeenl equon fo he nomlzed velocy ofle F nd emeue ofle β ''' ' ' F Re F F F 8 ' [ F F ]. '' β Ec P 9 Usng Esq. 6 nd 7 he oundy condons 5 wll ecome ' F F ' β β whee Re s Reynolds nume gven y: F f U Dvegen Chnnel: > U > Re. υ υ Convegen Chnnel : < U < And c Ec k U P c T w Reesen Ecke nume nd Pndl nume esecvely. 3. Soluon Pocedue To solve Eqs.9 nd wh ssoced oundy condons usng sndd ocedue fo VIM he coeconl funconl fo he couled sysem s gven y F β n n ''' ' F s F s F s Re F s F s n { } ds '' ~ ~ ' β s β Ec P F F n F β ~ ~ ~ ds whee s nd s F e Lgnge mulles fo velocy nd emeue ofle esecvely. We β cn ge ome Lgnge mulles s s eve fomul 3 cn e wen s s F nd β s! s so h he
6 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs F β n n F n s! ''' ' ' F s F s Re F s F s { } ds. '' ' β n s β Ec P F F ds 3 Usng oundy condons gven n Eq. we cn ge F A β B whee A nd B e consns o e deemned y usng oundy condons F nd β esecvely. e few eons of he soluon e gven y F 5 6 6 A RA A RA β B EP EP A EPA 3 3 6 EP RA EP RA EP A EP A 5 5 5 9 3 3 EP RA EP A EPA R EP R A 6 5 56 5 3 5 3 EP R A EP R A EP RA 58 376 88 5 6 3 EP RA EP A EP R A 8 8 7 3 3 EP RA EP R A 7 3 EP R A 655 8 6
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 65 F A R A 8 3 RA 8 3 R 96 3 R 95 3 6 3 RA 5 8 R A 56 5 A RA 3 3 A R 75 3 RA A RA 5 8 3 A 6 A R A 3 68 3 3 R 3 R A A 7 β 5 B EP EP A EPA 3 3 6 EP RA EP RA EP A EP A 5 5 9 6 EP A EP A EP R A 6 35 6 3 3 5 EPA R EP RA EP RA EP R A 56 8 35 35 8 8 Smlly ohe eons fo he soluon cn e oned.. Resuls nd dscussons Fg. shows he hyscl ehvo of he flow unde vyng ngle n cse of dvegng chnnel clely velocy s oseved o e decesng funcon of ngle. Fg. decs he oucomes of ncesng Reynolds nume wh fed vlue of ngle. I cn e seen fom Fg. h he ncemen n Re esuls n decemen n velocy ofle fo dvegng chnnel. Mmum of he velocy s oseved he cenelne of he chnnel fo oh cses.
66 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs Fg. : Von of F fo dffeen vlues of fo dvegng chnnel. Fg. : Von of F fo dffeen vlues of Re fo dvegng chnnel. Fg. 3: Von of F fo dffeen vlues of fo convegng chnnel. Fg. : Von of F fo dffeen vlues of Re fo convegng chnnel. Fg. 5: Von of β fo dffeen vlues Fg. 6: Von of of fo dvegng chnnel. β fo dffeen vlues of Re fo dvegng chnnel. Fg. 3 nd lluse he nfluence of ngle nd Reynolds nume on convegng chnnel esecvely. Behvo of flow fo chngng nd Re n convegng chnnel s que oose o he ehvo seen n dvegng chnnel. Bu he mmum velocy he cenelne of chnnel es s common fco n oh dvegng nd convegng chnnels. Also fo nd Re velocy s decesng funcon. Effecs of ngle Reynolds nume Re Pndl nume P Ecke nume Ec nd he on emeue ofle fo dvegng chnnel e esened n Fgs. 5-8 esecvely. I c e oseved clely h
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 67 β nceses wh ncese n ll Re P Ec. The mmum vlue s oseved o e he cenelne of he chnnel. Fg. 7: Von of β fo dffeen vlues of P fo dvegng chnnel. Fg. 8: Von of β fo dffeen vlues of Ec fo dvegng chnnel. Fg. 9: Von of β fo dffeen vlues of fo convegng chnnel. Fg. : Von of β fo dffeen vlues of Re fo convegng chnnel. Fg. : Von of β fo dffeen vlues of P fo dvegng chnnel. Fg. : Comson of numecl soluon nd VIM soluon fo β dvegng chnnel. Fo convegng chnnel von n emeue ehvo fo vyng flow mees s shown n Fgs. 9-. Effecs of nd Re e oose o ech ohe n cses of nowng nd wdenng chnnel
68 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs whle P nd Ec demonse sml effecs n oh cses. β whle nceses wh n ncese n P nd Ec. Also mmum vlue of deceses wh n ncese n nd Re β occus he cenelne of he chnnel. I s mon o show h he sees soluons gven n Eq. 7 nd Eq. 8 e convegen. Tle. gves he convegence of hese sees soluons oh fo dvegng nd convegng chnnels. Vlues of unknowns e oned o check he convegence of he soluon. Fom Tle. cn e seen h only 6 h ode soluon s enough o ge convegen soluon. Tle : Convegence of velocy nd emeue ofles fo 5 Re8 P. nd Ec.. Ode of omons Fo Dvegng Fo Convegng Chnnel '' A F B β Chnnel '' A F B β.96.85.97687.3383.7.598.8336.978 3.853.53979.888.937.87.53869.79875.983 5.858.53875.798577.98 6.857.53875.798567.98 7.857.53875.798567.98 umecl Soluon.857.53875.98.98 Sme olem s solved y usng well-known numecl mehod.e. Runge-Ku mehod RK-. Comsons of VIM soluon o numecl soluon fo velocy ofle nd emeue dsuon n oh dvegng nd convegng chnnels e esened n Fgs. 3-6. An ecellen geemen eween he soluons s found. Tle : Comson of numecl soluon nd VIM soluon fo dvegng chnnel fo 5 Re8 P. nd Ec.. F β VIM umecl VIM umecl.53875.53875..9765.9765.538.538..975.975.5355.5355.3.83.83.5398.5398..67593.67593.9868.9868.5.536659.536659.559.559.6.39939.39939.3936.3936.7.776.776.383.383.8.6353.6353.697.697.9.77.77.78.78 Tle. gves comson of VIM soluon nd numecl soluon fo he cse of dvegng chnnel whle n Tle.3 sme vlues e gven fo convegng chnnel. Boh velocy nd emeue ofles gve n ecellen geemen eween wo soluons. Fgs. 3-6 dec ghcl eesenon fo wo soluons. Plne lne gves he vlues fo numecl soluon whle doed lne gves he sme fo VIM soluon.
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 69 Tle 3: Comson of numecl soluon nd VIM soluon fo convegng chnnel fo 5 Re8 P. nd Ec.. F β VIM umecl VIM umecl.98.98..99596.99596.965.965..98375.98375.9.9.3.9679.9679.99.99..935.935.8859.8859.5.86858.86858.8387.8387.6.7889.7889.73.73.7.6733.6733.55.55.8.599.599.63.63.9.9558.9558.. Fg. 3: Comson of numecl soluon nd VIM soluon fo F fo dvegng chnnel. Fg. : Comson of numecl soluon nd VIM soluon fo β fo dvegng chnnel. Fg. 5: Comson of numecl soluon nd VIM soluon fo F fo convegng chnnel. Fg. 6: Comson of numecl soluon nd VIM soluon fo β fo convegng chnnel.
