Nield- Kuznetsov Functions of the First- and Second Kind
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1 IOSR Jourl of led Phscs IOSR-JP e-issn: Volue 8 Issue Ver. III M. - Ju. 6 PP osrourls.or S.M. lzhr * I. Gdour M.H. Hd + De. of Mhecs d Sscs Uvers of Ne rusc P.O. ox 55 S Joh Ne rusc CND EL 4L5 *O leve fro Uvers of U l-qur do of Sud rb. De. of Elecrcl d Couer Eeer Uvers of Ne rusc P.O. ox 55 S Joh Ne rusc CND EL 4L5 De. of Mhecs d Sscs Uvers of Ne rusc P.O. ox 55 S Joh Ne rusc CND EL 4L5 + Corresod uhor. hd@ub.c bsrc: The Neld-uzesov erl fucos of he frs- d secod-d re roduced d clssfed. Ech d s sub-clssfed s sdrd eerlzed or rerc deed o he source of ech fuco. Poer seres exressos re derved d Tbles of vlues of fucos of he frs-d re eered b evlu he derved seres. eords: Neld-uzesov Ierl fucos eber d r ODE. I. INTRODUCTION I her ele lss of he rso ler Neld d uzesov ] deosred he usefuless of r s equo h vrble erebl. The ese b hch he obed exc soluo o he over equos h he hel of r s fucos ed revvl of secl fucos h re of re ul he sud of flo hrouh vrble erebl ed. Ths cludes he rece eo b soe uhors o he use of eerlzed r s d eber s equos he sud of flo hrouh d over orous lers -4]. I he rocess of ob soluo o he resul boudr vlue roble Neld d uzesov ] roduced e erl fuco N x referred o s he Neld-uzesov fuco 5] hch rereses he bss for solv r s hooeeous dfferel equo h l or boudr codos d cos or vrble forc fuco. Ths fuco coues o be budl suded cf. 4-8] d he refereces here due o s useful roeres s vrous rereseos d exesos of s coce o reled fucos rs oher dfferel equos. or sce sce r s equo s secl cse of he eerlzed r s equo h s roduced b Sso d Hedle 9] sees url o exed he Neld-uzesov fuco o eerlzed fuco suble for solv he hooeeous eerlzed r s equo. eerlzed Neld-uzesov fuco s recel roduced o serve hs urose ] d hs bee successfull leeed he soluo of flo hrouh coose orous lers. The se les o eber s hooeeous equo here rerc Neld- uzesov fuco s recel roduced 4]. I he curre or e roduce d clssf he Neld-uzesov fucos rese lco hch he rse defe he fucos for vrble forc fucos d develo seres rereseo for he fucos. The seres rereseos ll he be used o eere relr Tbles of vlues of he Neld-uzesov fucos. II. POISEUILLE LO THROUGH POROUS CHNNEL I cse of Poseulle flo hrouh orous chel bouded b orous les s sho. belo r s equo h vrble erebl ] reduces o he follo ordr dfferel equo: u b... r Porous Ler. u b.. Rereseve Sech d u u d eff... DOI: osrourls.or 47 Pe
2 u u s he el veloc here x eff s he erebl h vres s fuco of he rsverse dreco o he flo x s he cos drv ressure rde s he bse vscos of he flud d eff s he effecve vscos of he flud rvers he orous ler. I he cofuro of. equo s o be solved for u subec o he codos: u b u b h vr ccord o eher of he fors: 4 5 here b b d re secfed o-eve rel ubers. The boudr vlue roble coosed of solv subec o codos d h he erebl dsrbuo ssf 4 or 5 s eerl h s ecouered rso ler lss. Equo s ler hooeeous ordr dfferel equo h vrble coeffces h deed o he dsrbuo of he vrble erebl. e ll selec erebl dsrbuos h reduce equo o secl for of equo h soluo exressble ers of erl fucos. Ths roch rovdes us h furher shs o he ure of he resul secl fucos d exded re of lcbl of hose fucos. III. PERMEILITY DISTRIUTIONS I h follos e ll dscuss hree fors of he vrble erebl dsrbuos h resul ell-o d clsscl dfferel equos hose soluos re erl fucos. Tble belo lss hree fors of vrble erebl fuco d he resul vrble coeffce dfferel equo h equo reduces o. or # Resul dfferel equo eff eff d u u d d u d u r s equo Geerlzed r s equo 4 eff 4 d u d 4 u ; eber s equo. Tble. Vrble Perebl ucos d he Resul Dfferel Equos. Soluos o he hooeeous rs of he equos Tble re ell-docueed he lerure d ve ers of u u ler cobos of o lerl deede fucos ve Tble oeher h her ross u u. Dfferel Equo u u u u r s equo Geerlzed r s equo s ; here DOI: osrourls.or 48 Pe
