Comparative Study of Finite Element and Haar Wavelet Correlation Method for the Numerical Solution of Parabolic Type Partial Differential Equations

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1 ISS , England, UK Journal of Informaon and Compung Scnc Vol., o. 3, 6, pp.88-7 Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal Soluon of Parabolc Typ Paral Dffrnal Equaons S. C. Shralash, P. B. Mualk Dsa, A. B. Dsh Dparmn of Mahmacs, Karnaaka Unvrsy Dharwad. Dparmn of Mahmacs, K. L. E. Collg of Engnrng and Tchnology, Chkod. (Rcvd Aprl 6, 6, accpd Jun 8, 6) Absrac. In hs papr, w prsn h comparav sudy of Haar wavl collocaon mhod (HWCM) and Fn Elmn Mhod (FEM) for h numrcal soluon of parabolc yp paral dffrnal quaons such as -D sngularly prurbd convcon-domnad dffuson quaon and -D Transn ha conducon problms valdad agans ac soluon. Th dsngushng faur of HWCM s ha provds a fas convrgng srs of asly compuabl componns. Compard wh FEM, hs approach nds subsanally shorr compuaonal m, a h sam m mng accuracy rqurmns. I s found ha hghr accuracy can b aand by ncrasng h lvl of Haar wavls. As Consquncs, avods mor compuaonal coss, mnmzs rrors and spds up h convrgnc, whch has bn jusfd n hs papr hrough h rror analyss. Kywords: Haar wavl collocaon mhod, parabolc quaon, Fn dffrnc mhod, Fn lmn mhod, Ha conducon problms.. Inroducon Dffrnal quaons hav numrous applcaons n many flds such as physcs, flud dynamcs and gophyscs c. Many racon dffuson problms n bology and chmsry ar modld by paral dffrnal quaons (PDEs). Ths problms hav bn nsvly sudd by many auhors lk Sngh and Sharma [], Guspp and Flppo [] n hr lraur and hr approma soluons hav bn accuraly compud povdd h dffuson coffcns, racon caons, nal and boundary condons ar spcfd n a drmnsc way. Howvr, s no always possbl o g h soluon n closd form and hus, many numrcal mhods com no h pcur.ths ar Fn Dffrnc, Spcral, Fn Elmn and Fn Volum Mhods and so on o handl a vary of problms. Many rsarchrs such as Kadalbajoo and Awas [3],F.D Mon[4] ar nvolvd n n dvlopng varous numrcal schms for fndng soluons of ha conducon problms appar n many aras of ngnrng and scnc. So, fndng ou fbl chnqus for gnrang h soluons of such PDEs s qu manngful. Rsarchrs Mdvdsk and Sgunov [5] and Doss.al [6] hav usd dffrn chnqus o compu h abov problms and smlar ons. Sngularly prurbd problmsappar n many branchs of ngnrng, such as flud mchancs, ha ransfr, and problms n srucural mchancs posd ovr hn domans. Thorms ha ls condons for h snc and unqunss of rsuls of such problms ar hroughly dscussd by Ross.al [7] and Gaml [8]. Th applcaon of FEM o varous ha conducon problms bgan hrough a papr by Znkncz and Chung n 965 [9]. Subsqunly, Wlson and ckl [] hav sudd m dpndn FE wh varaonal prncpl n hr work on ransn ha conducon problms wh Gurn s Varaonal prncpl [].Znkncz and Parkh [] drvd soparamrc fn lmn formulaons for -D ransn ha conducon problms o approma h soluon n spac and m. Argyrs.al [3,4] analyzd srucural problms by usng ral m-spac fn lmns. A parabolc m-spac lmn, an uncondonally sabl n h soluon of ha conducon problms hrough a quasvaraonal approach was usd by Tham and Chung [5]. Wood and Lws [6] compard h ha quaons for dffrn m-marchng schms. Howvr, s ncssary o choos vry small m-sps n ordr o ovrcom unwand numrcally nducd oscllaons n h soluon. From h pas fw yars, wavls hav bcom vry popular n h fld of numrcal appromaons. Among h dffrn wavl famls mahmacally mos smpl ar h Haar wavls. Du o h smplcy, Publshd by World Acadmc Prss, World Acadmc Unon

