A Numerical Simulation and New Traveling Wave Solutions of Convection-Diffusion Equation with Reaction

Transcription

1 Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Volu, Issu 4, ril 04, PP ISSN X Prin & ISSN Onlin Nurical Siulaion and Nw Travling Wav Soluions of Convcion-Diffusion Equaion wih Racion E. E. El-Bhadi Collg of Scinc and Huaniy Sudis, Mahaics Darn, Salan bin bdulaziz, Univrsiy,KS. M.. shabrawy *Couniy Collg, Cour Darn, Salan bin bdulaziz Univrsiy, KS *Racors Darn, Cour rou, oic Enrgy uhoriy, Egy bsrac: Throughou his ar w will discuss and analyz of h convcion-diffusion quaion which includd a racion r. Firs, w will aly an alrnaing-dircion ilici DI sch ain o ha roosd by Polzhav. Us of his ilici oraor-sliing sch allows h alicaion of a ridiagonal Thoas solvr o obain h soluion of sady convcion-diffusion quaion wih racion. ccording o h unsady cas, w will us h irovd ' / -ansion hod o consruc h ravling wav soluions, whr '/ saisfis a scond ordr linar ordinary diffrnial quaion. In his ar w will lor nw alicaions of his hod o so scial nonlinar convciondiffusion quaion wih racion. Kywords: convcion-diffusion-racion quaion, alrnaing-dircion ilici DI sch, '/ ansion, ravling wav soluions, linar and nonlinar diffrnial quaions. -. INTRODUCTION In his ar, w will discus and invsiga wo hods for solving a convcion diffusionracion CDR scalar ransor quaion. This quaion is racically ioran bcaus h woring quaions of any cass fall ino his fild. Tyical als ar h Hlholz quaion for odling rior acousics [], consiuiv quaions for odling h urbuln quaniis and ε [], and viscolasic consiuiv quaions for odling h ra srsss in non-nwonian fluid flows [3]. Th ain oins in his rsarch boils down o h following oins. Scion rsns h woring quaion, and hn an alrnaing-dircion ilici sch, siilar o ha of Polzhav [4], is usd o obain h sady/ransin CDR quaion in on dinsion. Throughou scions 3, 4, our hasis in his ar is on h drivaion of nw ravling wav soluions of unsady-sa nonlinar CDR quaion in on dinsion [5, 6]. Sinc h os of hnona in hysics and ohr filds of ahaics ar dscribd by nonlinar voluion quaions. nd whn w wan o undrsand h hysical aning of hnona, i was dscribd by nonlinar voluion quaions, and hn ac soluions for h nonlinar voluion quaions hav o b lord. For al, h wav hnona obsrvd in fluid dynaics [7, 8], lasa and lasic dia [9,0], c. Rcnly, Wang al [, ] inroducd a nw hod calld h h ' / -ansion hod o loo for h ravling wav soluions of nonlinar voluion quaions. Hr w shall us a nw irovd ' / -ansion hod [3-9]. Th ain ida of his hod is ha h ravling wav soluions of nonlinar quaions can b rssd by a olynoial in '/, whr, whr, saisfis h scond ordr linar ODE 0, whr,, and ar consans. Th dgr of his olynoial can b drind by considring h hoognous balanc bwn h highs ordr drivaivs and h nonlinar rs aaring in h givn nonlinar quaions [9-]. Th cofficins of his olynoial can b obaind by solving a s of algbraic quaions rsuling fro h rocss of using h roosd hod. In his ar, h RC Pag 345