7 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 5. Conclusons Ths e esens sudy of Jeffey Hmel flows wh he nsfe. Resulng nonlne equons e solved y Vonl Ieon Mehod. umecl soluon s lso oned usng Runge-Ku ode fou RK- mehod. Anlycl nd numecl soluons e comed. I s evden fom les nd ghs h ou esuls gee eceonlly well wh he numecl esuls. Besdes fom Fgs. -6 we cn conclude h: I. Fo dvegng chnnel > hee s decese n he velocy wh he ncese n ngle nd Reynolds nume Re. II. The effec of ngle s que oose fo convegng chnnel < o h fo dvegng chnnel. Thee s n ncese n he velocy fo convegng chnnel wh n ncese n nd Re. III. Effec of Pndl nume nd Ecke nume s sme fo dvegng nd convegng chnnel. Thee s n ncese n emeue fo oh he cses. Whle nd Re hve oose effecs fo dvegng nd convegng chnnel. Incese n emeue fo he cse of dvegng chnnel s oseved whle fo convegng chnnel emeue deceses. IV. VIM nd umecl esuls e n eceonl geemen fo oh dvegng nd convegng chnnel. Refeences [] G. B. Jeffey The wo-dmensonl sedy moon of vscous flud Phl. Mg. 6 95 55-65. [] G. Hmel Slfömge Bewgungen Zähe Flüssgkeen Jhese. Deusch.Mh. Veen 5 96 3-6. [3] S. Goldsen Moden Develomens n Flud Mechncs Clendon Pess: Ofod 938. [] L. Rosenhed The sedy wo-dmensonl dl flow of vscous flud eween wo nclned lne wlls Poc. R. Soc. A 75 9 36-67. [5] L.E. Fenkel On he Jeffey-Hmel soluons fo flow eween lne wlls Poc. R. Soc. A 67 96 9-38. [6] K. Bchelo An Inoducon o Flud Dynmcs Cmdge Unvesy Pess 967. [7] J. H. He Vonl eon mehod A knd of nonlne nlycl echnque some emles Inen. J. onln. Mech. 3 999 699-78. [8] J. H. He Vonl eon mehod fo uonomous odny dffeenl sysems Al. Mh. Comu 5-3. [9] J. H. He Vonl eon mehod- Some ecen esuls nd new neeons J. Com. Al. Mh. 7 7 3-7. [] J. H. He The vonl eon mehod fo eghh-ode nl oundy vlue olems Phys. Sc. 76 7 68-68. [] J. H. He An elemeny noducon of ecenly develoed symoc mehods nd nnomechncs n ele engneeng In. J. Mod. Phys. B 8 387-578. [] J. H. He Some symoc mehods fo songly nonlne equon In. J. Mod. Phys. 6-99. [3] J. H. He Vonl eon mehod- Some ecen esuls nd new neeons J. Comu. Al. Mh. 7 7 3-7. [] J. H. He nd X. Wu Vonl eon mehod: ew develomens nd lcons Comu. Mh. Al. 57 88-89. [5] J. H. He Vonl eon mehod A knd of non-lne nlycl echnque some emles Inen. J. onlne Mech. 3 999 699-78.
S. T. Mohyud-Dn U. Khn. Ahmed e l. onlne Sc. Le.A Vol.3 os.- 6-7 7 [6] J. H. He Vonl eon mehod fo uonomous odny dffeenl sysems Al. Mh. Comu. -3 5-3. [7] J. H. He nd X. H. Wu Consucon of soly soluon nd comcon-lke soluon y vonl eon mehod Chos Solons & Fcls 9 6 8-3. [8] M. A. Adou nd A. A. Solmn Vonl eon mehod fo solvng Buge s nd couled Buge s equons J. Com. Al. Mh 8 5 5 5. [9] M. A. Adou nd A. A. Solmn ew lcons of vonl eon mehod Physc D 5 8. [] S. Asndy umecl soluon of non-lne Klen-Godon equons y vonl eon mehod Inenonl Jounl fo umecl Mehods n Engneeng vol. 7 no. 7. 876 88 7. [] M. A. oo nd S. T. Mohyud-Dn Vonl eon echnque fo solvng hghe ode oundy vlue olems Al. Mh. Com. 89 7 99 9. [] M. A. oo K. I. oo S. T. Mohyud-Dn Vonl eon mehod fo solvng sh-ode oundy vlue olems Commun. onlne Sc. ume. Smul 9 57-58. [3] S. Momn nd S. Ausd Alcon of He s vonl eon mehod o Helmholz equon Chos Solons & Fcls 7 6 9. [] M. A. oo nd S. T. Mohyud-Dn Vonl eon mehod fo solvng hghe-ode nonlne oundy vlue olems usng He s olynomls In. J. onln. Sc. um. Sm 9 8-57. [5] S. T. Mohyud-Dn M. A. oo K. I. oo nd M. M. Hossen Vonl eon mehod fo e-fomuled l dffeenl equons In. J. onln. Sc. um. Sm 87-9. [6] S. T. Mohyud-Dn M. A. oo K. I. oo nd M. M. Hossen Soluon of sngul equons y He s vonl eon mehod In. J. onln. Sc. um. Sm 8-86. [7] M. Rfe nd H. Dnl Alcon of he vonl eon mehod o he Whhm-Boe-Ku equons Comu. Mh. Al. 5 7 79-85. [8] M. T nd M. Dehghn On he convegence of He s vonl eon mehod J. Comu. Al. Mh. 7 7-8.
P. D. Ael onlne Sc. Le.A Vol.3 os.- 7-77 Collz olem nd he homooy euon mehod P. Donld Ael Demen of Mhemcl Scences Tny Wesen Unvesy Bsh Colum Cnd VY Y Eml: del@wu.c Asc Collz hs used he olem of endng of em s enchmk fo vous numecl schemes dscussed n hs clsscl wok. In hs communcon we solve he sme olem usng he homooy euon mehod whch mkes use of one uly mee. An omum vlue of he mee s deemned nd s shown h wh hs vlue of he mee suffcenly ccue soluon s oned usng vey few ems n he euon soluon. Keywods: Collz olem endng of em homooy euon mehod uly mee numecl soluon. Inoducon The endng of em s clsscl olem of elscy whch does no dm closed fom nlycl soluon nd heefoe hs een used s enchmk fo vous echnques nlycl nd numecl o comue ome soluons. In cul Collz [] n hs celeed wok on numecl emen of dffeenl equons eensvely ulzed he sd olem o es he effcency nd ccucy of vous schemes he oneeed. Ths ws eseclly welcomed me when he dgl comues hd jus sed mkng n mc n he e of scenfc comung. Collz wok emoded n he ovemenoned monogh led o numeous lgohms whch could e edly mlemened on he comues nd soluons oned of olems whch wee hheo unsolved ecuse of lck of sule oweful comung devces. One of he lgohms dvoced y Collz conssed of noducng n uly mee n he l soluon h ssfes he oundy condons of he olem. The olem ws hen solved fesh wh he l soluon susued n he dffeenl equon. The uly mee ws susequenly deemned y equng h he sze of he newly oned soluon mched wh h of he l soluon. I ws found h he lgohm noduced y Collz no only gve n nlycl soluon n he closed fom u lso h he soluon so oned ws suffcenly ccue fo engneeng uoses. In cul Collz demonsed he usefulness of he lgohm y lyng o he olem of endng of em. He fuhe showed he flely nheen n he lgohm y consdeng s numeous vns nd onng ou he ccucy oned n ech vn. Ael [] ulzed Collz lgohm fo comung n ome soluon of he essue gden dven flow of hd gde flud n oous chnnel. He showed h he ome soluon geed well wh he ec numecl soluon. Fuhe he ome soluon could edc he ehvo of he soluon fo hose mees fo whch numecl soluon ws no fesle. The velocy feld deved y Ael [] usng he Collz mehod ws le ulzed y Ellh e l [3] o deve nlyclly he emeue dsuon n he chnnel fo he flow of hd gde flud. Once gn he nlycl soluon lled vey well wh he numecl soluon. Coygh Asn Acdemc Pulshe Ld. Jounl Homege: www.onlnescence.com
P. D. Ael onlne Sc. Le.A Vol.3 os.- 7-77 73 In he esen communcon we ncooe Collz de o on he soluon of he olem of he endng of em y lyng n he cone of he homooy euon mehod HPM noduced nd develoed y He []. The HPM s one of he moden nlycl mehods whch s hghly vesle nd llows ccue soluons of hghly nonlne hyscl olems n vey few ses. I hs een eensvely nvoked successfully ecenly n vous es of scenfc nd echnologcl monce see fo emle [5-]. As we show esenly n nellgen use of he HPM cn dsclly educe he comuonl ovehed nd yeld comc nlycl eessons fo he esuls h cn e edly evlued usng he hnd-held clculos.. Bendng of Bem Consde em of lengh l lced hozonlly nd clmed he wo ends. A wegh W s ched o he cene of he em whch cuses he em o e defleced veclly. If he cene of he em n he undsued oson s ken s he ogn s ognl oenon long he s of nd he vecl decon hough he ogn long he s of y-s hen cn e shown h y he deflecon dsnce ssfes he followng oundy vlue olem BVP n he nondmensonl fom y y y y. Hee me denoes he devve wh esec o. In he devon of equon whle clculng he cuvue dsnce he sques of he sloes of he em hve een negleced mlyng h equon s vld only fo smll deflecons. Rhe susngly BVP - does no dm closed fom soluon n ems of elemeny funcons. Collz mongs sevel schemes suggesed n ome soluon of he BVP - n he fom y A 3 whch ssfes he oundy condons of he olem. Pogessvely mng moe ccue soluons Collz gve hee eve schemes: y y n n y y y 5 n n n y y y 6 5 n n n he onle ehnd he ls scheme eng h 5 s n vege of. By mchng he vlues of y he zeoh nd fs eons Collz ws le o fnd he vlue of A. He fuhe demonsed h ceful shfng of he y n em cn go long wy n movng he ccucy of he soluon. Moved y Collz suggeson nd he success of he HPM mehod n onng he nlycl soluons of vous olems we hough uden o comne he wo des. As esul n he followng we noduce he Collz vn of he HPM.