2 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp h Haar wavls ar vry ffcv for solvng ordnary and paral dffrnal quaons. In h prvous yars, many rsarchrs lk Bujurk and Shralash.al [7,8, and 9] and [67], Harharan and Kannan[] hav workd wh Haar wavls and hr applcaons. In ordr o ak h advanags of h local propry, Chn and Hsao [], Lpk [,3] rsarchd h Haar wavl o solv h dffrnal and ngral quaons. Haar wavl collocaon mhod (HWCM) wh far lss dgrs of frdom and wh smallr CPU m provds mprovd soluons han classcal ons, s Islam.al[4], In h prsn work, w us FEM and HWCM for solvng ypcal ha conducon problms. Th organzaon of h prsn chapr s n h followng mannr; Haar wavls and opraonal mar of ngraon n h gnralzd form ar shown n scon. In scon 3 and 4, mhod of soluon of FEM and HWCM ar dscussd rspcvly. Scon 5 dals wh numrcal fndngs wh rror analyss of h ampls. Fnally, h concluson of h proposd work s dscrbd n scon 6.. Haar wavls and opraonal mar of ngraon Th scalng funcon h Th Haar wavl famly for, s dfnd as for h famly of h Haar wavls s dfnd as for, h ( ) (.) ohrws k k.5 for, m m k.5 k h ( ) for, m m ohrws In h dfnon (.), h ngr m l, l,,..., J, ndcas h lvl of rsoluon of h wavl and ngr k,,..., m s h ranslaon paramr. Mamum lvl of rsoluon s J. Th nd n (.) s calculad usng m k. In cas of mnmal valus m, k hn. Th mamal valu j.5 of s J.L us dfn h collocaon pons j, j,,...,, dscrz h Haar funcon h, n hs way, w g Haar coffcn mar H, j h( j) whch has h dmnson. For nsanc, J 3 6, hn w hav H6, Th opraonal mar of ngraon va Haar wavls s oband by ngrang (.) s as, (.) JIC mal for subscrpon: publshng@wau.org.uk

3 9 and S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Ph h ( ) d (.3) Qh Ph ( ) d (.4) Ths ngrals can b valuad by usng quaon (.) and hy ar gvn by k k k.5 for, m m m k k.5 k Ph ( ) for, m m m ohrws k k k.5 for, m m m k k.5 k for, Qh 4m m m m k for, 4m m Ohrws For nsanc, J 3 6, from (.5) hn w hav, Ph(6,6) 3 and from (.6) w g (.5) (.6) JIC mal for conrbuon: dor@jc.org.uk

4 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp also Qh(6,6) and for nsanc J 3 6, hn w hav Ch Ph d Ch(6,6) Fn Elmn Mhod for h umrcal Soluon of Parabolc quaons Cas. FEM n on dmnson: Th quaon can b wrn wh h gvn condons u u a cu c f (, ) n : To formula a FEM modl of h govrnng dffrnal quaon, h doman, M (=) lmns. A Typcal lmn s shown by,, a b (3.) s dvdd no whr ar h global cocordnas of h nd nods of h lmn. W bgn wh h wak formulaon by mulplyng h gvn quaon wh h s funcon w, w g b u u w (3.) a c u c f d a d b w u u a b (3.3) a c wu c w f d w w a d W assum fn lmn soluon n h form, n s s j j j u(, ) u L ( ) a b (3.4) JIC mal for subscrpon: publshng@wau.org.uk