2 E. E. El-Bhadi & M.. shabrawy ' / -ansion hod will lay an ioran rol in rssing h ravling wav soluions of h convcion-diffusion quaion which includd h racion r.. WORKIN EQUTIONS ND SOLUTION LORITHM W considr firsly h scalar wo dinsional convcion diffusion racion quaion V c 0 Whr V= V u, v rrsn h vlociy coonns along h and y dircions, rscivly. Ohr cofficins involv and c, which dno h diffusion cofficin and h racion cofficin, rscivly. For illusraiv uross, all hs valus ar assud o b consan. For siliciy, h invsigad quaion is subjc o h Dirichl-y boundary condiion g on, T. Equaion. wih givn boundary condiion consiu a closur robl rovidd ha h iniial daa of, y,0 ar rscribd. Th sragy w will considr for solving. is siilar o h DI alrnaing-dircion ilici sch of Polzhav. By viru of oraor sliing, calculaion of h aroiad soluion of Eq.. is accolishd in wo ss, firs is h rdicor s and h scond is h corrcor s; rscivly * * * * n n n u c v y yy.a n n n n * * * v y yy c u.b L us dfin u, v u, v,.3 c c Th abov wo-s DI sch for solving Eq.. can b rwrin as u.4a * * * c f v.4b n n n y yy c f Whr h sourc rs f and f ar givn by f n n v y f n yy..5a u.5b * * * For h unsady cas, h scalar convcion diffusion racion quaion in on dinsion is of h for u c 0.6 W aly h sidiscrizaion sch o aroia Eq..6. In h i-sing sch, n n w considr /, which yilds firs-ordr accuracy. Th rsuling quaion conaining only h saial drivaivs is n n n n uˆ ˆ c ˆ.7 Th dfiniions of uˆ, ˆ and ĉ ar uˆ u, ˆ, andcˆ c. Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 346

3 Nurical Siulaion and Nw Travling Wav Soluions of Convcion-Diffusion Equaion wih Racion Equaions.4a,.4b, and.7 ar nown as h sady-sa convcion diffusion racion quaions. his oin, w raliz ha h y o succss in solving Eq.. lis in h analysis of h following odl quaion: u c f.8 s is h cas whn a arial diffrnial quaion is siulad, w ai o obain highr rdicion accuracy. To his nd, w loy h gnral soluion for Eq..8, f a b.9 c whr a and b ar consans. Subsiuing Eq..9 ino Eq..8, w hav wo quaions for and, rscivly: u c.0 u c 0. Th abov wo quaions nabl us o drin and as follows: u u 4c,.. W will focus our anion now o h soluion of h nonlinar y of quaion.6 using h irovd ' / -ansion hod [3-9]. 3. DESCRIPTION OF THE ' / Suos ha a nonlinar quaion is givn by -EXPNSION METHOD P u, u, u, u, u, u,... 0, 3. whr u = u, is an unnown funcion and P is a olynoial in u = u, and is arial driva- ivs, in which h highs ordr drivaivs and nonlinar rs ar involvd. In h following w giv h ain ss of h ' / -ansion hod. S : Cobining h indndn variabls and ino on variabl, w suos ha u, u,, 3. Th ravlling wav variabl ris us o rduc Eq. 3. o an ODE for u=u, naly P u, u, u, u, u, u, S : Suos ha h soluion of ODE 3.3 can b rssd by a olynoial in ' / as follows: u..., 3.4 whr is calld h balanc nubr and saisfis h scond ordr linar diffrnial quaion in h for 0, 3.5 whr,,...,, ar consans o b drind lar,, 0. Using h gnral soluion of Eq. 3.5, w hav Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 347