7 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 3. Collz Vn of he HPM Rhe hn nsfeng y o 5 y o he lef hnd sde of equon we nsfe n y mulle of y. Thus we ewe equon s y y y 7 whee s sule uly mee whose vlue s o e deemned n n oe mnne n he couse of he soluon. Seng u he HPM fomulon fo equon 7 we we whee s he homooy mee. We seek soluon fo y n he fom y y y 8 n n y y y y y. 9 Susung fo y fom equon 9 no equon 8 nd equng lke owes of on oh sdes we on nd The oundy condons on y e The soluon of he zeoh ode soluon s y y y y y n >. n n n y y n >. n n cos y. 3 cos The hghe ode soluons cn e edly comued fom BVP - hough hey ecome ncesngly nvolved. Fo emle he fs ode soluon s gven y n 9 3 cos 3 6 sn cos y. 6 cos 3 6 cos sn cos The soluon fo y s smly
P. D. Ael onlne Sc. Le.A Vol.3 os.- 7-77 75 y y y y y 5 3 oe h we sll hve he uly mee n he soluon. We need o esme s vlue o ge he fnl soluon. Thee cn e nume of echnques o esme he vlue of. We decded o sele down fo h vlue of whch elvely sekng slzes he vlue of y. Accodngly n Fgue lmng ouselves o he hd ode ems n enson 5 we hve loed y gns. I cn e seen fom he fgue h y emns lmos sony fo.65. Thus we noe h he omum vlue of les somewhee eween he wo choces mde y Collz nmely nd.5. Fgue. Illusng he von of y wh fo he hd ode omon We ge Wh he vlue of deemned omlly we cn now ls he successve omons. y.897cos.65.88659 6 y.5377.9 cos.65.356.8775 sn.65.538.77733 3 7 y 6.5569.937358.3597.83 cos.65 3 5.63388.5.3 sn.65 9.99993 3.3883.685 3 8
76 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs y 53.75575.563 3.839.3373 3 6 8 3 5.85 cos.65.37 9.5796. 5963 7 9.75.6 sn.65 5.6679 6 97.3583 5.9598.63. 9 Accodngly y cn e omed y y 567.5673 6.8 3.6736.35996 6 8 3 5.85 cos.65 8.6593.93.59893.6 7 9.75 sn.65 567.77 6 38.3789 6.88. 63. Tle Illusng he von of y wh fo he fs hee omons. The ec soluon s gven n he ls column. \ 3 Ec.93855.937.935.9353.935.5.93576.9886.99638.99638.99639..9775.963.9393.9393.939.5.9885.99559.939.939.93..89693.8967.8937.8939.8939.5.873936.87965.87693.87695.87696.3.8596.85.8553.8557.8558.35.839.836.838.8386.8387..775.7775.777686.77769.777693.5.7338.7367.73687.7368.73685.5.68698.6967.69.69.693.55.6375.6387.6963.697.697.6.579.585587.5866.58636.58636.65.5978.563.56799.5689.568.7.5537.665.63.63.63.75.387779.39786.3958.3959.3959.8.36585.389.333.333.333.85.99.78.7.78.78.9.68.679.6899.68.68.95.8377.859.855.8553.8553
P. D. Ael onlne Sc. Le.A Vol.3 os.- 7-77 77 In Tle he vlues of y e esened gns he fs hee omons. Also gven n he ls column s he ec vlue of y oned numeclly y umeov s mehod. I cn e seen h hee s n ecellen geemen eween he hee-em soluon oned y he HPM mehod nd he ec numecl soluon.. Concluson In he esen wok we hve ncooed he Collz de of hvng n uly mee n he l soluon whn he fmewok of he HPM fo devng n ome soluon of oundy vlue olem. The olem of he endng of em unde he nfluence of wegh ched he cene of he em s consdeed. By ncoong n uly mee long he lnes of Collz suggeson fo onng n ome soluon evely wh he HPM nd esmng he vlue of uly mee y lookng he sony vlue of he deflecon n he cene n ccue nlycl soluon hs een deved whch s n ecellen geemen wh he ec numecl soluon. Refeences [] Collz L. umecl Temen of Dffeenl Equons Snge Velg Beln ew Yok 966. [] Ael PD. Flow of Thd Gde Flud hough Poous Fl Chnnel In. J. Enggn. Sc. 3 67-85. [3] Ellh R. Ael PD. Hy T. nd Asgh S. Effec of He Tnsfe on Thd Gde Flud n Fl Chnnel In. J. um. Meh. Fluds 63 87-859. [] He JH. Homooy euon echnque. Comu. Mehods Al. Mech. Eng. 78 999 57-6. [5] He JH. on-euve mehods fo songly nonlne olems. Beln: dsseon.de - Velg m Inene GmH 6. [6] He JH. Some Asymoc Mehods fo Songly onlne Equons In. J. Mod. Phys. B 6-99. [7] He JH. ew Ineeon of Homooy Peuon In. J. Mod. Phys 6 56-568. [8] Ael PD. Hy T. nd Asgh S. Homooy Peuon Mehod nd Asymmec Flow ove Sechng Shee In. J. onlne Sc. um. Sm. 7 6 399-6. [9] Asndy S. A umecl Soluon of Blsus Equon y Adomn s Decomoson Mehod nd Comson wh Homooy Peuon Mehod Chos Solons nd Fcls 3 7 57-6. [] Gho QK. Ahmed M. nd Sddqu AM. Alcon of Homooy Peuon Mehod o Squeezng flow of ewonn Flud In. Jou. onlne Sc. ume. Smul. 8 7 79-8. [] Chowdhuy MSH. nd Hshm I. Anlycl Soluons o He Tnsfe Equons y Homooy Peuon Mehod Revsed Phys. Le. A. 37 8-3. [] Ael PD. Eended Homooy Peuon Mehod nd Comuon of Flow s Sechng Shee Com. Mh. Al. 58 9-9. [3] He JH. Wu GC. nd Ausn F. The Vonl Ieon Mehod Whch Should Be Followed onlne Sc. Le. A -3. [] Ael PD. Homooy Peuon Mehod nd he ul Convecon Flow of Thd Gde Flud hough Ccul Tue onlne Sc. Le. A 3-5.