5 9 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons whr s h nal m and s h m nrval and, h wo lnar lmns ar gvn by n j &, L( ) & L( ). h h Th fn lmn soluon whch s connuous a spac s oband as whr s In mar form, w g L ( ) (3.5) (3.6) b b dl dl j Kj a d, F L Ljd Q (3.7) d d Th wak formulaon s a varaonal samn of h gvn problm n whch s ngrad agans a s funcon, and hnc afr dscrzaon, rsulng marcs can b asly solvd. Dscrzaon: Rwrng h fn lmn modl n h mar form (4.5) n h form (By akng M M M, K K ) K u M u F (3.8) b b dl dl j whr M j L Ljd, Kj d (3.9) d d Th smdscr quaons of a ypcal lmn for h choc of h lnar nrpolaon funcons ar h u u F (3.) 6 u h u F whr s h lngh of h lmn. For dffrn dffrnc (.. forward, backward and Crank-colson) schms, gnral form of -famly of h appromaon s gvn by h Whr s h m sp and s h nal m, and s n u(, ) u L s u L, a M j s j j j j j (3.) (3.) F F F, b b, (3.3) s, s s s s s Hr w usd Backward dffrnc schm o approma h soluon wh = and whch s sabl and ordr of accuracy s O( ). For M = -Elmn modl, h famly of m appromaon schms ar pu n h mar form as h h h h h h h h ( ) ( ) 3 h 3 6 h 6 3 h 3 6 h 6 u u F h h h h h h h h h h h h u 6 h 6 3 h 3 6 h 6 ( ) ( ) ( ) u F u 6 h 6 3 h 3 6 h 6 3 u F s 3 3 s s h h h h h h h h ( ) ( ) 6 h 6 3 h 3 h 6 h 6 3 h 3 (3.4) n K u M u F K K M b j c L Ljd, a a M j b j c L Ljd, a a s s s M K u M K u F F a s s s s s K M b K, K M b K s s s s JIC mal for conrbuon: dor@jc.org.uk

6 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp FEM conssncy, accuracy and sably: Th (3.) rprsns an -famly of appromaon, rror s n h soluon a ach m sp. If h rror s boundd, h soluon schm s assumd o b sabl. Th numrcal schm s conssn, whn h round off and runcaon rror nds o zro whn. Th sz of h m sp wll conrol boh accuracy and sably. Th numrcal soluon convrgs o h ac soluon whn h numbrs of lmns ar ncrasd and m sp s dcrasd. Th numrcal schm s convrgn f sasfs boh sabl and conssn condons. Cas. FEM n Two dmnsons: Th govrnng quaon for ransn ha conducon problms wh a dsrbud sourc b gvn by u u u K F c n y may (3.5) Subjcd o (3.6) whr s h mpraur funcon, s nal mpraur fld, h spcfd hrmal conducvy, h dnsy, h spcfc ha,, s a boundd doman wh a boundary wh h followng condons (On (3.7) Whr s boundary surfac mpraur, s h nnsy of ha npu, h ha ransfr coffcn, and h known funcons, s h ouward normal vcor of h boundary surfac, and h nvronmnal mpraur. Tm-doman dscrzaon: Ingrang h fld quaon (3.5) w r and usng condon (3.6), w oban (3.) Th Ingral quaon canno b consdrd analycally, so o approma h mpraur u(, y, ) by gvn funcons, dvd h m doman,t no qual nrvals, whr s a gvn m. W can approma L u(, y, ) u s whr k /and as a lnar funcon of m varabls as hn (3.) bcoms u(, y,) u (, y) u n c u, q s u(, y, ) u u s us F F(, y, ) u K q (On ) (3.8) n u K hu ua (On ) (3.9) n n ' ' q (,, ) (, ) (,. ) u y cu y K u y F d M 3 m m u(, y, ) um(, y) u m, y um, y m / T Whr M Pung (3.) no (3.), w g (,, ) m(, ) { m(, ) m(, ) m(, ) ( m) / } (,, ) cu y cu y K u y u y u y F y d m kk cum Fm(, y) m,,,..., M T h ua (3.) (3.) (3.3) JIC mal for subscrpon: publshng@wau.org.uk

