Numerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions
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1 IOSR Joural of Applid Chmisr IOSR-JAC -ISSN: Volum 9 Issu 8 Vr. I Aug. 6 PP Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Ima. I. Gorial parm of Mahmaics Collg of Educaio for Pur Scic / Ib Al-Haiham Baghdad Uivrsi Iraq Absrac: I his papr aalical umrical simulaio of h - ha quaio wih drivaiv boudar codiios has b prsd. h algorihm for h umrical simulaio for his quaio is basd o modifid dcomposiio mhod. h umrical mhod has b applid o solv a pracical ampls ad i's rsuls hav b compard wih ac soluio. Kwords: Modifid dcomposiio mhod - Ha quaio rivaiv boudar codiios. I. Iroducio hr ar ma w aalical approima mhods o solv iiial valu problms i h liraur. Amog hs Adomia s dcomposiio mhod [] hav b rcivd much aio i rc ars i applid mahmaics i gral ad asil hadl a wid a class of liar or oliar problms. h Adomia chiqu is basd o a dcomposiio of a soluio of a oliar fucioal quaio i a sris of fucios. Each rm of h sris is obaid from a polomial grad b a powr sris pasio of a aalic fucio. h mai advaag of h AM is ha i ca b applid dircl for all ps of fucioal quaios liar or oliar. Aohr impora advaag is capabl of gral rducig h siz of compuaio work whil sill maiaiig high accurac of h umrical soluio. I [] h auhor compard h Adomia composiio Mhod AM ad h alor sris mhod b usig som paricular ampls ad showd ha h Adomia s chiqu producd rliabl rsuls wih a fwr iraios whras h alor sris mhod suffrd from compuaioal difficulis. Bu i his papr w will show ha boh Adomia dcomposiio mhod ad alor sris mhod ar quivals ad hrfor hir covrgc is h sam i boh. iffr modificaios of his mhod ad hir applicaios ar giv i [4-9] II. Applig h Modifid Adomia s composiio Mhod of Soluio - h Ha Equaio I his scio w wa o dscrib how o us modifid dcomposiio mhod for solvig - h ha quaio wih drivaiv boudar codiios: f g g g g 4 g g g Whr f g 4 ad ar kow fucios is giv cosa. Now w us Adomia dcomposiio mhod h quaio ca b rwri as OI:.979/ Pag
2 Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios OI:.979/ Pag Whr h diffrial opraors ad h applig which is dfid as: d 4 h quaio bcom hrfor w ca wri 5 B Wazwaz [4] h modifid dcomposiio mhod is basd o h assumpio ha h fucio ca b dividd io wo pars aml ad. Udr his assumpio w s W suggs h followig modificaio III. Numrical Eampls Eampl : Cosidr h problm wih h followig drivaiv boudar ad iiial codiios: W appl h abov modifid dcomposiio mhod; w obai h h sris form is giv b:
3 Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios OI:.979/ Pag his is h ac soluio. Figur ad Figur show h plo of h ac soluio surfac ad h umrical soluio surfac for - ha quaio rspcivl. Figur : Eac soluio Figur : Numrical soluio Eampl : Cosidr - ha quaio wih drivaiv boudar codiios for h quaio : si si si si cos cos Now w appl h abov modifid dcomposiio mhod w obai: si h h sris form is giv b: si Which givs h ac soluio si. Figur ad Figur 4 show h plo of h ac ad h umrical soluio surfac for - ha quaio rspcivl.
4 Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios OI:.979/ Pag Figur : Eac soluio Figur 4: Numrical soluio Eampl : Cosidr h followig: si si cos si Now afr modifid dcomposiio mhod w obai: si h h sris form is giv b: si his is h ac soluio si. h plo of h ac soluio surfac is show i Figur 5 ad h umrical soluio surfac is show i Figur 6 for ha quaio Figur 5: Eac soluio Figur 6: Numrical soluio IV. Coclusio I his papr h rsuls obaid wih applid h modifid dcomposiio mhod of h - ha quaio wih drivaiv boudar codiios is accura. his algorihm is simpl ad as o implm. O h ohr had h calculaios ar as ad ak lss im ha i radiioal chiqus.
5 Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Rfrcs []. Adomia G. 986 Noliar Sochasic Opraor Equaios Acadmic Prss odo. []. Adomia G. 994 Solvig Froir Problms of Phsics: h composiio Mhod Kluwr Acadmic Publishrs ordrch. []. A. M. Wazwaz 998 A compariso bw Adomia dcomposiio mhod ad alor sris mhod i h sris soluio Appl. Mah. Comp [4]. A. M. Wazwaz A rliabl modificaio of Adomia dcomposiio mhod Appl. Mah. Compu [5]. A. M.Wazwaz A w algorihm for solvig diffrial quaios of a-emd p Appl. Mah. Compu [6]. A. M. Wazwaz A w mhod for solvig sigular iiial valu problms i h scod-ordr ordiar diffrial quaios Appl. Mah. Compu [7]. O. Kımaz S. Mirasdio glu A w smbolic compuaioal approach o sigular iiial valu problms i h scod-ordr ordiar diffrial quaios Appl. Mah. Compu [8]. Ima. I. Gorial "A Smi-Aalic Mhod for Solvig wo-imsioal Fracioal isprsio Equaio" Advacs i Phsics horis ad Applicaios.: p [9]. Chigul A. " Numrical Mhod for Solvig No-Homogous Ha Equaio wih rivaiv Boudar Codiios " Iraioal Mahmaical Forum Vol.6 o. 4: OI:.979/ Pag
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