Exponential B-Spline Solution of Convection-Diffusion Equations

Transcription

1 Appled Matematcs ttp://dxdoorg/436/am3469 Publsed Ole Jue 3 (ttp://wwwscrporg/oural/am) Expoetal B-Sple Soluto of Covecto-Dffuso Equatos Reza Moammad Departmet of Matematcs Uversty of Neysabur Neysabur Ira Emal: rezmoammad@gmalcom Receved September 5 ; revsed Aprl 3; accepted Aprl 3 3 Copyrgt 3 Reza Moammad Ts s a ope access artcle dstrbuted uder te Creatve Commos Attrbuto Lcese wc permts urestrcted use dstrbuto ad reproducto ay medum provded te orgal work s properly cted ABSTRACT We preset a expoetal B-sple collocato metod for solvg covecto-dffuso equato wt Drclet s type boudary codtos Te metod s based o te Crak-Ncolso formulato for tme tegrato ad expoetal B-sple fuctos for space tegrato Usg te Vo Neuma metod te proposed metod s sow to be ucodtoally stable Numercal expermets ave bee coducted to demostrate te accuracy of te curret algortm wt relatvely mmal computatoal effort Te results sowed tat use of te preset approac te smulato s very applcable for te soluto of covecto-dffuso equato Te curret results are also see to be more accurate ta some results gve te lterature Te proposed algortm s see to be very good alteratves to exstg approaces for suc pyscal applcatos Keywords: Expoetal B-Sple; Covecto-Dffuso Equato; Collocato; Crak-Ncolso Formulato; Ucodtoally Stable Itroducto Covecto-dffuso equato plays a mportat role te modelg of several pyscal peomea were eergy s trasformed sde a pyscal system due to two processes: covecto ad dffuso Te term covecto meas te movemet of molecules wt fluds wereas dffuso descrbes te spread of partcles troug radom moto from regos of ger cocetrato to regos of lower cocetrato Also ts equato descrbes advecto-dffuso of quattes suc as eat eergy mass etc Tey fd ter applcatos water trasfer sols eat trasfer drag flm spread of pollutats rvers dsperso of tracers porous meda Tey are also wdely used studyg te spread of solute a lqud flowg troug a tube log rage trasport of pollutats te atmospere flow porous meda ad may oters [-4] We cosder te tal-value problem for te oe-dmesoal tme-depedet covecto-dffuso equato u u u x L t T () t x x subected to te tal codtos u x x x L () ad wt approprate Drclet boudary codtos u t g t u L t g t t T (3) were te parameter s te vscosty coeffcet ad s te pase speed ad bot are assumed to be postve x g t ad g t are kow fuctos wt suffcet smootess It s ecessary to calculate te trasport of flud propertes or trace costtuet cocetratos wt a flud for applcatos suc as water qualty modelg ar polluto meteorology oceaograpy ad oter pyscal sceces We velocty feld s complex cagg tme ad trasport process caot be aalytcally calculated ad te umercal approxmatos to te covecto equato are dspesable [4] Tey are also mportat may braces of egeerg ad appled scece Terefore may researcers ave spet a great deal of effort to compute te soluto of tese equatos usg varous umercal metods Tere are may studes o te umercal soluto of tal ad tal-boudary problems of covecto-dffuso equato [3-4] clude fte dfferece metods Galerk metods spectral metods wavelet-based fte elemets B-sple metods ad several oters Tese equatos are caracterzed by o dsspatve advectve trasport compoet ad a Copyrgt 3 ScRes