7 94 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Hnc, h rlad boundary condons bcom; u, y u, y, (3.4) On (3.5) um K q(, y, m ) (On ) (3.6) n um K h um ua, y, m (On ) (3.7) n Fn lmn formulaon: Th fn lmn formulaon rlad o (3.3) o (3.7) s basd on an ndd varaonal prncpl. I can b sad as um u m um kk cu m d dy k hu mds y (3.8) Th Fn Elmn mhod s usful o oban h numrcal soluon of (3.8). For hs h doman s dvdd no a numbr of lmns. For ach lmn, h unknown funcon may b oband by, (3.9) Whr h shap s funcon, h nodal valu of u, y n h lmn, s h numbr of nods n an lmn. For hs job, a 4-nod quadrlaral lmn s usd and s a lnar funcon of and y.subsung (3.9) n (3.8), w g T T um / um K um um G Whr s h lmn numbr, s h sffnss mar and h quvaln nodal forc vcor, whch gvs o T T T T K kk c d dy h ds y y 3 Whr Hr dnos h nr boundary of lmn. 4. Haar wavl collocaon mhod for h numrcal soluon of parabolc quaons Consdr h parabolc quaon of h form (3.) wh h gvn condons, L, u(, ) ah( ) whr a s,,,..., ar Haar coffcns o b drmnd and &' ar dffrnaons wh rspc o 3 & rspcvly. m m,,,, F y F y d K u y m m s m k q(, y, ) u ds k hu (, y, ) u ds F u d dy saonary m m m m m m m u m K m, m u y u G k q ds k h u ds Q d dy T T T a m 3,,, 3 3 m G m 3 u m (4.) JIC mal for conrbuon: dor@jc.org.uk

8 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp a Ingrang h quaon (4.) w. r.. from o, w g Whr s h nal m and Ingrang h (4.) wc w. r.. Pu Thn (4.4) bcoms s w g n (4.4) and by gvn condons w g Dffrnang (4.5) w. r.. s h m nrval g ( ) ach ( ) g ( s ) g( s ) g( ) hn w hav (4.6) g ( ) ach ( ) g ( s ) g( s ) g( ) Subsung h prssons of (4.)-(4.6) n (.) and by solvng, w g h Haar wavl coffcns s usng Inac won s mhod []. Pung h valus of s n (4.5), o oban h Haar wavl collocaon mhod (HWCM) basd numrcal soluon of h problm (3.). Convrgnc analyss of h Haar wavls: a h j s Lmma: Assum ha lvl sasfs h subsqun nqualy (4.) (4.3) (4.4) (4.5) wh h boundd frs drvav on (, ), hn h rror norm K ( 3 ) j (, ) C 7 From h abov quaon, s clar ha h rror bound s nvrsly proporonal o h lvl of rsoluon of h Haar wavl. Ths promss h convrgnc of h Haar wavl appromaon whn s ncrasd. 5. umrcal Compuaons wh Error Analyss Ths scon dals wh h mplmnon of h FEM and HWCM as dscrbd n scon3 and 4 o fnd h numrcal soluon of som of h parabolc yp problms. Ts Problm. Frs consdr h quaon of h form Subjc o h condons u(,) sn, u(, ) and u(, ) FEM Soluon: Comparng h (5.) wh (3.), w g a, c, f, hn from (3.) s u(, ) ( ) a h ( ) u(, ) s s u(, ) a Ph ( ) u(, ) u(, ) u(, ) s s u(, ) a Qh ( ) u(, ) u(, ) u(, ) u(, ) u(, ) s s s u(, ) u(, ) g ( ) a Ch ( ) g ( ) g ( ) g ( ) s s s u(, ) a Qh ( ) u(, ) g ( ) g ( ) s s u(, ) a Qh ( ) u(, ) g ( ) g ( ) u(, ) L ( ) s s u u,, a (5.) JIC mal for subscrpon: publshng@wau.org.uk