3 eber equo The ross Tble re defed: u u u u uu uu u u Tble. Lerl deede fucos ssf he hooeeous equos. 6 The fucos d re he eerlzed r s fucos of he frs d secod ds resecvel defed ers of odfed essel s fuco I of he frs d of order b 9]: I I ] 7 I I ] here I 8 r r! r r s he fuco d. 9 The sdrd r s fucos of he frs d secod ds 8 obed b = =. d resecvel re secl cse of 7 d Soluos o eber s dfferel equos re he lerl deede uercll ssfcor r of fucos o s rbolc clder fucos -5] vld here d re rel ubers. IV. 4.. The cse of cos forc fuco: PRTICULR SOLUTIONS Prculr soluos re obed b he ehod of vro of reers d e he fors ve Tble oeher h he eerl soluos here c c c c c d c re rbrr coss: Dfferel Equo Prculr soluo Geerl soluo r s eq. u N u c c N Geerlzed r s eq u N s u c c N s eber eq. u N u c c N Tble : Prculr d eerl soluos of he hooeeous equos: orc fuco e o ou h Tble he = soluos o r s equo re recovered fro soluo o eerlzed r s equo. The fucos N N d N re he Neld-uzesov erl fucos of he frs d defed Tble 4 d rse he he forc fucos he hooeeous equos re cos. Ler hs or e defe he Neld-uzesov fucos of he secod d hch rse he he forc fuco s vrble fuco of. 4.. The cse of vrble forc fuco: DOI: osrourls.or 49 Pe
4 he he cos Tble relced h vrble fuco f eerlzed r s d eber s equos re o ffeced b f u u of Tble. Prculr soluos re flueced b soluos o he hooeeous rs of r s d re obed b ler cobos of f d c be obed b he ehod of vro of reers s follos. Tble 5 ves he rculr soluos re ers of he Neld-uzesov fucos of he secod d ve Tble 6 here e ssued h f. Ierl uco Sdrd Neld-uzesov fuco of he frs d Ierl or d rs Dervve N N Geerlzed Neld-uzesov fuco of he frs d N N Prerc Neld-uzesov fuco of he frs d N N Tble 4. The Neld-uzesov ucos of he rs d Dfferel Equo Prculr soluo Geerl soluo r s eq u u c c Geerlzed r s eq u s u c c s eber eq u u c c Tble 5: Prculr d eerl soluos of he hooeeous equos: forc fuco f The Neld-uzesov ucos of he secod d d her frs dervves c hus be defed s ve Tble 6 belo: DOI: osrourls.or 5 Pe
5 DOI: osrourls.or 5 Pe Ierl uco Ierl or d rs Dervve Sdrd Neld- uzesov fuco of he secod d ] Geerlzed Neld- uzesov fuco of he secod d ] s Prerc Neld- uzesov fuco of he secod d Tble 6. The Neld-uzesov ucos of he rs d V. VLUES T ZERO The eerlzed r s fucos hve bee sho o hve he follo oer seres exsos 9]: ] 4 ]. 5 Equos -5 re evlued = o eere vlues Tble 7 belo. or he vlues of r s fucos d frs dervves = e use = d = -5. I ddo vlues of d hve bee reored he lerure 5] d e he follo exressos resecvel:
6 Vlues of he Neld-uzesov fucos = re obed fro her defos Tbles 4 d 6. These vlues re ve Tble 8 belo. r s ucos Geerlzed r s ucos 6 eber s ucos Tble 7. Vlues of r s Geerlzed r s d eber ucos d Dervves Zero Neld-uzesov ucos of he rs d Neld-uzesov ucos of he Secod d N ; N ; N ; N ; s N ; N ; Tble 8. Vlues of he Neld-uzesov ucos d Dervves Zero VI. POER SERIES REPRESENTTION Evluos of he Neld-uzesov fucos ve vlues of he deede vrble re essel for solv l d boudr vlue robles. e derve oer seres exressos for hese fucos. 6.. Seres Exressos for he Sdrd Neld-uzesov ucos: The follo sced seres exressos for N d hve bee derved elsehere cf. 7]: DOI: osrourls.or 5 Pe