2 934 R MOHMADI dsspatve dffusve compoet We dffuso s te domat factor aall umercal profles go well O te cotrary most umercal results exbt some combato of spurous oscllatos ad excessve umercal dffuso we advecto s domat trasport process Tese beavors ca be easly explaed usg a geeral Fourer aalyss lttle progress as bee made to overcome suc dffcultes effectvely Usg extremely fe mes s oe suc alteratve but s ot prudet to apply t as t s computatoally costler So a great effort as bee made o developg te effcet ad stable umercal tecques Nguye ad Reye [5] preseted a space-tme leastsquares fte-elemet sceme for advecto-dffuso equato Coda [7] cosdered several fte-elemet metods for solvg te dffuso-covecto-reacto equato ad sowed tat Taylor-Galerk metod as a stablzato effect smlar to a sub grd scale model wc s tur related to te troducto of bubble fuctos Dega [] developed several dfferet umercal tecques for solvg te tree-dmesoal advecto-dffuso equato wt costat coeffcets ad compared tem wt oter metods lterature Tese tecques are based o te two-level fully explct ad fully mplct fte dfferece approxmatos Dega [3] developed a ew practcal sceme desgg approac wose applcato s based o te modfed equvalet partal dfferetal equato (MEPDE) Dega ad Moebb [] preseted ew classes of g-order accurate metods for solvg te two-dmesoal usteady covecto-dffuso equato based o te metod of les approac Tese metods are secod-order accurate ad tecques tat are trd order or fourt order accurate Dega [4] derved a varety of explct ad mplct algortms based o te wegted fte dfferece approxmatos dealg wt te soluto of te oe-dmesoal advecto equato Dega ad Saker [] obtaed te soluto of Caucy reacto-dffuso problem va varatoal terato metod It s well-kow tat a good terpolatg or approxmatg sceme addto to te stadard requests as good approxmato rate low computatoal cost u- mercal relablty sould possess te capablty of reproducg te sape of te data It as bee foud te lterature tat usg pecewse polyomal fuctos leads to better covergece results ad smpler proofs ta usg polyomals Recetly Sple ad B-sple fuctos togeter wt some umercal tecques ave bee used gettg te umercal soluto of te dfferetal equatos Kadalbaoo ad Arora [4] used Taylor-Galerk metods togeter wt te type of sples kow as B- sples to costruct te approxmato fuctos over te fte elemets for te soluto of tme-depedet advec- to-dffuso problems Mttal ad Ja [] dscussed collocato metod based o redefed cubc B-sples bass fuctos for solvg covecto-dffuso equato wt Drclet s type boudary codtos Te ma obectve of ts study s to develop a user fredly ecoomcal ad stable metod wc ca work for ger values of Péclet umber for covecto-dffuso equato by usg redefed cubc B-sples collocato metod I te curret paper we develop te collocato metod by usg te expoetal B-sple fucto for u- mercal soluto of te covecto-dffuso equato Our ma am s to mprove te accuracy of B-sple metod by volvg some parameters wc eable us to obta te classes of metods Our metod s a modfcato of cubc B-sple metod for soluto of () Applcato of our metod s smple ad comparso wt te exstg well-kow metods s accurate Wle solvg tal boudary value problems partal dfferetal equatos te procedure s to combe a sple approxmato for te space dervatve wt a Crak-Ncolso fte dfferece approxmato for te tme dervatve Te combato of a fte dfferece ad a expoetal sple fucto tecques provde better accuracy ta te fte dfferece metods Terefore te tme dervatve s replaced by fte dfferece represetato ad te frst-order space ad secod-order space dervatves by expoetal B-sple Also we study stablty