PIECEWISE N TH ORDER ADOMIAN POLYNOMIAL STIFF DIFFERENTIAL EQUATION SOLVER 13

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1 Abrac PIECEWISE N TH ORDER ADOMIAN POLYNOMIAL A piecewie h order Adomia polyomial olver for iiial value differeial equaio capable of olvig highly iff problem i preeed here. Thi powerful echique which employ Adomia polyomial i how o obai early exac oluio for he bechmark cae udied. The high accuracie a reduced compuaioal co i obaied by varyig he iegraio ime ad he order of he polyomial. The ime ep i he curre olver i o he order of o 3 ime larger ha hoe ued by oher adard echique uch a fourh-order Ruge-Kua. I wa foud ha he piecewie coiuou polyomial icreae he imeep requireme a he order of polyomial icreae. The curre algorihm uilize high-order Adomia polyomial o advace he oluio wihi he iff regio wih relaively large ime ep. The algorihm wa validaed agai a firorder yem wih ad wihou large iffe ad a ecodorder, o-liear flame propagaio. Reul were compared wih exac oluio uig he Lamber W-fucio ad fourh-order Ruge-Kua mehod. Good agreeme wa obaied for he flame-propagaio predicio. I wa foud ha while obaiig relaively large ime ep, accuracie o he order of -5 were obaied. Iroducio STIFF DIFFERENTIAL EQUATION SOLVER Amir A. Mobaher, Alabama A&M Uiveriy; Araoo Poor Biazar, Uiveriy of Alabama i Huville; Zhegao Deg, Alabama A&M Uiveriy A umber of applicaio i ciece ad egieerig demad oluio of iff differeial equaio. Siffe i geerally defied a he oluio pace ha coai (very) large gradie. Such example iclude problem i chemical kieic, amopheric ciece, biochemiry, elecroic ad auomaic corol yem, o ame a few. I flame propagaio, coceraio of chemical pecie ca decay a differe rae; hu, a kieic-reacio differeial equaio decribig he pecie coceraio uually ha a broad rage of ime coa. The oluio o hi yem of ordiary differeial equaio i domiaed by he pecie ha have he hore ime coa []. The iffe of he problem require ha he error iroduced durig he compuaio be damped by he algorihm. The ep ize ha o be exremely mall. Larger ep ize may caue umerical accuracy ad abiliy problem. I geeral, iff problem may be olved by ome differeial equaio olver if he compuaioal co i o a iue. For large-cale egieerig problem ivolvig flame propagaio [], he oluio of iff problem become a maer of efficiecy. Currely, may elaborae cheme for he oluio of iff differeial equaio exi [3]. Perhap he mo widely acceped echique i he Gear [4] mehod. Oher mehod uch a he Si-Co-Taylor-Like [5] mehod ad he Muliep Ruge-Kua mehod [6] have bee ued by oher auhor. Commercial ofware program uch a Malab [] ue modificaio o Ruge-Kua o addre iffe. Uil recely, oe overlooked approach i he mehod propoed by Adomia for he oluio of o-liear ad liear ordiary differeial equaio [7-]. Thi approach wa alo employed o addre a claic fluid-dyamic problem wih high accuracy [], [3]. A umber of oher phyical problem have bee recely olved by hi mehodology [4-6]. The Adomia Decompoiio Mehod require erm-by -erm differeiaio ad iegraio of he baic differeial equaio. Alhough hi approach produce highly accurae oluio o o-liear differeial equaio, i ha ome drawback: ) I require laboriou work by he uer, due o ucceive differeiaio ad iegraio of he reulig Adomia Polyomial; ) For iff problem, he oluio ca oly be parially coverge []; ad 3) he approach i problem-pecific [7]. The goal of hi curre udy wa o formulae a algorihm ha would addre he aforemeioed iue. The auhor refer o he curre formulaio a Piecewie h order Adomia Polyomial Differeial Equaio Solver or, i hor, Adomia Differeial Equaio Solver (ADES). The word piecewie mea o break up he domai io ub-domai i order o obai a erie oluio i each ubdomai uilizig Adomia polyomial of a pecified order. Here, a ew mehodology for he oluio of iff differeial equaio will be preeed. I will be how ha he curre echique will obai early exac oluio while maiaiig large (a much a 3 ime larger ha he adard fourh-order Ruge-Kua echique). The curre echique wa validaed agai hree e example wih a highly iff aure ad i wa foud ha he oluio obaied via hi echique wa highly accurae while he ime required o obai he oluio wa miimal. PIECEWISE N TH ORDER ADOMIAN POLYNOMIAL STIFF DIFFERENTIAL EQUATION SOLVER 3