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 Anlyc soluons fo nonlne Schödnge-ye sysem M.M. K E. Ad Aghdm Demen of Engneeng Islmc Azd Unvesy Ald Koul nch Golesn Povnce In Demen of Mechncl Engneeng Unvesy of Mohghegh Adl Adl In Asc An nlyc sudy on nonlne Schödnge-ye sysem clled he genelzed Zkhov sysem s esened n hs e. The G'/G-enson mehod nd he E-funcon mehod e emloyed o consuc ec vellng wve soluons of hs sysem n foms of he hyeolc funcons nd he gonomec funcons. Some of soluons gned fom ech of he oosed mehods hve een comed nd vefed ogehe. MSC o.: 35Q55; 35C7; 35Q6; 37 Keywods: onlne Schödnge equons; Genelzed Zkhov sysem; G'/G-enson mehod; E-funcon mehod; Solon; Peodc soluons. Inoducon In he fs of hs fou lecues on wve mechncs Schödnge woe: Susung fom nd 8 n nd elcng y ψ... we on 8π m ψ E V ψ. h A smlfcon n he olem of he mechncl wves consss n he sence of oundy condons. I hough he le smlfcon fl when I fs cked hese equons. Beng nsuffcenly vesed n mhemcs I could no mgne how oe von fequences could e whou oundy condons. Le on I ecognsed h he moe comlced fom of he coeffcens.e. he ence of Vyz kes chge so o sek of wh s odnly ough ou y oundy condons nmely he selecon of defne vlues of E. D. Ewn Schödnge Fou Lecues on Wve Mechncs. Delveed he Royl Insuon London on 5 7 nd Mch 98 Hee we e: he ove equon ved n Knowledge Sce nd s hee o sy. Who could hve hough hough h some 85 yes le eole e sll hnkng ou solvng fse nd moe ccuely? * Coesondng uho.: Tel.: 98 9 79 36 Eml ddess: k.mehd@gml.com Coygh Asn Acdemc Pulshe Ld. Jounl Homege: www.onlnescence.com
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 79 As you know u o now vous mehods hve een ulzed o eloe dffeen knds of soluons of hyscl models desced y nonlne l dffeenl equons PDEs. In he numecl mehods [] sly nd convegence should e consdeed so s o vod dvegen o noe esuls. Howeve n ecen yes vey of effecve nlycl nd sem-nlycl mehods hve een develoed o e used fo solvng nonlne PDEs such s he vonl eon mehod VIM [3] he homooy euon mehod HPM [5] E-nfny heoy [6] he mee-enson mehod [7] he sne-cosne mehod [8] he nh mehod [8] he homooy nlyss mehod HAM [9] he homogeneous lnce mehod [] he nvese sceng mehod [] nd ohes. Lkewse He nd Wu [3] oosed sgh-fowd nd concse mehod clled he E-funcon mehod o on soly wve soluons eodc soluons nd comc-lke soluons of EEs. The sc de of he E-funcon mehod ws esened n He s monogh []. The mehod wh he d of Mle o Ml hs een successfully led o mny knds of EEs [5-3]. Lely he G'/G-enson mehod fs noduced y Wng e l. [] hs ecome wdely used o sech fo vous ec soluons of EEs [7 8 5-3]. The esuls evel h he wo ecen mehods e oweful echnques fo solvng PDEs n ems of ccucy nd effcency. Ths s mon snce sysems of PDEs hve mny lcons n engneeng. In he necon of lse-lsm he sysem of Zkhov equons lys n mon ole. The Zkhov sysem s ye of nonlne Schödnge equons. Ths sysem ced mny scenss wde nees nd enon. The genelzed Zkhov sysem cn e gven s ψ ψ γ ψ ψ ψυ. c υ υ ψ.. whee he el unknown funcon υ s he flucuon n he on densy ou s equlum vlue nd he comle unknown funcon ψ s he slowly vyng enveloe of hghly osclloy elecon feld. The mees γ nd c e el numes whee c s ooonl o he elecon sound seed. The coeffcen γ s el consn h cn e osve o negve nume. The genel fom of. nd. coves mny genelzed Zkhov sysems sng n vous hyscl lcons. The well-known Zkhov sysem ZS hs een fs deved y Zkhov [3] o desce he necon eween Lngmu dsesve nd on cousc omely nondsesve wves n lsm. Le hs ecome commonly cceed h he ZS s genel model o goven necon of dsesve nd non-dsesve wves. When γ nd hs sysem s educed o he clsscl Zkhov sysem of lsm hyscs. When he sound seed c he soclled susonc lm he Zkhov sysem ecomes he cuclly nonlne Schödnge equon. If we se c nd he genelzed Zkhov sysem ecomes [83] ψ ψ γ ψ ψ ψυ.3 υ υ ψ.. U o now mny numecl nd nlycl mehods hve een oosed o sech some yes of soluons of he ZS. Fo emle Pyne e l. [33] desgned secl mehod fo D ZS. They used unced Foue enson n he scheme o elmne he lsng eos. Glssey [3] esened n enegy-esevng fne dffeence scheme fo he ZS n one dmenson nd oved s
8 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs convegence n [35]. Also Chng e l. [36] led consevve dffeence scheme fo he genelzed Zkhov sysem. Ths scheme cn e mlc o sem-elc deendng on he choce of mee. They lso oved he convegence of he mehod. In wo ohe numecl sudes Bo e l. [37] nd Jn e l. [3] oosed wo me-slng secl echnques o solve he genelzed ZS. In ecen yes sevel nlycl mehods hve een used o fnd ec vellng wve soluons of he genelzed ZS. El-Wkl e l. [8] led he eended nh nd he sne-cosne mehods o look fo ec eodc nd solon soluons of he sysem. Lely Tghzdeh e l. [38] mlemened he nfne sees mehod fo onng ec soluons of Eqs..3 nd.. In nohe sudy Al- Muhmeed nd Adel-Slm [39] used n moved Jco ellc funcon mehod o deve ec soluons of he ZS wh he d of he homogenous lnce ncle. Consdeng ll he ndsensly sgnfcn ssues menoned ove he ojecve of hs e s o nvesge he vellng wve soluons of he genelzed Zkhov sysem Eqs..3 nd. sysemclly y lyng he E-funcon nd he G'/G-enson mehods. Some of soluons oned y ech of he menoned mehods hve een vefed ogehe.. Descon of he wo mehods.. The G'/G-enson mehod Suose h nonlne PDE sy n wo ndeenden vles nd s gven y P u u u u u u. whee P s olynoml n s gumens whch nclude nonlne ems nd he hghes- ode devves. Inoducng wve von defned s u U k c. Eq.. educes o he odny dffeenl equons ODE P U kcu ku k U k c U k cu.3 n whch k nd c e consns o e deemned le. Accodng o he G'/G-enson mehod G' s ssumed h he vellng wve soluon of Eq..3 cn e eessed y olynoml n s G follows: m G' U m. G whee nd fo... m e consns o e deemned le G ssfes secondode lne odny dffeenl equon LODE: d G dg G d d.5
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 8 whee nd e y consns. Usng he genel soluons of Eq..5 we hve < > sn cos cos sn snh cosh cosh snh ' C C C C C C C C G G.6 nd follows fom. nd.5 h ' ' ' ' ' ' ' ' m m G G G G G G G G G G U G G G G G G U.7 nd so on. Hee he me denoes he devve wh esecve o. To deemne u elcly we ke he followng fou ses: Se. Deemne he nege m y susung Eq.. long wh Eq..5 no Eq..3 nd lncng he hghes-ode nonlne ems nd he hghes-ode l devve. Se. Susue Eq.. wh he vlue of m deemned n Se long wh Eq..5 no Eq..3 nd collec ll ems wh he sme ode of G G' ogehe; he lef-hnd sde of Eq..3 s conveed no olynoml n G G'. Then se ech coeffcen of hs olynoml o zeo o deve se of lgec equons fo c k nd fo m.... Se 3. Solve he sysem of lgec equons oned n Se fo c k nd fo m... y use of Mle. Se. Use he esuls oned n he ove ses o deve sees of fundmenl soluons u of Eq..3 deendng on G G' ; snce he soluons of Eq..5 e well known o us we cn on ec soluons of Eq...
8 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs.. The E-funcon mehod Accodng o he clssc E-funcon mehod s ssumed h he soluon of ODE.3 cn e wen s U d n nc g m m f e n e m c f e c... e f... d g e d e g..8 whee c d f nd g e osve neges whch e unknown o e fuhe deemned nd n nd m e unknown consns. 3. The Genelzed Zkhov Sysem 3.. Alcon of he G'/G-enson mehod Le us ssume he vellng wve soluon of Eqs..3 nd. n he fom θ ψ e u υ υ θ q k 3. whee u nd υ e el funcons he consns q nd k e o e deemned le. Susung 3. no Eqs..3 nd. we hve 3 k u uυ q u γ u 3. k υ k u. 3.3 In ode o smlfy ODEs 3. nd 3.3 negng Eq. 3.3 once nd kng negon consn o zeo nd negng yelds u υ c 3. whee c -negon consn. Inseng Eq. 3. no 3. we on k u 3 c q u γ u. 3.5 Accodng o se consdeng he homogeneous lnce eween u nd 3 u n Eq. 3.5 gves
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 83 m 3m 3.6 so h m. 3.7 G' Suose h he soluons of 3.5 cn e eessed y olynoml n G s follows: G' u 3.8 G whee nd e consns whch e unknown o e deemned le. Susung Eq. 3.8 long wh Eq..5 no Eq. 3.5 nd collecng ll ems wh he sme G' G' owe of ogehe he lef-hnd sde of Eq. 3.5 s conveed no olynoml n. G G Equng ech coeffcen of hs olynoml o zeo yelds se of smulneous lgec equons fo k q c nd. Solvng he sysem of lgec equons wh he d of Mle 3 we on he followng: k ± γ c c Ψ q 3.9 whee c nd e y consns nd Ψ s gven s Ψ 8γ 8c 3γ γ 8 c 8γ By usng Eq. 3.9 eesson 3.7 cn e wen s G u 3. G n whch k ± γ nd γ >. Inseng he genel soluon of.6 no Eq. 3. we ge he genelzed vellng wve soluons s follows: Cse A. >.