for te ew metod ad wll sow tat t s ucodtoally stable We use te expoetal B-sples bass tat leads to te tr-dagoal system wc ca be solved by te well kow algortm Numercal examples are preseted wc demostrate tat te preset sceme wt expoetal B-sples gves more accurate approxmatos ta te sceme usg cubc B-sples Smlar to te metod [] our metod s a user fredly ecoomcal ad stable metod wc ca work for ger values of Péclet umber for covecto-dffuso equato Ts paper s arraged as follows I Secto te B- sples bass for te space of expoetal sple s gve ad some terpolato results for te expoetal sple terpolate are stated I Secto 3 te costructo of te expoetal B-sple collocato metod for te soluto of te covecto-dffuso equato s descrbed Te stablty aalyss of te preseted metod s dscussed Secto 4 I Secto 5 umercal expermets are coducted to demostrate te effcecy of te proposed metod ad cofrm t s teoretcal beavor Tese computatoal results sow tat our proposed algortm s effectve ad accurate comparso wt te lterature Fally results of expermets ad te coclusos are cluded Secto 7 Let deotes te value of x at te odal Copyrgt 3 ScRes

3 R MOHMADI 935 pots x tat s x Te we use te remag parts of te paper max Expoetal Sple Fuctos Let I L be te gve terval ad be te partto of I defed as x x x x x L wt mes spac- L g Te fucto Sx t s sad to be a cubc sple over f Sx t C L ad S x t restrcted t o x x s a cubc polyomal for Te cubc sple Sx t s well kow to ave te followg aal ogue te beam teory Cosder a smply supported beam wt supports x t u Te Sx t te deflecto of te beam s a soluto to te dfferetal equato EID S xt M for x x x Here E = Youg s modulus I = Crosssectoal momet of erta M = Bedg momet ad D deotes te secod order dfferetal operator Dfferetatg te above equato twce we obta te followg two pot boudary value problems o eac subterval x x 4 DS x t for x x x S x t u S x t u t Z S x t S x Z were S ad are uquely determed from te codto S S C L for gve S ad S L 4 Here D s te fourt order dfferetal operator Te cubc sple Sx t so defed as a tedecy to exbt uwated udulatos Te above aalogy suggests tat te applcato of te uform teso betwee te supports mgt be a remedy to te problem Te te beam equato becomes EID Sxt M for x x x Let be te set of teso para- meters defed o eac subterval x x Suppose te te above cosderato le ads EI to defe te expoetal sple x t as te soluto to te boudary value problems o eac subterval x x 4 D D x t for x x x x tu x t x t x t u (4) (5) wt mples tat D x x x x yet to be determed Note tat 4 wc gves te cubc sples; wle mples tat D x x x x wc gves te lear sples Followg [56] te soluto of te above boudary value problem s xt s s x x x x s xxu x x u were s are prescrbed o-egatve real umbers ad x s a expoetal sple of order four les te spa xe x e x Tus geeral we ca defe te expoetal sple k of order k as te fucto xc L wose restrcto o o-empty terval x x for k3 x x spa x x e e les te Amog te varous classes of sples te polyomal sple as bee receved te greatest atteto prmarly because t admts a bass of B-sples wc ca be accurately ad effcetly computed It as bee sow tat te expoetal sple also admts a bass of B- sples wc ca be defed as follows Let B x be te B-sple cetered at x ad avg a fte support o te four cosecutve tervals x k x k k Accordg to McCart [7] te B x ca be defed as were ad g xx x x x gx x xx x B xgxx xx x gx x xx x oterwse gxbx s x g x a b xce d e x x (6) (7) Copyrgt 3 ScRes