2 Mehodology Coider he differeial equaio yem of he form u & f (, u) where f(,u) i ay liear or o-liear fucio of order. Adomia polyomial are employed o coruc he oluio pace for u a: u + L ( f ) L ( )d where L - i he iegraio operaor. The oluio of he yem of differeial equaio i Equaio () i he expreed a: () () Thi example coider a fir-order iff yem wih o o-liear erm. Thi problem ha bee examied by Ahmad [5] ad i refereced here agai i Equaio (4). u& u + 99v v Adomia polyomial ake he form of u wih (4) v Subeque value of r ad are preeed i Equaio (6), ri ri + 99i i i (5) (6) wih he oluio u, v preeed i Equaio (7). where u i, u i () i he precribed iiial codiio ad u i,+ L - (A i, (u i,, u i,,,u i, ) i he + h erm of he i h equaio ad A i, i he Adomia Polyomial [7]. Thi mehod, propoed by Adomia, iegrae he Adomia polyomial from o. For iff problem, he oluio i valid oly up o a cerai ime afer which he oluio begi o diverge. I pracical erm, polyomial of order ad higher are eeded o approximae he oluio i he high gradie regio. Obviouly, hi i beyod compuaioal mea ad efficiecy. To addre hi iuaio, he curre echique divide he domai o m ub-domai hereafer referred o a Piecewie Adomia Polyomial. The oluio advace by ierig he ed-codiio a each ub-domai a iiial codiio for he ubeque ub-domai. The oluio wihi each ub-domai i approximaed by lower-order Adomia polyomial (ypically order of o ). Reul ad Dicuio I hi ecio, hree well-defied problem ha are highly iff ad have bee refereced by oher auhor i bechmarkig iff differeial equaio will be examied. Example : Fir-Order Liear Syem (3) v Table compare he relaive error for hree differe echique of fourh-order Ruge-Kua (RK4), Si-Co- Taylor-Like (SCTL) oluio ad he curre formulaio of he Adomia Differeial Equaio Solver (ADES). A how i Table, RK4 geerae relaive error i he rage of -3 o -4 wihi he iff regio ad SCTL geerae error of -5 o -, while ADES geerae error of order -. I mu be oed ha he curre echique ue a of.3, which i 3 ime higher ha hoe of RK4 ad SCTL. Table. Compario of Relaive Error for Three Techique r!! Time() Exac RK4 SCTL ADES (7) INTERNATIONAL JOURNAL OF MODERN ENGINEERING VOLUME 3, NUMBER, FALL/WINTER

3 Example : Fir-Order Syem wih large Siffe Thi example alo coider a liear problem, hough wih large iff regio. The ordiary differeial equaio (ODE) yem ca be decribed a u& 998u 998v (8) 999u 999v To obai he ADES oluio, he iiial codiio were e equal o vecor r ad. (9) Sice all coefficie of r ca be facored ou, he geeral expreio for r ad, a i Equaio (), ca be wrie a: ri 999ri 998i () i 999ri 999i Ad he oluio u, v i give by Equaio (). r! () v! Table compare he relaive error bewee RK4 ad ADES. The relaive error geeraed by RK4 wihi he iff regio i i he rage of -3 o -5, while he curre mehod geerae error of order le ha -5 i all he regio. Example 3: Flame Propagaio The fial example i hi ecio coider he problem of Flame Propagaio, which i alo olved by he Malab differeial equaio olver RK4. The problem i decribed here agai for referece i Equaio () u& v uv v where δ.. ( u v) u v δ δ wih () The exac aalyical oluio o he flame model i give by (3) Table. Compario of Relaive Error Bewee RK4 ad ADES Time () Exac RK4 ADES < < < < < < < < < < Ulike cae ad, hi example ivolve ecod-order o-liear erm. The corucio of Adomia Polyomial for hi problem i how i Equaio 4-7. r r r r ( r ) r ( r ) + ( r ) (4) (5) (6) where fucio. ad he fucio W(z) i he Lamber W r3 r r r 3 ( r ) + ( r ) + ( r ) (7) PIECEWISE N TH ORDER ADOMIAN POLYNOMIAL STIFF DIFFERENTIAL EQUATION SOLVER 5