9 96 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons By pung h, M 4, and by assmblng h mar lmns, usng M M f, u,sn / 4,sn / 4,sn 3 / 4, and omng h frs and las row and columns (du o h boundary condons), w g, u u u Hnc h soluons ar u.4, u 3.65, u 4.4. For hghr valus of, FEM basd numrcal soluons ar prsnd n h Tabl & and n h Fg.. Tabl. Comparson of FEM, FDM and HWCM wh Eac soluons for =6 of h Ts Problm. =(/3) FEM FDM Eac HWCM Tabl. Error analyss of h Ts Problm wh /. HWCM Soluon: Assum ha (FEM) (FDM) (HWCM) E-.544 E E E E E E E-.657 E E E E L L L.6 E E-4.66 E E-4 u(, ) ah( ) Ingrang h quaon (5.) w. r.. from o, w g s Whr h nal s m and s s h m nrval Ingrang h (3.6) wc w. r.. from o, w g s 7.4 E E- u(, ) ( ) a h ( ) u(, ) s s (5.) (5.3) JIC mal for conrbuon: dor@jc.org.uk

10 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp HWCM Eac FEM FDM u Fg.. Comparson of HWCM, FEM & FDM wh Eac soluons for =3 of h Ts Problm.. Pu n (5.5) and by usng gvn condons w g Thn (5.5) bcoms u(, ) aqh ( ) sn ach ( ) Dffrnang (5.6) w. r.. hn w hav u(, ) aqh ( ) ach ( ) Subsung h prssons of (5.3) & (5.7) n (5.) w hav aqh ( ) ach ( ) ah ( ) u(, s ) (5.4) (5.5) (5.6) (5.7) (5.8) By solvng (5.8) usng Inac won s mhod [5], w g h Haar wavl coffcns s = [38.74,.36, -., 3.74, -.9, -.5, 3.34,.44, -7.54, -3.8, -.7, -.46,.89,.5, 4.37 & 5.96]. Subsung h valus of u(, ) a Ph ( ) u(, ) u(, ) u(, ) s s u(, ) a Qh ( ) u(, ) u(, ) u(, ) u(, ) u(, ) a s s s u(, ) u(, ) a Ch ( ) s s n (5.6), o oban h numrcal soluon of h problm (5.) and s prsnd wh Fn lmn mhod (FEM) and Fn dffrnc mhod (FDM) soluons n comparson wh h ac soluon u(, ) sn n h Tabl for =6 and Fg. for =3. Th rror analyss for supror valus of s shown Tabl wh /. a JIC mal for subscrpon: publshng@wau.org.uk

11 98 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Ts Problm ow consdr h quaon of h form wh h gvn condons u (,), u(, ) and u(, ) Du o h nal condon, h FEM gvs h rval soluon as dscussd n scon 3. Th soluon of (5.9) s oband usng h mhods prsnd n scon 4, Haar coffcns s = [4.37, -.73, -.6, -.89, -.37, -.3, -.73, -.43, -.5, -.4, -.4, -.8, -.8, -.47, -.87 & -.6] and h corrspondng HWCM soluon s prsnd n comparson wh h FDM and ac soluon n 3 ( ) n u(, ) 6 sn n h Tabl 3 for =6 and Fg. for =3. Th 3 n 3 6 n n rror analyss for hghr valus of s gvn n Tabl 4 wh /. Tabl 3. Comparson of FDM and HWCM wh Eac soluons for =6 of h Ts Problm.. =(/3) FDM Eac HWCM Tabl 4. Error analyss of h Ts Problm. wh /. (FDM) (HWCM) 8.6 E-.584 E E E E E E E u u,, L 7.47 E E-4 L E E-4 a (5.9) JIC mal for conrbuon: dor@jc.org.uk

12 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp HWCM Eac FDM. u Fg.. Comparson of HWCM & FDM wh Eac soluons for =3 of h Ts Problm.. Ts Problm 3. consdr h quaon of h form (4.45), (5.) Wh h gvn condons u(,) cos ( ), u(, ) ( ) and u(, ) FEM Soluon: Comparng h (5.) wh (3.), by pung h, M 4, and by assmblng h mar M M f, u,sn / 4,sn / 4,sn 3 / 4, and omng h frs and las row and lmns, usng u u u,, columns (du o h boundary condons), w g u u u Hnc h soluon su.749, u 3.387, u For hghr valus of, FEM basd numrcal soluons ar prsnd n h Tabl 5 & 6 and n h Fg. 3. Tabl 5. Comparson of FEM, FDM and HWCM wh Eac soluons for =6 of h Ts Problm.3. =(/3) FEM FDM Eac HWCM JIC mal for subscrpon: publshng@wau.org.uk