7 DOI: osrourls.or 5 Pe!!!! N 8 f f N f!! 9!!!! cos dx d s. Seres 9 volves f d N hch us be deered before s deered. r f he ore covee defo of ve Tble 6 s obed. ollo he rocedure descrbed 7] e ob he follo seres rereseo for :!!!! Seres Exressos for he Geerlzed Neld-uzesov ucos: The eerlzed r s fucos re ve equos -5 bove. e use hese equos o derve seres exressos for N d defed Tbles 4 d 6 s follos. r 4 d 5 s: 5 ] 6 d uo dffere d er 5 d 6 e ob:
8 DOI: osrourls.or 54 Pe 7 ] 8 ] 9 ] ]. Us 5- he exressos for N d Tbles 4 d 6 e ob: N ] ] ] ]. 6.. Seres Exressos for he Prerc Neld-uzesov ucos: The follo exressos develoed -5] for he eber fucos d :!!!! 4!! 5!! ; ; ; 9
9 d re s ve Tble 7. Us equos -9 he defos of ve Tbles 4 d 6 e ob he follo seres exressos: N d N!!!! 4 ] ]. 4!! VII. TLES O VLUES O THE NIELD-UZNETSOV UNCTIONS O THE IRST IND Poer seres 8 d 4 hve bee evlued us Mle h full couol ccurc h s hou se uer l o he uber of ers used. The follo Tbles of vlues re eered for h se of.. N Tble 9. Vlues of N us seres 8 N N N N Tble. Vlues of N us seres N N N DOI: osrourls.or 55 Pe
10 Tble b. Vlues of N us seres N ; N ;. 5 ; Tble. Vlues of N us seres 4 VIII. CONCLUSION I hs or e roduced redefed d clssfed hree Neld-uzesov fucos of he frs-d d hree of he secod-d d shoed hscl suo hch he rse. The Poer seres exressos ere derved for ll sx fucos d relr Tbles of vlues for he Neld-uzesov fucos of he frs-d ere obed. Ths or ses he se for furher sudes of he roeres lcos d he develoe of ore Neld-uzesov erl fucos. REERENCES ] D.. Neld d. V. uzesov The effec of rso ler beee flud d orous edu: sher flo chel Trsor Porous Med ] M.S. bu Zoo lo hrouh d over orous lers of vrble hcesses d vrble erebl PhD hess Uvers of Ne rusc S Joh N.. Cd 5. ] M.S. bu Zoo T.L. lderso d M.H. Hd lo hrouh lered orous cofuro h eerlzed vrble erebl I. J. of Ehced Reserch Scece Techolo & Eeer Vol. 5 Issue ] M.S. bu Zoo T.L. lderso d M.H. Hd eber s Ihooeeous Dfferel Equo h Il d oudr Codos I. J. Oe Probles Co. Mh. Vol. 9 No. Jue 6 es. 5] M.H. Hd d M.T. el O he Nx erl fuco d s lco o he r s ohooeeous equo led Mh.Cou. 7# ] M.H. Hd d M.T. el Noe o he soc seres soluo o r s hooeeous equo I. J. Oe Probles Co. Mh. 4# ] S.M. lzhr I. Gdour d M.H. Hd sced seres soluo o r s hooeeous boudr vlue roble I. J. Oe Probles Co. Mh. 9# ] S.M. lzhr I. Gdour d M.H. Hd Rereseos d Couo of he Neld- uzesov Ierl uco ISOR Jourl of led Phscs Vol. 8 Issue Ver II ] C.. Sso d V.. Hedle exeso of r's equo. SIM J. l. Mh. Vol ]. Gl J. Seur d N.M. Tee s d ccure couo of heeber rbolc clder fuco x IM Jourl of Nuercl lss Vol ] J.C.P. Mller Prbolc Clder ucos Cher M. broz d I.. Seu eds. Hdboo of Mhecl ucos h oruls Grhs d Mhecl Tbles Nol ureu of Sdrds led Mhecs Seres - 55 Teh Pr Deceber 97. ] J. Seur d. Gl Prbolc Clder ucos of Ieer d Hlf- eer Orders of No-eve rues Couer Phscs Coucos Vol ] N.M. Tee Nuercl d soc secs of Prbolc Clder ucos J. Couol d led Mhecs Vol ] N.M. Tee. Secl fucos: roduco o he clsscl fucos of hecl hscs Ne Yor NY; Joh le & Sos ] N. M. Tee Prbolc Clder ucos NIST Hdboo of Mhecl ucos Cher. - 9 r. J. Olver Del. Lozer Rold. osver d Chrles. Clr eds. U.S. De. Coerce sho DC M Cbrde Uvers Press. N DOI: osrourls.or 56 Pe
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