4 936 R MOHMADI a cos cos s cos cos s b cos s cos b cos s e cos s e e cosse os s cos c 4 cos s cos d 4 c Eac bass fucto B x s twce cotuously dfferetable Te values o f Bx B x ad B x at te odal pots x s are sow Table Let Sx t be te cubc sple ad x t be te expoetal sple Now we cosder to te followg results wc ave bee establsed by Pruess [8] addtoal kots outsde te rego postoed at: x x ad x x Te set of expoetal B-sples B B B forms a bass for fuctos defed over te problem doma L Te expoetal B-sples (Equato (7)) ad t s frst ad secod dervatves vas outsde te terval x x A approxmate soluto u x t to te aalytcal soluto u x t wll be sougt form of a expaso of B-sples: u x t t B x (8) were are tme depedet parameters to be determed from te expoetal B-sple collocato form of Equato () Usg te expresso (8) ad Table odal value u ad ts frst ad secod dervatves at te kots x are obtaed terms of te elemet parameters by: s Teorem u u x cos s 6 4 D S max k Zk 3 s Sml ar to te above teorem we ave aoter com- cos s Teorem cos u u x cos s 6 4 D S max k k 3 cos Tese teorems togeter wt te de Boor-Hall error cos s pao result due to McCart [9] estmate Preter leads to te followg corollary 4 Corollary If u x t L te tere exts a stat depedet of suc tat u u co x 4 D u 3 Numercal Metod Te rego L s parttoed t o fte elemets of equal legt by kots x su c tat x x x x L Let B be te expoetal B-sples wt bot kots x ad Table Expoetal B-sple bass values s cos s s s s cos cos s (9) () () To apply te proposed metod dscretzg te tme dervatve te usual fte dfferece way ad applyg B x B x x x x x x B x s cos s s cos s cos cos cos s cos s s cos s s s cos s cos s Copyrgt 3 ScRes

5 R MOHMADI 937 Crak-Ncolso sceme to () we get currece relato oce te tal vectors ave bee com ted from te tal ad boudary c o u u ux u pu odt s x k 3 Treatmet of Boudary Codtos () uxx uxx I order to elmate te parameters ad from te system we ave used te boudary codt os (3) From (3) we ave were k s te tme step u x t gt ux t g t Substtutg te approxmate soluto u for u ad puttg te values of te odal values u ts dervatves Expadg u terms of approxmate expoetal usg Equatos (9)-() at te kots Equato () B-sple formula from (9) at x puttg (9) yelds te followg dfferece equato wt te var- we get ables s (3) cos s (4) s g t were cos s k s kcos Smlarly at x puttg (9) we get 4kcos s s s cos s (5) k cos s s g t k skcos cos s 4kcos s Solvg te obtaed equatos we get te values of k skcos terms of Smlarly ca be expressed terms of 4kcos s 3 Te Ital State s k cos s Te tal vector ca be determed from te tal codto u x x wc gves equato k s kcos 3 ukows For te determato of te 4kcos s ukows relatos at te kot are used Te system (3) cossts of lear equatos u x g ux g of 3 ukows T To obta a uque soluto to ts system two add- 4 Stablty Aalyss toal costrats are requred Tese are obtaed from We ave vestgated stablty of te preseted metod te boudary codtos Imposto of te boudary by applyg vo-neuma metod Now we cosder te codtos eables us to elmate te parameters tral soluto (o Fourer mode out of te full soluto) at ad from te system Elmatg ad a gve pot x te sy stem (3) s reduced to a tr-dagoal system of lear equatos wt ukows Ts exp (6) tr-dagoal system ca be solved by Tomas algortm were β te mode umber s te elemet sze ad Te tme evoluto of te approxmate soluto u x t s determed by te tme evoluto of te Now by substtutg (6) (3) ad symlfyg te vector wc s foud repeatedly by solvg te re- equato we obta 4 k 4k cos cos4 s s s s 4 k 4k cos cos 4 s s s s (7) Copyrgt 3 ScRes