13 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Tabl.6. Error analyss of h Ts Problm.3 wh. (FEM) (FDM) (HWCM) E E-.658 E E E E E- 6.9 E E E E E L L L E E E E-.44 E E-4 / HWCM Eac FEM FDM. u Fg. 3. Comparson of HWCM, FEM & FDM wh Eac soluons for =3 of h Ts Problm.3. HWCM Soluon: Wh h gvn condons u(,) cos ( ), u(, ) and u(, ) ( ) As n prvous ampls, h soluon of (4.45) s oband wh h Haar coffcns s = [-7., 3.6, 6.55, 5.8, 9.69, 7.85, 8.5, 6.74, 6.7, 4.3, 3.84, 4., 4., 4.5, 3.85 &.8] and h consqun HWCM soluon s compud and prsnd n comparson wh h FEM, FDM and ac soluon a JIC mal for conrbuon: dor@jc.org.uk

14 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp 88-7 ( ) u(, ) cos s shown n Tabl 6 wh. n h Tabl 5 for =6 and Fg. 3 for =3. Th rror analyss for hghr valus of Ts Problm 4. Consdr sngularly prurbd convcon-domnad dffuson quaon Whr / u u u (, ),, (, ) wh h gvn condons u (,), u(, ) and u(, ). and (5.) As n prvous Ts Problms, h soluon of (5.) s oband wh h Haar coffcns s = [3.4, -58.6, -5.3, , -9.7, -.38, -4.3, , -3.6, -3.7, -.4, -.7, -.4, -3.5, -4. & -9.47] and h rlad HWCM soluon s abulad n comparson wh h FDM and ac soluon ( ) u(, ) n h Tabl 7 for =6 and Fg. 4 for =3 for hghr valus of s gvn n Tabl 9 wh for dffrn. /. Th rror analyss for Tabl 7. Comparson of FDM and HWCM wh Eac soluons for =6 of h Ts Problm.4 for.8. =(/3) FDM Eac HWCM Tabl 8. Comparson of FDM and HWCM wh Eac soluons for =6 of h Ts Problm.4 for.9. =(/3) FDM Eac HWCM ( ).8 a JIC mal for subscrpon: publshng@wau.org.uk

15 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons of. / Tabl 9. Error analyss of h Ts Problm.4 wh for dffrn L L L L L L (FDM) (HWCM) (FDM) (HWCM) (FDM) (HWCM) E E E-.483 E E E E E-.88 E-.86 E E-.744 E E- 9.8 E-.549 E-.637 E-.3 E E E E E E-.96 E E E E E E E-3.58 E E E E E E E-3 Th rror analyss for hghr valus of s gvn n Tabl wh / for hghr valus Ts Problm 5. ow consdr h wo dmnsonal problm as, u K u F c, (n, y, 3, y 3 ) (5.) whr c, K.5, L 3, F, subjc o h boundary and nal condons as u(, y, ) u(,, ) u( L, y, ) u(, L, ) o; u(, y) 3.. (5.3) Th analycal soluon for h (5.) s, (,, ) sn( / 3)sn( / 3) p[ / 3 ] n n j u y A n j y K n j n j whr A 43 [ ] / nj n. Du o h symmry, only on quadran of h soluon doman s formd by lmns n h problm. Som rsuls ar shown n Tabls and. Whr Tabl gvs h dsrbuon of mpraur, y,.5,.5,.h wh and wh analycal soluon. Tabl gvs h varaon of mpraur a Tm sp, Th Rsuls of HWCM ar basd on Scon 4. Th dsrbuons of mpraur wh analycal soluons for Ts problm 5 ar gvn n abl. JIC mal for conrbuon: dor@jc.org.uk