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 8 ξ ξ ξ ξ snh cosh cosh snh C C C C u 3. whee ± γ ξ nd > γ. Susung Eq. 3. no he nsfomons 3. nd 3. leds o he followng hyeolc funcon soluons of Eqs..3 nd.: snh cosh cosh snh q e C C C C ξ ξ ξ ξ ψ 3. nd c u υ 3.3 n whch c nd e y consns; nd q should e eclled fom Eq. 3.9. ow o on some secl cses of he ove genel soluons we se C ; hen he soluons 3. nd 3.3 educe o nh nh q e q e ± γ ξ ψ 3. nd nh nh c c v ± γ ξ 3.5 nd when C ou genel soluons 3. nd 3.3 led o
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 85 coh coh q e q e ± γ ξ ψ 3.6 nd. coh coh c c v ± γ ξ 3.7 Cse B.. < ξ ξ ξ ξ sn cos cos sn C C C C u 3.8 n whch ± γ ξ nd > γ. Inseng Eq. 3.8 no he nsfomons 3. nd 3. we on he genelzed gonomec funcon soluons of Eqs..3 nd. s follows: sn cos cos sn q e C C C C ξ ξ ξ ξ ψ 3.9 nd c u υ 3. n whch c nd e y consns; nd q should e eclled fom Eq. 3.9. Smlly o deve some secl cses of he ove genel soluons we choose C ; hen he soluons 3.9 nd 3. led o
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 86 n n q e q e ± γ ξ ψ 3. nd n n c c v ± γ ξ 3. nd when C ou genel soluons 3.9 nd 3. educe o co co q e q e ± γ ξ ψ 3.3 nd. co co c c v ± γ ξ 3. 3.. Alcon of he E-funcon mehod In ode o deemne vlues of c nd f n Eq..8 we lnce he lne em of he hghes ode u wh he hghes ode nonlne em 3 u n Eq. 3.5; we hve... ] e[... ] 3 e[ f c c f c u 3.5... ] e[... ] 3 e[ 3 3 f c f c c u 3.6
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 87 whee c e deemned coeffcens only fo smlcy. Blncng he hghes ode of he Efuncon n Eqs. 3.5 nd 3.6 we hve whch leds o he esul nd 3 f c 3c f 3.7 f c. 3.8 Smlly o deemne vlues of d nd g we lnce he lne em of he lowes ode n Eq. 3.5... d e[ 3 g d ] u 3.9... d e[ g ] u 3... d 3 e[ 3d g ] 3.3... d e[ g ] whee d e deemned coeffcens fo smlcy. Blncng he lowes ode of he E-Funcon n Eqs. 3.9 nd 3.3 we hve whch leds o he esul 3g d 3d g 3.3 g d. 3.3 We cn feely choose he vlues of nd q. Fo smlcy we se f c nd d g so Eq..8 educes o e e u. 3.33 e e Susung Eq. 3.33 no Eq. 3.5 nd mkng use of Mle we ve C3 e3 C e C e C A C e C3 e 3 C e 3.3 n whch [ e e ] 3 A 3.35
88 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs nd he C n e coeffcens of e n. Equng o zeo he coeffcens of ll owes of e n yelds se of lgec equons fo k q nd c. Solvng he sysem of lgec equons wh he d of Mle 3 we on he followng. Cse. k k q q c q k γ γ k 3.36 Susung Eq. 3.36 no 3.33 nd nseng he esul no he nsfomons 3.3 nd 3. we on he genelzed soly wve soluons of Eqs..3 nd. u e γ γ e k 3.37 nd q ψ e 3.38 γ γ e e k u u υ c q k 3.39 whee k nd s n y mee whch cn e deemned y nl nd oundy condons. The cse n whch k s n mgny nume he oned soly soluon 3.37 cn e conveed no eodc soluon [5] we we k K whee K s el nume. Usng he nsfomon ζ ζ K e cos ζ sn ζ e cos ζ sn ζ. 3. nd susung Eq.3. no 3.37 yelds u ζ mcos ζ m sn ζ 3.
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 89 γ γ whee m. K If we look fo eodc o comc-lke soluon he mgny n he denomno of Eq. 3. mus vnsh whch eques m K γ γ 3. Solvng fom Eq. 3. we on ± K. 3.3 γ γ Susung Eq. 3.3 no Eq. 3. esuls n eodc soluon whch eds nd u u ζ ± K sec ζ 3. γ γ q ψ ± K sec ζ e 3.5 γ γ K υ sec ζ c 3.6 γ γ whee ζ K nd c q K. Cse. γ k k ± q q γ γq γk c 8 γ γ 3.7 By he sme mnulon s llused n he evous cse we cn fnlly on he genelzed soly wve soluons of Eqs..3 nd. s e e u 3.8
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 9 nd q e e e ± γ γ ψ 3.9 whee ± k γ γ nd > γ γ γ γ γ γ γ υ 8 k q u c u 3.5 whee nd e fee mees; fo emle f we se ± n Eq. 3.8 he soluon educes o [ ] snh cosh u. 3.5 Cse 3. ± q c q q k γ γ 3.5 nd consequenly we ge e e e e u 3.53 whee ± γ nd > γ nd e fee mees; fo emle f we u n 3.53 ou genel soluons educe o nh u 3.5 nh q e ψ 3.55
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 9 nh q c u γ υ. 3.56 If we ke n soluon 3.53 we on coh u 3.57 coh q e ψ 3.58 coh q c u γ υ. 3.59 If we se n 3.5-3.56 nd ecll q fom Eq. 3.9; lso f we u n 3.57-3.59 nd ecll q fom Eq. 3.9 hen cn e esly conveed o he sme soluons 3.-3.7 esecvely. Cse. ± 3 6 8 q c q q k γ γ 3.6 By he sme ocedue s llused ove we cn fnlly on he followng ec soluons: e e e e u 3.6 whee 8 ± γ nd > γ. nd e fee mees; fo emle f we se n he soluon 3.6 we ge nh u 3.6
9 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs q ψ nh e 3.63 nh u υ c q γ If we u n he soluon 3.6 we hve. 3.6 u coh. 3.65 If we se n 3.6-3.65 nd ecll q fom Eq. 3.9 hen cn e esly conveed o he sme soluons 3.-3.7 esecvely. Remk. The cul cses 3.-3.7 3.5-3.59 nd 3.6-3.65 of he genel soluons oned y he oosed mehods hve een comed nd vefed ogehe. Remk. We hve vefed ll he oned soluons y ung hem ck no he ognl equons.3 nd. wh he d of Mle 3.. Conclusons To sum u he uose of he sudy s o show h ec soluons of nonlne Schödnge-ye sysem known s he genelzed Zkhov equons cn e oned y he G'/G-enson mehod nd he E-funcon mehod. Some of he cul cses of he oned genel soluons y he oosed mehods hve een comed nd vefed ogehe. ew nd moe genel ec soluons no oned y he evously vlle mehods e lso found. I cn e seen h he E-funcon mehod yelds moe genel soluons n comson wh he ohe mehod. Ovell he esuls evel h he G'/G-enson nd E-funcon mehods e oweful mhemcl ools o solve nonlne l dffeenl equons PDEs n he ems of ccucy nd effcency. Ths s mon snce sysems of PDEs hve mny lcons n engneeng. Refeences [] E. Ad Aghdm M.M. K Vldon of lowy model usng eemenl esuls n moong condon wh he chnge of comesson o nd engne seed Eemenl Theml nd Flud Scence 3 97 9. [] J.H. He G.C. Wu F. Ausn The vonl eon mehod whch should e followed onlne Scence Lees A 3. [3] J.H. He Vonl eon mehod-some ecen esuls nd new neeons J. Comuonl & Aled Mhemcs 7 7 3-7. [] J.H. He ew neeon of homooy euon mehod In. J. Mod. Phys. B 8 6 56 8.