6 938 R MOHMADI were Followg [] by substtutg we get 4 k cos cos 4 s s 4k s s 4 k cos cos4 s s k P e ad cos (8) were P e s called Péclet umber (8) 4 4P e cos 4 s s s 4 4P cos 4 s s e s 4 6P cos e 4 4 s s s 4 6P e 4 cos 4 s s s (9) Sce umerator (9) s less ta deomator terefore ece te metod s ucodtoally stable It meas tat tere s o restrcto o te grd sze e o ad k but we sould coose tem suc a way tat te accuracy of te sceme s ot degraded 5 Numercal Illustratos I ts secto some umercal examples are preseted to evaluate te performace of te proposed metod We cosder sx covecto-dffuso equatos wc te exact solutos are kow to us To llustrate our preseted metod ad to demostrate t s applcablty computatoally computed solutos for dfferet values of k ad are compared wt exact solutos at grd pots ad wt te results exs tg metods All pro- ru Matematca 6 Te computed ab- grams are solute errors ad maxsmume absolute errors umercal solutos are lsted Tables For te sake of comparso followg [] some mportat o-dmesoal parameters umercal aalyss are defed as follows: Courat umber: Te Courat umber s defed as k Cr Dffuso Number: Te dffuso umber s defed k as S Grd Péclet Number: Te Péclet umber s defed as Cr Pe S Numercal results cofrm tat te Péclet umber s g te covecto term domates ad we te Péclet umber s low te dffuso term domates Example 5 Cosder te covecto-dffuso equato [] u u u x t T t x x wt ad subect to te tal codtos x x exp Te teortcal soluto of ts problem s exp u x t x t Te boudary codtos are obtaed from te teortcal soluto I all computatos we take α = β = 9 = ad k = Copyrgt 3 ScRes

7 R MOHMADI 939 so tat Cr S ad Pe 5 Te absolute errors are tabulated Tables ad 3 for dfferet tme levels We observe tat our preseted metod s more accurate comparso wt Mttal ad Ja [] ad Douglas metods but less accurate to Ismal et al [] metod However It s evdet tat our metod s ucodtoally stable wle te metod [] s codtoally stable Example 5 Cosder te covecto-dffuso equ- ato [] u u u x t T t x x wt 35 ad subect to te tal codtos x x exp Te aalytcal soluto of ts problem s exp u x t x t Te boudary codtos are obtaed from te aalytcal soluto I all computatos we take ad k = so tat Cr 35 S ad Pe 599 Te absolute errors for dfferet tme levels are lsted Tables 4 ad 5 We observe tat our preseted metod s more accurate comparso wt exstg metods Also for ger value of P e our metod produces more accurate results comparso wt well kow metods Example 53 Cosder te followg equato [3] u u u x t T t x x subected to te followg tal codto x xexp 8 Te teortcal soluto of ts problem s Table Te absolute errors of our metod ad te metod [] for example 5 Our metod Mttal ad Ja [] x t t t 5 t t t 5 655E 868E 958E 73E 7 9E 7 58E E 9 346E 9 439E 9 54E 7 93E 7 36E 6 9 3E 9 36E 9 37E 9 537E 7 89E 7 E 6 Table 3 Te absolute errors of Douglas metod ad Ismal et al [] metod for example 5 Douglas metod Ismal et al [] x t t t 5 t t t 5 33E 4 77E 4 E 4 E 6 E 6 333E E 4 7E 4 5E 3 888E 6 33E 5 44E E 4 6 3E 4 883E 4 E+ 444E 6 888E 6 Table 4 Te absolute errors of our metod ad te metod [] for example 5 Our metod Mttal ad Ja [] x t t t 5 t t t 5 36E 3 86E 3 59E 3 6E 95E 45E 5 6E 45E 3E 9E 988E 73E 9 93E 65E 4E 99E 8E 33E Table 5 Te absolute errors of Douglas metod ad Ismal et al [] metod for example 5 Douglas metod Ismal et al [] x t t t 5 t t t 5 56E 7 37E 7 563E 7 56E 38E 565E 5 837E 7 38E 6 9E 6 839E 38E 9 9E E 6 8E 6 395E 6 33E 9 83E 9 397E 9 Copyrgt 3 ScRes