16 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp HWCM Eac FDM.5. u Fg. 4. Comparson of HWCM & FDM wh Eac soluons for =3 of h Ts Problm 4 for HWCM Eac FDM.4 u Fg. 5. Comparson of HWCM & FDM wh Eac soluons for =3 of h Ts Problm 5.4 for.9. JIC mal for subscrpon: publshng@wau.org.uk

17 4 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Tabl.. Error analyss of h Ts Problm.4 wh for hghr valus of. (FDM) (HWCM) (FDM) (HWCM) (FDM) (HWCM) E-.87 E E E-5.76 E E E E E E E E E E E-.556 E E E E-.845 E E-.397 E-5.55 E-.397 E E E E E E E-3 / 7.6 E E E E E E-7 Tabl.. Th dsrbuon of mpraur wh analycal soluon for Ts Problm.5. Mhods FEM.5h HWCM.5h Eac L L L L L L.h.h Ts Problm 6. Consdr h ransn ha conducon problm T T T, y Subjc o h boundary condons, for, T T, y,,,,, T (, y, ), T (,, ) y T(, y,), y and h nal condons. Analycal soluon s m n m, n,3,5,... mn. 4 (5.4). (5.5) 44 W chck for msh of lnar rangular lmns o modl h doman, and analyz h Sably and accuracy of h Crank-colson mhod for.5 whch s uncondonally sabl. For h hghr valus of, w ak cr ma JIC mal for conrbuon: dor@jc.org.uk

18 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp U U U U U Th boundary condons of h problm ar gvn by. Haar wavl collocaon mhod and Fn lmn basd mhod numrcal soluons ar oband for h dffrn valus of of h Ts Problm 6, Tmpraur agans msh and Tm sp., y,.5,.5,. ar shown n Tabl Rsuls ar wh Crank-colson schm and shown n Tabl 3. Tabl. Th dsrbuon of mpraur wh analycal soluon for Ts Problm.6. Mhod =5 = =5.h =.5 ar FEM.5h.h.5h HWCM.h h h h EXACT Ts Problm 7. Lasly, consdr h -D Parabolc problm, u u u y, y Subjcd o h condons, y u(, y,) sn,, y ; u, y,,, y,. Wh h analycal soluon, n 4 n ( n ) u(, y, ) sn y sn p n n 4 Errors of h Ts Problm 7 wh ar gvn n Tabl 4. (5.6) (5.7) JIC mal for subscrpon: publshng@wau.org.uk

19 6 S. C. Shralash al.: Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal S oluon of Parabolc Typ Paral Dffrnal Equaons Tabl 3. Rsuls ar wh Crank-colson schm and =.5 of h Ts Problm.6. od FEM HWCM EXACT Tabl 4 Errors of h Ts Problm.7 wh. Mhod.. h. h.5 h.5 h. h.5 h.5 FEM O.63E-.63E-.63E-.63E-3.58E-3.57E-3 HWCM.44E-.E-.46E-.84E-4.3E-4.E-4 EXACT.47E-.4E-.46E-.89E-4.35E-4.E-4 6. Concluson In hs paprr, w appld h Haar wavl collocaon mhod (HWCM) for h numrcal soluon of parabolc s of dffrnal quaons. I has bn wll dmonsrad ha whl applyng h nc proprs of Haar wavls, h parabolc yp paral dffrnal quaons b abl o b solvd convnnly and accuraly by usng HWCM sysmacally. In h frs Ts Problm FEM & FDM gvs br rsuls han h HWCM. Whl n h scond Ts Problm, FDM rsuls closr o HWCM whr FEM gvs h rval soluon du o h nal condon. Thrd Ts Problm shows ha h FEM & FDM gvs h pabl prformanc as compard o HWCM. In h fourh Ts Problm du o h valu of h rsuls ar vard, as h valu of s lss han, h FDM rsuls ar br han HWCM. Th HWCM rsuls closr o h FDM as h valu of s closr o. For h hghr valus of, h HWCM rsuls ar br han h FDM. Th las four.. -D Ts Problm shows h robusnss of h HWCM ovr FEM whn compard wh ac soluon. Th major advanags of h HWCM ar s smplcy and small compuaon coss: s du o h sparcy of h ransform marcs and o h small quany of sgnfcan wavl coffcns. Hnc h Haar wavl collocaon mhod s compv n comparson wh h classcal mhods. 7. Rfrncs P. Sngh, Kapl K. Sharma, umrcal appromaons o h ranspor quaon arsng n uronal varably, Inrnaonal Journal of Pur andappld Mahmacs, 69 (), Guspp Ponrll, Flppo d Mon, Mass dffuson hrough wo-layr mda: An Applcaon o h drug-lung JIC mal for conrbuon: dor@jc.org.uk