M.M. K onlne Sc. Le.A Vol.3 os.- 78-9 93 [5] A. Yıldıım S. Sıydın S.T. Mohyud-Dn Homooy Peuon Mehod fo Boundy Lye Flow on Connuous Sechng Sufce onlne Scence Lees A 385 39. [6] J.H. He Alcon of E-nfny heoy o ology Chos Solons & Fcls 8 6 58-89. [7] S.Q. Wng J.H. He onlne oscllo wh dsconnuy y mee-enson mehod Chos Solons & Fcls 35 8 688 69. [8] S.A. EL-Wkl M.A. Adou A. Hend ew eodc nd solon soluons of nonlne evoluon equons Aled mhemcs nd comuon 97 8 97 56. [9] S. Asndy Homooy nlyss mehod fo he Kwh equon onlne Anlyss: Rel Wold Alcons 37-3. [] M.M. K E. Ad Aghdm Unsedy Flow nd He Tnsfe Anlyss on Ssko Flud Theml Scence 3 cceed- n ess DOI REFERECE:.98/TSCI33K [] E. Fn H. Zhng A noe on he homogeneous lnce mehod Phys. Le. A 6 998 3-6. [] M.J. Alowz H. Segu Solons nd nvese sceng nsfom SIAM Phldelh 98. [3] J.H. He X.H. Wu E-funcon mehod fo nonlne wve equons Chos Solons & Fcls 33 6 7 8. [] J.H. He on-peuve Mehods fo Songly onlne Polems Dsseon. de-velg m Inene GmH Beln 6. [5] X.H. Wu J.H. He Soly soluons eodc soluons nd comcon-lke soluons usng he E-funcon mehod Comues nd Mhemcs wh Alcons 5 7 966-986. [6] S. Zhng E-funcon Mehod: Soly Peodc nd Ronl Wve Soluons of onlne Evoluon Equons onlne Scence Lees A 3-6. [7] M.M. K Anlyc soluons fo genelzed foms of he nonlne he conducon equon onlne Anlyss: Rel Wold Alcons 68 69. [8] M.M. K Anlyc soluons fo nonlne vn of he -dmensonl Cmss Holm KP equon Ausln Jounl of Bsc nd Aled Scences 5 566-577. [9] M.M. K A. Khjeh ew elc soluons fo he Vkhnenko nd genelzed fom of he nonlne he conducon equons v E-funcon mehod In. J. onlne Scences & umecl Smulon 9 37-38. [] M.M. K e l. Modfed Kudyshov mehod fo fndng ec soly wve soluons of hghe-ode nonlne equons Mh. Mehods Al. Sc. 3 3-9. [] A. Bohnf M.M. K ew eodc nd solon soluons y lcon of E-funcon mehod fo nonlne evoluon equons J. Comuonl & Aled Mhemcs 9 9 58-67. [] A. Bohnf M.M. K M. Vhd Lsem ew eodc nd solon wve soluons fo he genelzed Zkhov sysem nd -dmensonl zhnk ovkov Veselov sysem Chos Solons & Fcls 9 66 65. [3] A. Bohnf M.M. K Solon nd Peodc soluons fo 3-dmensonl nonlne evoluon equons y E-funcon mehod Alcons nd Aled Mhemcs: Inenonl Jounl AAM 5 59-69. [] M. Wng X. L J. Zhng The G'/G-enson mehod nd velng wve soluons of nonlne evoluon equons n mhemcl hyscs Phys. Le. A 37 8 7 3. [5] S. Zhng W. Wng J. Tong A genelzed G'/G-enson mehod nd s lcon o he -dmensonl Boe-Ku equons Al. Mh. Comu. 9 9 399-. [6] H.A. Zedn ew clsses of soluons fo sysem of l dffeenl equons y G'/G- enson mehod onlne Scence Lees A 3 9 38. [7] M.M. K A. Bohnf R. Az Alcon of G'/G-enson mehod o Regulzed Long Wve RLW equon Comues nd Mhemcs wh Alcons 68 7.
9 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs [8] M.M. K R. Bghezdeh Alcon of G'/G-Enson Mehod o onlne Vns of he -Dmensonl Cmss-Holm-KP Equon Mddle-Es Jounl of Scenfc Resech 95 6-6. [9] A. Bohnf A. Zm M.M. K Ec Tvellng Wve Soluons fo he Genelzed Shllow We Wve GSWW Equon Mddle-Es Jounl of Scenfc Resech 3 3-35. [3] R. Az The G'/G-enson mehod fo Tzzéc ye nonlne evoluon equons Mhemcl nd Comue Modellng 5 83-85. [3] V.E. Zkhov Zh. Eks. Teo. Fz. 6 97 75 [Sov. Phys. JETP 35 97 98]. [3] S. Jn P.A. Mkowch C. Zheng umecl smulon of Genelzed Zkhov sysem Jounl of Comuonl Physcs 376 95. [33] G.L. Pyne D.R. cholson R.M. Downe umecl soluon of he Zkhov sysem J. Comu. Phys. 5 983 8 98. [3] R. Glssey Aome soluons o he Zkhov equons v fne dffeences J. Comu. Phys. 99 377. [35] R. Glssey Convegence of n enegy-esevng scheme fo he Zkhov equons n one sce dmenson Mh. Comu. 58 99 83. [36] Q. Chng H. Jng A consevve dffeence scheme fo he Zkhov equons J. Comu. Phys. 3 99 39. [37] W. Bo F.F. Sun G.W. We umecl mehods fo he genelzed Zkhov sysem J. Comu. Phys. 9 3 8. [38]. Tghzdeh M. Mzzdeh F. Fhooz Ec Soluons of he Genelzed- Zkhov GZ Equon y he Infne Sees Mehod Alcons nd Aled Mhemcs: An Inenonl Jounl AAM 5 7 77. [39] Z.I.A. Al-Muhmeed E.A.B. Adel-Slm Genelzed Jco Ellc Funcon Soluon o Clss of onlne Schödnge-Tye Equons Mhemcl Polems n Engneeng Acle ID 575679 do:.55//575679.
A. Ghon J. Se-djf M. Gchzn onlne Sc. Le.A Vol.3 os.- 95- An dve secl vonl eon mehod fo solvng nonlne nl vlue olems A. Ghon * J. Se-djf M. Gchzn Demen of Aled Mhemcs School of Mhemcl Scences Fedows Unvesy of Mshhd P. O. Bo 59 Mshhd 9775 In E-ml ddess: s.ghon@yhoo.com A. Ghon Asc The successve eons of he vonl eon mehod VIM n solvng nonlne nl vlue olems IVPs my e vey comle so h he esulng negls n s eve elon my no e efomed nlyclly. In ode o comleely elmne hs escon n hs e n dve secl VIM s oosed fo solvng he IVPs. The oned esuls hee demonse ecellen efomnce of hs lgohm. Keywods: Vonl eon mehod; Secl collocon; onlne nl vlue olems Inoducon In mny ccumsnces nonlne odny dffeenl equons wll model he dynmcl ehvou of cen mechncl sysems wheey ec soluons o closed fom of nlycl soluons e vey dffcul o on. In genel we hve o ely on numecl negon some cul nsfomons lnezon o dscezon n ode o on he ome soluons. Also hee hs een much enon devoed o sech fo ee nd moe effcen mehods fo deemnng soluons ome o ec nlycl o numecl o hese knds of nonlne equons see e.g. [-] nd he efeences heen. The vonl eon mehod VIM lys n mon ole n ecen eseches. Ths mehod s oosed y He [5-] s modfcon of genel Lgnge mulle mehod []. I hs een shown h hs ocedue s oweful ool fo solvng vous knds of olems e.g. see [-5]. We eleve h n esy-o-use lgohm cn e oosed fo solvng he IVPs. Theefoe he segy h wll e usued n hs wok ess mnly on eslshng n effecve lgohm sed on he VIM nd he secl collocon scheme e.g. see [6] fo onng hghly ccue ome soluon of he IVPs. The emle nlyzed n hs e shows h he develoed lgohm s vey effecve o solve he IVPs s comed o he Ml ode5 solve. Legende omon Ths secon s devoed o noducng Legende funcons nd eessng some sc oees of hem. Le P e he sndd Legende olynoml of degee. The shfed Legende olynomls o Λ [ T ] e defned y P T Coygh Asn Acdemc Pulshe Ld. Jounl Homege: www.onlnescence.com
96 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs d P T P! T d T... nd sses he ecusve elon PT PT PT T. The se of he shfed Legende olynomls P T s comlee L Λ -ohogonl sysem n Λ.e. Λ P T PT j d δ T j 3 whee δ s he Konecke symol. Thus fo ny v L Λ we hve h j v vt PT vt v PT d. T Λ We denoe y j j he nodes of he sndd Legende-Guss-Loo LGL neolon on he nevl [ ]. The coesondng weghs e w j j. The nodes of T he shfed LGL neolon on Λ e denoed y T j j j. The T coesondng weghs e wt j wj j. Le π Λ e he se of olynomls of degee mos. Also le u v T nd v e he nne oduc nd he nom of sce L Λ T esecvely. The shfed LGL neolon I v π s deemned y I T v T j T j We cn end I T v s whee T Λ v j. 5 T I v v P 6 v T T IT v PT T v PT T. 7 T T T ow le us consde model IVP s follwos wh hs ssumon h he olem hs he unque soluon on he nevl Λ :