8 94 R MOHMADI x t ux t exp t 4 t Te boudary codtos are obtaed from te teortcal soluto I all computatos we take 8 ad k = so tat Cr 8 S ad Pe 8 Te absolute errors for dfferet tme levels are tabulated Table 6 Te results dcate tat te errors our metod more or less s sml ar to [] ad [3] for t = t = ad t 5 respectvely but te errors our metod for t are muc less ta [3] Example 5 4 Co sder te followg equat o [46 4] u u u x L t T t x x Te aalytcal soluto of ts problem s xp x t u x t e were t tal codto ad subected to te followg x x exp Te boudary codtos are obtaed from te aalytcal soluto I ts example frst smlar to [4] we take L = 5 5 ad k = so tat Cr S ad Pe Te absolute errors for dffe ret tme levels are tabulated Table 7 ad compared wt results [] Te results dcate tat te errors our metod more or less s sm lar to [] Te we compare our metod wt te metods [4] Table 8 were k ad t so tat Cr S Pe Te results sow tat our metod s cosderably accurate com- parso wt te metods [4] Followg [] we com- put umercal soluto at t 468 ad wt ad k so tat P e Te absolute errors are tabulated Table 9 ad compared wt results [] wc are lsted Table Te Tables sow tat our metod s stable besde ts our metod s a ccurate compar so wt te metod [] Table 6 Te absolu te errors of our metod ad te metods [3] for example 53 Our metod Mttal ad Ja [] Dega [3] x t t t 5 t t t t 5 87E 9 7 7E 9 73E 9 996E E 9 3 4E 5 9 6E 8E 8 8E 8 E 9 9E 8 76E 8 3 E 5 54E 9 3 5E 8 35E 8 39E 7E 8 7E 8 3E 5 84E E 8 36E 8 96E 9 333E 8 367E 8 9E 5 77E E 8 47E 8 359E 9 378E 8 459E 8 7E 5 33E E 8 44E 8 55E 9 4E 8 538E 8 7E 5 559E 7 374E 8 374E 8 694E 9 399E 8 589E 8 5E 5 383E 8 8E 8 8E 8 857E 9 3 6E 8 579E 8 E 5 8E E 8 65E 8 93E 9 55E 8 436E 8 E 5 3E 9 Table 7 Te absolute errors of our metod ad te metod [] for example 54 (wt k ) Our metod Mttal ad Ja [] x t t t t 4E 6 E 3 6E 6 E 3 5E 6 36E 5 E 6 365E 3 8E E 8E 5 756E 4 E 5 E 9 8E 5 6E E 5 E 8 463E 5 4E E 6 74E 8 47E 6 77E E 5 345E 7 378E 5 353E E 6 3E 6 7E 6 4E E 6 9E 6 898E 6 E 6 Copyrgt 3 ScRes

9 R MOHMADI 94 Table 8 Approxmate ad exact solutos for exa mple 54 ( wt = k = ad t = ) x Our metod Mttal ad Ja [] Kadalbaoo ad Arora [4] Exact Table 9 Te absolute errors of o ur metod for example 5 4 x t 4 t 6 t 8 t t 9E 7 835E 8 33E 8 873E 9 546E 47E 7 47E 7 34E 8 9E 9 437E 3 6E 7 3E 7 5E 8 68E 9 46E 4 79E 7 6E 7 659E 8 E 8 E E 7 83E 7 75E 8 45E 8 59E E 7 78E 7 77E 8 66E 8 73E 9 7 6E 7 47E 7 74E 8 65E 8 9E E 7 83E 7 55E 8 37E 8 74E 9 9 4E 7 E 7 3E 8 86E 9 39E 9 Table Te a bsolute errors of te metod [] for example 54 x t 4 t 6 t 8 t t 35E 7 849E 8 355E 8 E 8 649E 438E 7 49E 7 378E 8 689E 9 766E 3 67E 7 4E 7 557E 8 7E 8 345E 4 73E 7 6E 7 73E 8 6E 8 78E E 7 86E 7 79E 8 95E 8 3E E 7 83E 7 8E 8 E 8 4 3E E 7 5E 7 77E 8 99E 8 474E E 7 9E 7 564E 8 6E 8 46E 9 9 5E 7 5E 7 38E 8 945E 9 7E 9 For te sake of comparso wt Mttal we compute umercal soluto at t = ad 6 wt L ad k so tat Cr 5 S 5 ad P e Te observed results are lsted Table ad compared wt [] Ta ble Te umercal approxmatos seem to b e good agreemet wt te exact solutos Morever te results dcate tat our metod s accurate compar so wt te metod [] Example 55 Cosder te followg equato [9] u u u x t T t x x wt subected to te followg tal codto x x exp5 S S S dffuso umber Te teortcal soluto o f ts problem s Copyrgt 3 ScRes