4 E. E. El-Bhadi & M.. shabrawy 4 c sinh c cosh 4 c sin c cos 4 c cosh 4 c sinh 4, c cos, 4 4 c sin Th unwrin ar in Eq. 3.4 is also a olynoial in '/, bu h dgr of which is qual or lss han -. Th osiiv ingr can b drind by considring h hoognous balanc bwn h highs ordr drivaivs and nonlinar rs aaring in ODE 3.3. S 3: By subsiuing 3.4 ino 3.3 and using 3.5, collcing all rs wih h sa ordr of ' / oghr, and hn quaing ach cofficin of h rsuling olynoial o zro yilds a s of algbraic quaions for,,...,, and. S 4: ssuing ha h consans,,...,, and can b obaind by solving h algbraic quaions in S 3, sinc h gnral soluions of h scond ordr linar ODE 3.5 is wll nown for us, hn subsiuing,...,,, and h gnral soluions of Eq. 3.5 ino, w hav or ravlling wav soluions of h nonlinar voluion Eq. 3.. In h subsqun scions, w will illusra h validiy and rliabiliy of his hod in dail wih so nonlinar voluion quaions in ahaical hysics h balanc nubrs of which ar no osiiv ingrs. 4. PPLICTION TO THE CDR EQUTION In his scion, w will aly h ' / -ansion hod o consruc h ravling soluions for CDR [5,6] u F 0 4. Whr h racion r F is a sourc. W rar ha gologiss, civil nginrs, ahaicians, and so on, frqunly us diffrn rinology in dscribing h hnona bodid in quaion 4.. s found in [5], if h racr is radioaciv wih dcay ra c, hn F c and w obain h linar CDR Eq..6. If h racr is a biological scis wih logisic growh ra, whr r is h F r growh consan, R is h carrying caaciy, and is a osiiv quaniy. Thn w sar wih h CDR quaion convcion-diffusion quaion wih growh in h for u 0 4. Whr, and ar nonzro consans. Th ravling wav variabl blow,,, 4.3 Pris us o convr Eq. 4. i n o an ODE for, in h for u 0, 4.4 whr is a consan. Suos ha h soluion of h ODE 4.4 can b rssd by a olynoial in '/ as follows:..., 4.5 R Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 348

5 Nurical Siulaion and Nw Travling Wav Soluions of Convcion-Diffusion Equaion wih Racion Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 349 whr saisfis Eq By using 4.5 and 3.5, i is asily drivd ha..., Considring h hoognous balanc bwn and in 4.4, basd on 4.6 and 4.7, w r q uird ha. I should b noicd ha is no a osiiv ingr. Howvr, w ay choos h soluion of Eq. 4.4 in h for, 4.8 whr is a ral consan o b drind lar and saisfis Eq I is asy o dduc ha Subsiuing i n o 4.4 a n d collcing all rs wih h sa owrs of / ' oghr, w obain 4. 3 u 4. 4 u u Solving h abov algbraic quaions , w hav 0 4 u 4.6 Subsiuing 4.6 ino 3.7 yilds 4.7 whr

6 E. E. El-Bhadi & M.. shabrawy 4 u. 4.8 Using Eq. 3.6 w dduc afr so rducion ha c 4.9 c c Whr c and c ar arbirary consans. Subsiuing 4.9 i n o 4.7, w obain h nw ac ravling wav soluion of Eq. 4. as follows: c,, 4.0 c c whr and ar givn abov. If w chos c c in 4. hn w obain h nvlo soliary wav soluions of Eq. 4.0, anh 4. Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 350 s a scial Cas: if w u = in Eq. 4., hn w hav 6,, u 5, nd hrfor is soluion, in his cas, as h for 6 c c c Whr, 4.3 u No ha Eq. 4.3 rrsns h soliary wav soluion of h convcion-diffusion-racion quaion 4. in h cas whn =. 5. CONCLUSIONS Throughou his rsarch w hav rsnd an alrnaing-dircion ilici sch, which is siilar o Polzhav s sch, o rduc D convcion-diffusion-racion quaion ino sadysa on-dinsional on. Th ain discussion hr is o find a nw soluion of unsady-sa CDR using ' / -ansion hod. Sinc h ' / -ansion hod was rsnd by Wang, his hod has bn irovd by svral auhors. Howvr, h alicaion of his hod was sill liid o hos quaions h balanc nubrs of which ar osiiv ingrs. In his ar, w lor a nw alicaion of h ' / -ansion hod and obain nw ys of ac ravlling wav soluions o a ind of nonlinar CDR quaion. This ar rsns a widr alicabiliy for handling nonlinar voluion quaions using h -ansion hod. Th nw y of ac ravlling wav soluion obaind in his ar igh hav significan iac on fuur rsarchs. I is worhy of furhr sudy. CKNOWLEDEMENTS Th auhor grafully acnowldg Rsarch Cnr, Salan bin bdul-ziz Univrsiy, Kingdo of Saudi rabia, for suoring and ncouragn during his wor. REFERENCES [] Harari I. and Hughs T. J. R., Fini ln hods for h Hlholz quaion in an rior doain: Modl robls, Cou. Mhods l. Mch. Eng. 87, [] Ilinca F. and Pllir D., Posiiviy rsrvaion and adaiv soluion for h ² odl of urbulnc, IJ. 36,