A. Ghon J. Se-djf M. Gchzn onlne Sc. Le.A Vol.3 os.- 95-97 d u f u d u u Λ 8 d whee u nd f e ssumed o e connuous n Λ. The Legende seudosecl mehod d sed on he LGL ons fo solvng 8 s o seek u π Λ such h d u T k d u u. f T k u T k k 9 ow we need n eve mehod o solve he olem 9. In hs wok we shll ly he elc VIM n n dve mnne. 3 The secl vonl eon mehod The de of he VIM s vey smle nd sghfowd. To eln he sc de of he VIM fs we consde Eq. 9 s: L [ u ] [ u ] g whee L wh he oey L v when v denoes he lne oeo wh esec o u s nonlne oeo wh esec o u nd g s he souce em. Accodng o [5] we hen consuc he followng elc VIM fo Eq. : L[ u u ] A[ u ] wh he nl condon n u n n n n u whee u s he nl guess he nl guess cn e feely found fom solvng s coesondng lne equon L [ u ] o L [ u ] g nd he susc n denoes he n -h eon nd d A[ un ] L[ un ] [ un ] g un f un. 3 d Accodngly he successve omons u n n of he VIM wll e edly oned y choosng ll he ove-menoned mees. Consequenly he ec soluon my e oned y usng
98 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs u lm u. n n ow we e n oson o consuc he numecl lgohm sed on he VIM fo 9. To do hs n vew of nd we deve he followng sle scheme fo solvng 9 whch s clled he secl VIM L[ un T k u un u n T k n. ] A[ u n T k ] k 5 As oned ou efoe hs s n elc och nd unde cen condons hs unque soluon. Hee n ode o decly clcule he unknown u T k we gve smle mlemenon y endng u y he shfed Legende olynomls whch leds o sle lgohm. The olynoml j u j PT j neoles he ons T j u j j h s T u. The vlue of he neolng olynoml's h devve he nodes s P T j T j whee he k j h elemen of he dffeenon m D s T D u. These dffeenon mces e suded n [67] whee lso elc eessons fo hem e gven. Genelly n ode o solve 9 usng 5 he neolng olynoml s equed o ssfy he equon he neo nodes m :.e. T m u m Im: u whee I m : denoes he m ow of he deny m nd he devve vlue s T m Dm: u whee D m: denoes he m ow of he dffeenon m D. The nl condon h nvolves he vlue of he neolng olynoml cn e hndled y usng he fomul T u I: u whee I : denoes he fs ow of he deny m. Susung he ove m elons he soluon veco u cn e found y solvng he followng m equon: Lu Lun A[ un ] 6 n whee he m L nd he veco A[ u n ] e s elow: I : I: u u L A[ un ]. 7 Lm: Lm: u dg [ u ] g T m ow n esy nd elle wy of ensung he vldy of omons fo lge s o deemne he soluon n sequence of nevls of whch e sujec o connuy condons he end s ons of ech nevl. Thus defne he followng se of dsjon nevls ] wh I Δ s [ s s K s [ s s ] [ T Δ s Δs s s s... K whee K T nd ] Assume h he ons s... s e he LGL nodes on he nevl I s Δ wh s s s nd. We heefoe hve he followng ecewse-secl VIM PSV fo solvng 9: s s
A. Ghon J. Se-djf M. Gchzn onlne Sc. Le.A Vol.3 os.- 95-99 L u Lu [ u ] 8 s n s n A s s n whee L s noed s n 7 nd m u denoes he ls comonen of s n u s n m I : us u s nm As [ u s n]. 9 Lm: us dg [ us ] g s m Thus sng fom he nl omon u we cn use he ecuence fomul 8 o successvely on decly u n fo n. I should e emhszed h n sml wy he PSV lgohm s lso lcle o sysems of odny dffeenl equons. An dve segy In hs secon he followng dve segy [] s oosed fo he PSV lgohm whch we summze s APSV. Ths echnque smles comuon nd sves me nd wok s wll e oseved le n hs e. Le u n e he soluon of he PSV fomul wh he se sze Δ n nd u ~ n he soluon wh he se sze Δ n /. Tkng he dffeence of u n nd ~ u n he locl eo esmo of u n.e. ~ Es u n un s defned. Ths vlue s n esmon of he mn of he locl dscezon eo of he mehod. Addonlly le k e he dmenson of he ODE sysem nd Aol nd Rol he use-secfed solue nd elve eo olences. The olences occung n ech se e denoed y Tol Aol Rol. u.... Tkng n k k Es e k Tol s mesue we fnd n oml se sze Δ o y comng e o. Thus we on he oml se sze s Δo Δn. e whee fo e fce we use nd n n m{ n mn} nd fo e > fce nd n mn{ n m}. Ths s of couse no he es choce fo ll olems. The new se sze n new Δn Δn.mn fcmm fcmn fc. e Δ s oned y usng e wh n s ode of olynoml nsed of ode of conssency. The negon of he gowh fcos fc nd fc mn o he elon evens fo oo lge se m
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs ncese nd conue o he sfey of he code. Addonlly usng he sfey fco fc mkes sue h e wll e cceed n he ne se wh hgh oly. The se s cceed n cse h e fce ohewse s ejeced nd hen he ocedue s edone. In oh cses he new soluon s comued wh Δ new s se sze. 5 An llusve emle In ode o lluse he effcency of he APSV desced n hs e we esen one emle whch wll efom n Ml 7 wh doule ecson n Tosh A8 Wndows XP Pofessonl InelR CoeTM Duo Pocesso T7. The foced Duffng equon [8]: 3 c Acos models he dmed moon of mss on nonlne sng dven y eodc focng funcon of mlude A. We ke nl condons. 5 nd solve he equon when c.5 nd A. 3. The chnge of vles u nd u nsfoms he second-ode IVP of o he followng nonlne sysem of he fs-ode dffeenl equons: u u u.5 3 u u cu u Acos u. 3 In ode o show he ecency of he ove dve mechnsm conollng he uncon eo we solve he ove sysem usng he ove-menoned APSV lgohm. In he fmewok of he APSV lgohm fo smlcy we now u he lne oeo L u' he nonlne oeo f u nd he souce em g. The numecl esuls cn e oseved n Tle. Tle The numecl esuls oned fom solvng he ove nvesged emle usng he APSV lgohm fo mn 5 m 5 fc e. fc. 9 fc mn. 5 nd fc m. 5. Algohm T Aol Rol o. of ses CPU me s APSV 3 3 76.5 ODE5 3 3 7889.68 APSV 5 3.3 ODE5 3737 9.8 APSV 65 8. ODE5 96 67.3 In Tle we ls he cosed nume of ses leled s o. of ses fo some dffeen vlues of L Aol nd Rol nd he coesondng cosed CPU elsed me leled s CPU me. We cn oseve fom Tle h he APSV lgohm coss oh less comuonl me nd vey smlle ses hn he Ml ode5 solve wh he sme olences fo hs secfc equon. 6 Concluson
A. Ghon J. Se-djf M. Gchzn onlne Sc. Le.A Vol.3 os.- 95- In hs e we oosed n dve ecewse secl vonl eon mehod APSV. As shown n hs e he develoed echnque mkes smle comuon nd sves me nd wok. The numecl esuls demonse he effcency of he suggesed scheme. They lso show h he oosed mehod coss oh lesse comuonl me nd vey smlle ses hn he Ml ode5 solve. Alhough we only consdeed model olem n hs sudy he develoed APSV lgohm s lcle fuhe o mny ohe olems. Refeences [] B. Guo Z. Wng Legende-guss collocon mehods fo odny dffeenl equons Adv. Comu. Mh. 3 9 9-8. [] E. He G. Wnne Solvng odny dffeenl equons. II Snge-Velg Beln 99. [3] U. Asche L. Pezold Comue Mehods fo Odny Dffeenl Equons nd Dffeenl-Algec Equons SIAM Phldelh PA 997. [] A. Ghon S. Momn An effecve vonl eon lgohm fo solvng Rcc dffeenl equons Al. Mh. Le. 3 9-97. [5] J.H. He Some symoc mehods fo songly nonlne equons Inen. J. Moden Phys. B 6-99. [6] J.H. He Vonl eon mehod fo dely dffeenl equons Comm. on-lne. Sc. ume. Smulon 997 35-36. [7] J.H. He Aome soluon of nonlne dffeenl equons wh convoluon oduc non-lnees Comu. Mehods. Al. Mech. Engng. 67 998 69-73. [8] J.H. He Aome nlycl soluon fo seege flow wh fconl devves n oous med Comu. Mehods. Al. Mech. Engng. 67 998 57-68. [9] J.H. He Vonl eon mehod- knd of non-lne nlycl echnque: some emles Inen. J. on-lne Mech. 3 999 699-78. [] J.H. He Vonl eon mehod- some ecen esuls nd new neeons J. Comu. Al. Mh. 7 7 3-7. [] M. Inoku H. Sekne T. Mu Genel use of he Lgnge mulle n non-lne mhemcl hyscs n: Vonl Mehods n he Mechncs of Solds Pegmon Pess ewyok 978. [] S. Momn S. Ausd Alcon of He s vonl eon mehod o Helmholz equon Chos Solons Fcls 7 6 9-3. [3]. Heşnu V. Mnc A modfed vonl eon mehod fo songly nonlne olems onlne Sc. Le. A 83-9. [] J.H. He G.C. Wu F. Ausn The vonl eon mehod whch should e followed onlne Sc. Le. A -3. [5] A. Ghon J. Se-djf Convegence of He s vonl eon mehod fo nonlne oscllos onlne Sc. Le. A l 379-38. [6] L. Tefehen Secl Mehods n MATLAB SIAM Phldelh PA. [7] J. Wedemn S. Reddy A ml dffeenon m sue ACM T. Mh. Sofwe 6 65-59. [8] J. Dvd Logn A Fs Couse n Dffeenl Equons Snge ew Yok 6.