10 94 R MOHMADI Table Te absolute errors of our metod for example 54 x t t t 3 t 4 t 5 t E 6 68E 7 6E 9 5E 5E 4 E+ 448E 5 96E 5 E 6 86E 8 67E 5E 5 E 4 89E 5 53E 5 86E 6 85E 8 E 9 44E 5 7E 4 947E 5 53E 5 5E 6 9E 7 5 3E 6 539E 5 3E 4 993E 5 E 5 788E E 8 357E 6 47E 5 E 4 963E 5 49E E 55E 7 7E 7 84E 5 94E 5 95E 5 Table Te absolute errors of te me tod [] for example 54 x t t t 3 t 4 t 5 t 6 5 9E 6 7E 7 36E 9 9E 53E 4 E+ 46E 5 E 5 8E 6 99E 8 79E E 5 4E 4 8E 5 6E 5 93E 6 83E 8 7E 9 453E 5 E 4 953E 5 68E 5 53E 6 34E E 6 54E 5 4E 4 995E 5 35E 5 79E 6 3 7E 8 48E 6 438E 5 4E 4 975E 5 79E E 58E 7 5E 7 88E 5 935E 5 96E 5 u x t exp 5 x t S t S S Te boudary codtos are ob taed from te teor- tcal soluto We compute maxmum absolute errors wt 5 k so tat Cr S 4 Pe 5 Te results are computed for dfferet tme levels ad lsted Table 3 Also we comp are our results wt te results [9] Te results dcate tat te errors our metod more or less s smlar to [] but te errors our metod are muc less ta te errors [9] Example 56 Cosder te followg equato [3] u u u x L t T t x x wt ad subected to te followg tal codto π π x exp5x cos x 5s x Te aalytcal soluto of ts problem s t π ux t exp 5 x exp 4 t π π cos x5s x Te boudary codtos are obtaed from te aaly- tcal soluto We compute maxmum abso lute errors wt k ad k so tat C r S Pe Te observed results are tabulated Table 4 ad compared wt results [3] Ts res ults sow tat our metod s cosderable accurate compare wt te metods [3] 6 Dscussos I ts artcle a expoetal B-sple sceme as bee proposed to solve te covecto-dffuso equato wt Drclet s type boudary codtos ad as bee effcetly llustrated To tackle ts te proposed sceme of expoetal B-sple space ad te Crak-Ncolso sceme tme ave bee combed By takg dfferet values of parameter we ca obta varous classes of metods But all computatos we coose 75 wc s te optmum case of our metod Stablty of ts metod as bee dscussed ad sow tat t s ucodtoally stable Te performace of te curret sceme for te prob lem as bee measured by comparg wt te exact solutos For comparso purposes te re sults of te sceme are exbted for varous values of te correspodg parameters Com parsos of te com- puted results wt exact solutos sowed tat te sceme s capable of solvg te covecto-dffuso equato ad producg gly accurate solutos wt mmal computatoal effort It was see tat te pro- posed sceme approxmate te exact soluto very well Te produced results were see to be more accurate ta Copyrgt 3 ScRes

11 R MOHMADI 943 Table 3 Maxmum absolute errors for example 55 Our metod Mttal ad Ja [] Cawla et al [ 9] t t t 5 t t t 5 t E 6 83E 8 7E 8 7E 6 388E 8 45E 8 8E 5 Our metod k Table 4 Absolute errors for example 5 6 wt = Our metod k Mttal ad Ja [] k Mttal ad Ja [ ] k Moebb ad Dega [3] k x t t t 5 t t t t 5 t t 745E 6 5E 7 35E 99E 6 8 6E 6 56E 7 49E 6E 6 8E 6 7E 5 6E 6 37E 548E 6 89E 5 3E 6 357E 564E 6 77E 6 3 6E 5 8E 6 53E 9E 5 95E 5 3E 6 587E 3E 5 47E 6 4 8E 5 6E 6 59E 89E 5 339E 5 59E 6 743E 94E 5 67E 6 5 3E 5 5E 6 384E 9E 5 5E 5 97E 6 64E 3E 5 9E E 5 794E 7 389E 49E 5 7E 5 83E 7 E 4E 5 3E 5 7 5E 4 69E 6 38E 9 5E 5 4E 4 683E 6 46E 9 55E 5 84E 5 8 5E 4 48E 5 34E 9 54E 5 6E 4 55E 5 36E 9 556E 5 55E 5 9 3E 4 95E 5 449E 9 4E E 4 5E 5 495E 9 4E 5 34E 5 some avalable results gve te lterature Ts tecque as bee see to be very good alteratve to some exstg oes solvg pyscal problems represeted by te olear partal dfferetal equatos REFERENCES [] R C