Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI * Sda Agss REJU * Mahmacs/Compr Scc Dparm Fdral Uvrs of Tcholog Ma 9 Ngr Sa Ngra; * corrspodg ahor; Naoal Op Uvrs of Ngra Vcora Islad Lagos Ngra droomawar@ahoo.com srj@o.d.g Absrac Ths sd prss a w vara aalss ad smlaos of h wodmsoal rgd wav qao rmarkabl dffr from h dffso qaos sdd arlr sdd. Th objcv fcoal ad h damcal rgd wav ar pald o form a fco calld h Hamloa fco. From hs fco w obad h cssar codos for h opmal solos sg h mamm prcpl. B applg h Forr solo o h frs ordr dffral qao h aalcal solos for h sa ad corol ar obad. Th solos ar smlad o gv vsal phscal rprao of h wavs ad h mrcal vals. Kwords Mamm prcpl Hamloa fco Forr srs rgd wav qao ad Smlas Irodco Cosdr a opmao problm: m J m{ [ ddd hp://ljp.acadmcdrc.org 4
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU 44 sbjc o h damcal cosra rgd wav qao: Th cosrad qao s formlad b agmg qaos ad b palg h qaos sg h pal fco µ fco o b drmd War ad Rj 7: [ m{ m J }ddd µ Th Hamloa fco Th cosrad formlao of qaos ad s achvd b palg h cosrad qao o oba h Hamloa fco as Sgh al 4. Th Hamloa fco s dfd: H T 4 whr T T s coos pal fco Th mamm prcpl Th frs ordr cssar codos for opmal; also kow as h mamm prcpl o h Hamloa fco 4 s qvall prssd as: H 5 W o b dfo ak o Sgh al. bd h drvav wh rspc o h pal : H H 6
Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 also b dfo: H 7 H 8 mplg ha: 9 Dffrag qao 7 wh rspc o -varabl lds: Eqag qaos 6 ad rsl a frs ordr paral dffral qao: Eqao shows ha h ra of chag of opmal corol s h opmal sa. Sppos ha qao has a Forr srs solo as proposd b Dchaa al. h: ss α ss B from qao w ms hav ha: ss 4 Comparg h α wh h qaos 4 ad show ha α. Forr srs rprsao of h rgd wav problm Ths sco formlas h wo-dmsoal rgd wav qao problm cosoac wh h Forr srs solos. W o ha h followg dr mras as drvd from sco abov ar who coradcos: s s 5 45
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU { ss 6 ss ss 7 8 ss 9 Sbsg all h drvd Forr solos 5 hrogh o 9 o h damcal cosra qao lds: ss ss ss ss ss Eqao s comparabl qval o qao. Hc facorg lds h hrd ordr dffral ssm: Thrfor a qval rprsao o h cosra qao s: Th qadrac objcv fcoal has qvalc Forr srs solo s hr gv vcor gral form: ad m{ [... d [... d} Ths vw of qaos ad h opmao problm has qvalc vcor gral formlaos: m{ [... d [... d} 4 sbjc o h damcal cosra qao: 46
Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 47... 5 Comparg qaos 4 h cosrad qvalc s dfd: d d... [... [ m{ [ µ }... d 6 L s rcosdr h gral qao 5. Th h hrd ordr paral dffral qao s gv b: 7 Th characrsc o qao 7 s wr smpl: 8 Solvg h characrsc qao Th roos of h characrsc qao 8 ar: 8 5 6 [4 8 5 8 5 6 [4 8 5 / 6 [ 9 8 5 6 [4 8 5 8 5 6 [4 8 5 / 6 6 [
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU [ 5 8 5 8 6 6 [4 6 / [4 6 5 8 5 8 Th roos ar a ral ad a par of cojgas compl roos. B D-Movr s horm w rasform h cojga roos o a s of ral qvalc roos. B h choc of o qvalc ral roo w hav wo ral roos. Ths spprssg h varabls ad w hav:.. A B c ss Wh qao bcoms: mplg ha: A.. A B ss ss B Takg h paral drvav of qao wh rspc o -varabl lds:.. A B c 5 Oc mor b sg qao 5 w oba:.. A B ss 6 From qao 4 qao 6 lds: B ss ss 7 Pg qao 7 o 4 lds: A ss ss 8 From qaos 7 ad 8 s o dffcl o s ha qao bcoms:.. ss ss 4 48
Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 ss ss c B b dfo. Thrfor w ddc ha c ; whc. Wh hs fal ddco w hav: ss ss 9 ss ss Frhrmor sc h ra of chag of h opmal corol s h sa h: ss ss 4 ss ss Eqaos 4 ad 4 ar or dsrd aalcal opmal corol ad sa rspcvl. 4 Th smlao of h aalcal opmal solos of qao Th followg fgrs blow corrspod o dffr smlas obad dr varos dmsoal spacs Fgr -8. Th mshs ad h argm ra rma cosa. Fgr. Th corol smla a Fgr. Th sa smla a 49
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU Fgr. Th corol smla a 5 Fgr 4. Th sa smla a 5 Fgr 5. Th corol smla a Fgr 6. Th sa smla a Fgr 7. Th corol smla a Fgr 8. Th sa smla a 5
Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 Fgr 9. Th corol smla a Fgr. Th sa smla a Fgr. Th corol smla a 5 Fgr. Th sa smla a 5 Fgr. Th corol smla a 4 Fgr 4. Th sa smla a 4 5
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU Fgr 5. Th corol smla a 5 Fgr 6. Th sa smla a 5 Fgr 7. Th corol smla a 6 Fgr 8. Th sa smla a 6 Aalss of Smlad Ops Th abl blow abl gvs s h gral mrcal smlad vals for h problm P. If w ak h ppr mos mrcal vals as or mrcal solos from ach of h Smlas abl smmars h ops. 5
Loardo Elcroc Joral of Praccs ad Tchologs ISSN 58-78 Iss 9 Jl-Dcmbr 6 p. 4-54 Tabl. Smlad ops opmal vals s/ Dmsoal spac Corol Corol 6-8 6-8 5 6-8 -7 6-8 -7 4 6-7 -7 5-7 4-7 6 4 6-8 -7 7 5-7 5-7 8 6-7 5-7 Gral comms o h smlas ad h ops vals. A crsor obsrvao of abl shows h corol ad h sas a dffr sraa maa sabl. Ths clarl porras h fac ha h corols ad sa rajcors covrg.. Th aalcal opmal Sas ad corols ar domal bodar solos; whl som poros of h corols ad Sas ar parall sabl wh som block wavs scard far blow h mor cospcos dsrbd dg bodars.. Th Smlas rval mor rgd wavs propagao; spcall a h rgms ar h dgs of h Smlas wh h profls ar a plas 5 4 ad 6. 4. Whl ohr dgs of h Smlas hb cospcos hghr dsrbacs h ohr rm ds ar rmarkabl a lowr sas of agaos. Ths aks mor corollg rg o sabl h corols ad sas a h dsrbd bodars ha a h ohr rlavl dsrbd dgs. 5. Som Smlas rsmbls som sbdco os wh br aalcal dph characrg som kow coc damcal procsss. Ths phoma hb h phscal prom afrmah of dsoros wh h sraa ar a h spac lvls ad 6. 6. Tabl shows ha h opmd vals progrssvl bcom largr as h ss of h spaal plas ar crasd. 7. I vw of h mrcal vals dmosrad o ma cocld ha h corol ad sa characr rlavl sgfca vals. I ohr words h wav s wavlghs ar of shor magds cosrg som hgh packags of rg. 5
Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU Coclsos W hav obad h aalcal opmal solos of h wo-dmsoal rgd wav qao sg h Hamloa forms. Th Smlas gv s vsal rprao of h opmd solos. Th aalcal solos wold hac frhr compaoal procsss sg h ECGM algorhm owards h drvao of h opmal corols for sas ad corols a varos spaal plas. Rfrcs [ Dchaa P. Zachma D. W. Paral dffral qaos. Mcgraw-Hll pblshg 986. [ Ibjgba M. A. Oma P. O corol opraor ad som of s applcaos. J. Mah. Aal. Applc. 984 p. -7. [ Rj S. A. Compaoal Opmao Mahmacal phscs Ph.D. Thss Uv. of Ilor Ilor Ngra 995. [4 Sgh M. A. Tl A. J. Ssm Dcomposo Opmao ad Corol; Prgamo To Los Frac 978. [5 War V. O. Opmal Corol of Ergd Wav qaos sg h Edd Cojga Grad Mhod ECGM. Ph.D. Thss Fdral Uv. of Tcholog Ma Ngra 4. [6 War V. O. Rj S. A. Corol Opraor for h Two-Dmsoal Ergd Wav Eqao. Corol Opraor for h Two-Dmsoal Ergd Wav Eqao Loardo Joral of Scc Romaa ISSN 58- Iss 9 Jl-Dcmbr 6 p. -44. 54