20 Journal of Informaon and Compung Scnc, Vol. (6) o.3, pp sn, Inrnaonal Journal of Ha and Mass Transfr, 5 (7), M. K. Kadalbajoo, A. Awash, A numrcal mhod basd on crank-ncolson schm for Burgrs quaon, Appl. Mah. Compu., 8 (6) F. d Mon, Transn ha conducon n on-dmnsonal compos slab:a naural Analyc approach, Inrnaonal Journal of Ha and Mass Transfr, 43 (), R. I. Mdvdsk, Y. A. Sgunov, Mhod of umrcal soluon of on-dmnsonal mulfron Sfan problms, Inzhnrno-Fzchsk Zhurnal, 58 (989) L. J. T. Doss, A. K. Pan, S. Padhy, Galrkn mhod for a Sfan-yp problm n on spac dmnson, umrcal Soluon for Paral Dffrnal Equaons, 3 (4) (998) G. Roos, M. Syns, L. Tobska, umrcal Mhods for Sngularly Prurbd Dffrnal Equaons, Sprngr- Vrlag, 996. M. E. Gaml, A Wavl-Galrkn mhod for a sngularly prurbd convcon-domnad dffuson quaon, Appl. Mah. & Compu., 8 (6) O. C. Znkwcz and Y.K.Chung, Fn lmn n h soluon of fld problm, Th Engnr,, 57-5 (965). E.L.Wlson and R.E.ckll, Applcaon of h fn lmn mhod o ha conducon analyss, ucl.eng Ds., (966). M. E. Gurn, Varaonal prncpl for lnar nal-valu problm, Q. J. Appl. Mah.,, 5-56 (964). O.C.Znkwcz and C.J.Parkh, Transn fld problms: wo dmnsonal and Thr dmnsonal analyss by soprmrc fn lmns, In. j. numr. Mhods ng.,, 6-7 (97) J. H. Argyrs and A. S.L. Chn, Applcaon of fn lmn n spac and m, Ing. Archv. 4, (97). J. H Argyrs and D. W. Scharpf, Fn lmn n m and spac, Aronau, J. Roy. Arona. Soc., 73, 4-44 (973). L. G. Tham and Y. K. Chung, umrcal soluon of ha conducon problms by Parabolc m-spac lmn,in.j.numr.mhods ng., 8, (98). W. I. Sood and R. W. Lws, A comparson of m-marchng schms for htransn ha conducon quaon. In.j.numr.mhods ng., 9, (975).. M. Bujurk, C. S Salmah, S. C. Shralash, umrcal Soluon of Sff Sysms from onlnar Dynamcs Usng Sngl-rm Haar Wavl Srs, onlnar Dyn (8) 5: M. Bujurk, S. C. Shralash, C. S. Salmah, Compuaon of gnvalus and soluons of rgular Surm- Louvll problms usng Haar wavls, J. Compu. and Appl. Mah. 9 (8) 9-. M. Bujurk, S. C. Shralash, C. S Salmah, An Applcaon of Sngl-rm Haar Wavl Srs n h Soluon of onlnar Oscllaor Equaons, J. Compu. And Appl. Mah. 7 () G. Harharan, K. Kannan, A comparson of Haar wavl and Adoman dcomposon mhod for Solvng ondmnsonal racon-dffuson quaons, I. J. Appl. Mah. & Compu., () 5 6 JIC mal for subscrpon: publshng@wau.org.uk