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 Soluon of me-fconl econ dffuson equon y usng homooy nlyss mehod A. A. Rg K. M. Hemd Mohmed S. Mohmed nd M. A. Ad El Slm Mhemcs Demen Fculy of Scence Al-Azh Unvesy s Cy Co Egy Mhemcs Demen Fculy of Scence Tf Unvesy Tf Sud A Eml:m_s_mohmed@yhoo.com Asc. The Homooy nlyss mehod HAM s used o on n ome soluon of he nonlne me fconl econ - dffuson equon. Convegence of he soluon nd effecs fo he mehod e dscussed whn comng he oned esuls wh ec soluon of he coesondng nonlne l dffeenl equon whch ndced h he oosed mehod s vey effecve nd smle. I lso suggess h oh he Homooy euon mehod HPM Adomn decomoson mehod ADM nd vonl eon mehod VIM e secl cses of he HAM. The HAM conns cen uly mee whch ovdes us wh smle wy o djus nd conol he convegence egon nd e of convegence of he sees soluon. Keywods: Homooy nlyss mehod Adomn mehod homooy euon mehod vonl eon mehod uly mee fconl clculus - Inoducon. onlne l dffeenl equons PDEs e encouneed n such vous felds s hyscs chemsy ology mhemcs nd engneeng. Mos nonlne models of el lfe olems e sll vey dffcul o solve ehe numeclly o heoeclly. In hs e we consde he nonlne fconl l dffeenl equon fconl econ - dffuson equon of genel fom Whee s he dffuson coeffcen nd s nonlne funcon eesenng econ knecs. I s neesng o oseve h fo educes o he mefconl Fshe equon whch ws ognlly oosed y Fshe [ ] s model fo he sl nd emol ogon of vle gene n n nfne medum. If we se gves se o he me-fconl Fzhugh gumo equon whch s n mon nonlne econ-dffuson equon nd led o model he nsmsson of neve mulses [ ] lso used n ology nd he e of oulon genecs n ccu heoy [5]. Whle he Fzhugh gumo equon educes o he el ewell Whehed equon. Coygh Asn Acdemc Pulshe Ld. Jounl Homege: www.onlnescence.com
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 3 - Bsc defnons. In hs secon we gve some defnons nd oees of he fconl clculus. The fconl clculus s nme fo he heoy of negls nd devves of y ode. Vous defnons of fconl negon nd dffeenon e found n [-] such s Gunwld-Lenkov's defnon Remnn-Louvlle defnon nd Cuo's defnon nd genelzed funcon och. Fo he uose of hs e he Cuo's defnon of he fconl dffeenon wll e used kng he dvnge of Cuo's och h he nl condons fo fconl dffeenl equon wh Cuo's devves ke on he donl fom s fo nege-ode dffeenl equon. Defnon. Ael funcon s sd o e n he sce f hee ess el nume such h whee nd s sd o e n he sce f nd only f Defnon. The Remnn-Louvlle fconl negl oeo of ode of funcon s defned s s he well known gmm funcon. Some of he need hee e s follows: oees of he oeo whch we wll 3 Defnon.3 The fconl devve of n he Cuo's sense s defned s follows The followng e wo sc oees of Cuo's fconl Devve [] Le hen s well defned nd Le nd hen
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs 3- The Homooy nlyss mehod HAM. The HAM [] s led o he nonlne homogeneous fconl econ - dffuson equon wh nl condons. Then he suded dffeenl equon s consdeed s Whee s nonlne oeo fo hs olem nd denoe ndeenden vles s n unknown funcon. By mens of he HAM one fs consucs he zeo-ode defomon equon Whee s he emeddng mee s n uly mee s n uly funcon s n uly lne oeo s n nl guess. Ovously when nd holds h Lo [-] ended n Tylo sees wh esec o he emeddng mee s follows: Whee Assume h he uly lne oeo he nl guess he uly mee nd he uly funcon e seleced such h he sees s convegen hen we hve fom Le us defne he veco Dffeenng mes wh esec o hen seng nd dvdng y h he mh-ode defomon equon Whee
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 5 And The mh-ode defomon Eq. ecomes lne nd cn e esly solved eseclly y mens of symolc comuon sofwe such s Mhemc Mle Ml. -Tme- fconl econ - dffuson equon of he fshe ye. Alcon Fsly we egn y Fshe's equon. Wh nl condon Accodng o he HAM we choose he uly oeo s Wh oey whee s consn. We defne nonlne oeo s In ode o oey he ule of soluon eesson nd he ule of he coeffcen eodcy [6] he uly funcon cn e deemned unquely nd ow he soluon of he mh-ode defomon equons fo ecomes So he fs hee ems of he soluon e
6 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs Then we cn conclude h Usng he Geomec sees s [6] when he sees ends o nfny nd soluon ecomes ndeenden of nd kes he followng fom mus e less hn he A he soluon s he sme s [9] nd [] whch s closed fom wh he ec soluon of he equon. The numecl vlues of he soluon wh dffeen vlues of nd ndeed of he ec soluon lluse n Tle [] see lso Fg []. HAM ec soluon HAM HAM
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 7 Fg [] dffeen vlues of Alcon Consde he Fshe's equon. Wh nl condon Smlly choosng Then So fs hee ems of he sees e
8 ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs Then we cn conclude h As he sees ends o nfny usng Geomec sees s [6] whee mus e less hn he soluon ecomes ndeenden of n he followng fom
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 9 A he soluon s he sme s [9] nd [] whch s closed fom wh he ec soluon of he equon. A he soluon s he sme s [7] nd when he soluon s he sme s [8]. Tle [] lluses he numecl vlues of he soluon wh dffeen vlues of see lso Fg [] [3]. HAM ec soluon HAM HAM Fg [] ec soluon nd ome soluon Fg [3] ec soluon nd ome soluon 5- onlne dffuson equon of he Fshe ye. The equon hs fom
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs Wh nl condon Smlly on choosng Then So fs hee ems of sees e s follows
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 Then we cn conclude h Usng he Geomec sees s [6] such h nfny he soluon ecomes ndeenden of mus e less hn when he sees ends o nd kes he followng fom A he soluon s he sme s [9] nd [] whch s closed fom wh he ec soluon of he equon. A he soluon s he sme wh [7] [8]. Tle [3] lluses he numecl vlues of he soluon wh dffeen vlues of Fg [] [5]. lso see
ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs HAM ec soluon HAM HAM Fg [] ec soluon nd ome soluon Fg [5] ec soluon nd ome soluon 6- The genelzed Fshe equon. The genel fom of he suded equon hs he fom Wh nl condon Smlly choose And So fs hee ems of soluon e s follows
A. A. Rg K. M. Hemd M. S. Mohmed e l. onlne Sc. Le.A Vol.3 os.- -5 3 Then we cn conclude h As he sees ends o nfny usng Geomec sees s [6] whee mus e less hn he soluon ecomes ndeenden of n he followng fom
A ISS 76-75: onlne Scence Lees A- Mhemcs Physcs nd Mechncs he soluon s he sme s [9] nd [] whch s closed fom wh he ec soluon of he equon. A he soluon s he sme wh [7]. Tle [] lluses he numecl vlues of he soluon wh dffeen vlues of nd. HAM ec soluon HAM HAM Fg [6] ec soluon nd ome soluon Fg [7] ec soluon nd ome soluon Concluson In hs e he Homooy nlyss mehod HAM s led o on he soluon of mefconl econ dffuson equon. The esuls show h HAM s oweful nd effcen echnques n fndng ec nd ome soluons fo nonlne fconl l dffeenl equons. The HAM ovdes us wh convenen wy o conol he convegence of omon sees whch s fundmenl qulve dffeence n nlyss eween HAM nd ohe mehod. Thus he uly mee lys n mon ole whn he fme of he HAM. Mhemc hs een used fo comuons n hs e. Refeences [] S.J. Lo. Beyond euon: noducon o he Homooy nlyss mehod CRC Pess Boc Ron: Chmn& Hll 3. [] RA. Fshe. The wve of dvnce of dvngeous genes. Ann Eugene.