The Analysis of the Two-dimensional Diffusion Equation With a Source

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1 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p 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

2 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

3 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p 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

4 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

5 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p 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: [ [4 8 5 / 6 [ [ [4 8 5 / 6 6 [

6 Th Aalss of h Two-dmsoal Dffso Eqao Wh a Sorc Vcor Ooma WAZIRI Sda Agss REJU [ [4 6 / [ 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.

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

7 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p 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

8 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

9 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p 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

blow abl gvs s h gral mrcal smlad vals for

10 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

11 Loardo Elcroc Joral of Praccs ad Tchologs ISSN Iss 9 Jl-Dcmbr 6 p Tabl. Smlad ops opmal vals s/ Dmsoal spac Corol Corol 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 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 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

12 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

Linearization Technique and its Application. to Numerical Solution of Bidimensional Nonlinear. Convection Diffusion Equation

Linearization Technique and its Application. to Numerical Solution of Bidimensional Nonlinear. Convection Diffusion Equation Appled Mahemacal Sceces Vol. 8 04 o. 5 74-750 HIKARI Ld www.m-hkar.com hp://d.do.org/0.988/ams.04.70 Learzao Techqe ad s Applcao o Nmercal Solo of Bdmesoal Nolear Coveco Dffso Eqao Marco Dosee de Campos