7 Nurical Siulaion and Nw Travling Wav Soluions of Convcion-Diffusion Equaion wih Racion [3] Croch M. J., Davis. R., and Walrs K., Nurical Siulaion of Non-Nwonian Flow Elsvir, NwYor, 984. [4] Polzhav V. I., Nurical soluion of h sys of wo-dinsional unsady Navir Sos quaions for a corssibl gas in a closd rgion, Fluid Dyn., 967. [5] J. David Logan, Transor odling in hydrogochical syss, ISBN: ,Inrdiscilinary lid Mahaics, Volu [6] Mario Ohlbrgr. Posrior Error Esias For Cnrd Fini Volu roiaions of Convcion-Diffusion-Racion Equaions. Mah. Modl. and Nur. naly. Vol 35 No [7] Ray K, Bhaacharj JK. Sanding and ravlling wavs in h shallow-war circular hydraulic ju. Phys L 007;37:4 8. [8] Inan IE, Kaya D. Eac soluions of so nonlinar arial diffrnial quaions. Physica 007; 38:04 5. [9] Osiov V. n ac soluion for a fracional disclinaion vor. Phys L 994;93:97 0. [0] Jordan PM, Puri. no on ravling wav soluions for a class of nonlinar viscolasic Mdia, Phys L 005;335:50 6Jordan PM, Puri. [] Wang ML, Li XZ, Zhang JL. Th ' / -ansion hod and ravlling wav soluions of nonlinar voluion quaions in ahaical hysics. Phys L 008;37:47 3. [] Wang ML. Soliary wav soluions for varian Boussinsq quaions, Phys L 995;99:69 7. [3] Huiqun Zhang. Nw alicaion of h ' / -ansion hod, Coun Nonlinar Sci Nur Siula, [4] Shng Zhang, Ling Dong, Jin-Mi Ba, Ying-Na Sun. Th ' / -ansion hod for nonlinar diffrnial-diffrnc quaions, Phys L [5] Ling-iao Li, Wang Ming-Liang. Th ' / -ansion hod and ravlling wav soluions for a highr-ordr nonlinar Schrodingr quaion, lid Mahaics and Couaion [6] Yulan Ma, Bangqing Li. Nw alicaion of ' / -ansion hod o a nonlinar voluion Equaion, lid Mahaics and Couaion [7] Zayd E. M. E., Khald. rl. Th ' / -ansion hod for finding ravlling wav soluions for nonlinar arial diffrnial quaions in ahaical hysics, J. Mah. Phys [8] Shng Zhang, Ling Dong, Jin-Mi Ba, Ying-Na Sun. Th ' / -ansion hod for a discr nonlinar Schrodingr quaion, PRMN J. Phys Vol. 74 No [9] Fign Kangalgil, Faa yaz. Nw ac ravlling wav soluions for h Osrovsy quaion,phys L [0] Wang ML, Zhou YB, Li ZB. licaions of a hoognous balanc hod o ac soluions of nonlinar quaions in ahaical hysics, Phys L 996;6: [] Wang ML, Li XZ, Zhang JL. Th ' / -ansion hod and ravlling wav soluions of nonlinar voluion quaions in ahaical hysics, Phys L 008;37:47 3. UTHORS' BIORPHY Dr. Eid lsaid Eid Elbhadi obaind B.Sc. ahaics in 989, M.Sc. alid ahaics in 996 and Ph.D alid ahaics, fluid Mchanics scially couaional fluid chanics in 003 fro h Darn of ahaics, Zagazig Univrsiy, Egy. Dr. Mohad bdl Faah shabrawy obaind B.Sc. Cour Scinc in 996, M.Sc. Cour Scinc in 00 and Ph.D Cour Scinc, Iag rocssing, Mdical iag rocssing in 00 fro h Darn of Cour Scinc, Suz canal Univrsiy, ssisan Profssor in Racors Darn, Nuclar Rsarch Cnr, oic Enrgy uhoriy Egy. Inrnaional Journal of Scinific and Innovaiv Mahaical Rsarch IJSIMR Pag 35