4 Ad he geeral form of he above equaio i repreeed i Equaio (8) wih he oluio how i Equaio (9). (8) (9) Figure depic he oluio for hi problem wih h.5. I i how ha a, he oluio udergoe a very eep gradie. Table 3 compare he oluio of he flamepropagaio problem wih hree differe mehod: he Piecewie Adomia (PAP) olver, he exac oluio obaied by Maple ofware wih Lamber W-fucio ad fourh-order Ruge-Kua mehod. The oluio ime domai lied i Table 3 i from o 8 ecod, where he iff regio exi i hi ime domai. For 5, i ca be ee ha he relaive error i wihi accepable compuaioal limi ad he oluio quickly coverge o a early exac oluio wih he Lamber W-fucio. Soluio u Figure. The Soluio for he Flame-Propagaio Problem wih h.5 Cocluio ri i i i i v ( r ) I hi paper, a robu, accurae ew echique wa preeed for he oluio of iff differeial equaio. The algorihm i baed o he Adomia polyomial. I wa how ha a much a 3 ime higher ha hoe of r r!! Soluio u v. Time 5 5 Time (ec) adard echique of fourh-order Ruge-Kua ca be ued o obai early exac oluio o highly iff differeial equaio. Oe characeriic of he curre algorihm i hi abiliy o vary he order of polyomial o be ued o icreae he accuracy. Table 3. Compario of Relaive Error for Differe Order of Adomia Polyomial Time (ec) Referece Exac Wih L-W Fucio PAP (5) RK [] Kuo, K. K., (986). Priciple of Combuio. Joh Wiley. [] Moler, C. (.d.). Siff Differeial Equaio, hp:// iff-differeial-equaio.hm [3] Biography for Siff Differeial Equaio. (3). Rerieved from hp://mah.fullero.edu/mahew/ 3/iffde/SiffDEBib/Lik/ SiffDEBib_lk_3.hm [4] Gear, C. W., & Pezold, L. R. (984), ODE yem for he oluio of differeial algebraic yem, SI- AM Joural o Numerical Aalyi,, INTERNATIONAL JOURNAL OF MODERN ENGINEERING VOLUME 3, NUMBER, FALL/WINTER

5 [5] Ahmad, R., & Yaacob, N. (5). Si-Co-Taylor- Like Mehod for Solvig Siff Differeial Equaio. Joural of Fudameal Sciece,, [6] Chapra, S. C., & Caale, R. P., (). Numerical Mehod for Egieer. (6 h ed.). McGraw Hill. [7] Adomia, G. (989). Noliear ochaic Syem Theory ad Applicaio o Phyic. Kluwer Academic. [8] Adomia, G. (983). Sochaic yem. Academic Pre, Lodo. [9] Adomia, G., (984). A ew approach o o-liear parial differeial equaio, Joural of Mahemaical Aalyi ad Apolicaio,, [] Adomia, G. (99). A Review of he decompoiio mehod ad ome rece reul for oliear equaio, Compuer & Mahemaic wih Applicaio,, -7. [] Adomia, G. (994). Solvig Froier Problem of Phyic: The decompoiio Mehod. Kulvar Academic, Boo. [] Mobaher, A., Oviedo, R. R., & Vu, B. (). Applicaio of Mehod of Decompoiio o Claical Fluid Dyamic Problem. Fifh Miiippi Sae Coferece o Differeial Equaio & Compuaioal Simulaio. Sarkville, MS. [3] Wag, L. (4). A ew algorihm for olvig claical Blaiu equaio, Applied Mahemaic ad Compuaio, 57, -9. [4] Babolia, E., Biazar, J., & Vahidi, A. R. (4). A ew compuaioal mehod for Laplace raform by decompoiio mehod. Applied Mahemaic ad Compuaio, 5, [5] Dehgha, M. (4). Applicaio of Adomia Decompoiio Mehod for wo dimeioal parabolic equaio ubec o o-adard boudary pecificaio, Applied Mahemaic ad Compuaio, 57, [6] Luo, X., Wu, Q. & Zhag, B. (6). Revii o Parial Soluio i he Adomia Decompoiio Mehod: Solvig Hea ad Wave Equaio, Joural of Mah. Aalyi ad Applicaio, 3, ZHENGTAO DENG i a profeor of Mechaical Egieerig a Alabama A&M Uiveriy. Dr. Deg may be reached a zhegao.deg@aamu.edu Biographie AMIR A. MOBASHER i a Aociae Profeor of Mechaical Egieerig a Alabama A&M Uiveriy. Dr. Mobaher may be reached a amir.mobaher@aamu.edu ARASTOO POOR BIAZAR i a eior reearch ciei of he Deparme of Amopheric Sciece a Uiveriy of Alabama i Huville. Dr. Biazar may be reached a biazar@c.uah.edu PIECEWISE N TH ORDER ADOMIAN POLYNOMIAL STIFF DIFFERENTIAL EQUATION SOLVER 7