Mttal ad R K Ja Redefed Cubc B-Sples Collocato Metod for Solvg Covecto-Dffuso Equatos Appled Matematcal Modellg Press do:6/apm9 [] M A Cela T F Russell I Herrera ad R E Ewg A Eulera-Lagraga Localzed Adot Metod for te Advecto-Dffuso Equato Advaces Water Resources Vol 3 No 4 99 pp 87-6 do:6/39-78(9)94- [3] M Dega Wegted Fte Dfferece Tecques for te Oe-Dmesoal Advecto Dffuso Equato Appled Matematcs ad Computato Vol 47 No 4 pp do:6/s96-33()667-7 [4] M K Kadalbaoo ad P Arora Taylor-Galerk B- Sple Fte Elem et Metod for te Oe-Dmesoal Advecto-Dffuso Equato Numercal Metods for Partal Dfferetal Equatos Vol 6 No 5 pp 6-3 [5] H Nguye ad J Reye A Space Tme Least-Squares Fte-Elemet Sceme for Advecto-Dffuso Equato Computer Metods Appled Mecacs ad Egeerg Vol pp do:6/45-785( 84 )9-4 [6] Rzwa-Udd A Secod-Order Space ad Tme Nodal Metod for te Oe-Dmesoal Covecto-Dffuso Equato Computers & Flud s Vol 6 No pp do:6/s45-793(96)39-4 [7] R Coda Comparso of Some Fte-Elemet Metods for Solvg te Dffuso-Covecto-Reacto Equato Computer Metods Appled Mecacs ad E- geerg Vol 56 No 998 pp 85- do:6/s45-785(97)6-5 [8] M M Cawla M A Al-Zaad ad D J Evas Geeralzed Trapezodal Formulas for Covecto-Dffuso Equatos Iteratoal Joural of Computer Matematcs Vol 7 No 999 pp 4-54 do:8/ [9] M M Cawla M A Al-Zaad ad M G Al-Aslab Exteded Oe Step Tme-Itegrato Scemes for Covecto-Dffuso Equatos Computers & Matematcs wt Applcatos Vol 39 No 3-4 pp 7-84 do:6/s898-(99)334-x [] H N A Ismal E M E Elbarbary ad G S E Salem Restrctve Taylor s Approxmato for Solvg Covecto-Dffuso Equato Appled Matematcs ad Computato Vol 47 No 4 pp do:6/s96-3 3()67- [] S Karaa ad J Zag Hg Order ADI Metod for Solvg Usteady Covecto- Dffuso Problems Jour- al of Computatoal Pyscs Vol 98 No 4 pp -9 do:6/cp4 [] M Dega Numercal Soluto of te Tree-Dmesoal Advecto-Dffuso Equato Appled Matematcs ad Computato Vol 5 No 4 pp 5-9 do:6/s 96-33(3)93- [3] M Dega O te Numercal Soluto of te Oe- Dmesoal Covecto-Dffuso Equato Matema- tcal Problems Egeerg Vol 5 No 5 pp 6-74 do:55/mpe 56 Copyrgt 3 ScRes

12 944 R MOHMADI [4] M Dega Quas-Implct ad Two-Level Explct Fte-Dfferece Procedures for Solvg te Oe-Dme- soal Advecto Equato Appled Matematcs ad Computato Vo l 67 No 5 pp do: 6/amc4667 [5] D K Salkuye O te Fte Dfferece Approxmato to te Covecto-Dffuso Equato Appled Mate- matcs ad Computato V ol 79 No 6 pp do: 6/amc5 78 [6] I Dag D Irk a d M Tomb ul Least-Squares Fte-E le- Metod for te Advecto-Dffuso Equato Ap- met pled Matematcs ad Computato Vol 73 No 6 pp do:6/amc5454 [7] H Karaa Implct Fte Dfferece Tecques for te Advecto-Dffuso Equato Usg Spreadseets Advaces Egeerg Software Vol 37 No 9 6 pp 6-68 do:6/advegsoft63 [8] H Karaa A Trd-Order Upwd Sceme for te Advecto-Dffuso Equato Usg S preadseets Advaces Egeerg Software Vol 38 No 7 pp do:6/advegsoft66 [9] H Karaa Ucodtoal Stable Explct Fte Dfferece Tecque for te Advecto-Dffuso Equato Usg Spreadseets Advaces Egeerg Software Vol 38 No 7 pp 8-86 do:6/advegsoft68 [] M Dega ad A Moebb Hg-Order Compact Boudary Value Metod for te Soluto of Usteady Covecto-Dffuso Problems Matematcs ad Computers Smulato Vol 79 No 3 8 pp do:6/matcom845 [] M Dega ad F Saker Applcato of He s Varatoal Iterato Metod for Solvg te Caucy Reacto- Dffuso Problem Joural of Computatoal ad Appled Matematcs Vol 4 No 8 pp d o:6/cam736 [] H F Dg ad Y X Za g A New Dfferece Sceme wt Hg Accu racy ad Absolute Stablty for Solvg Covecto-Dffuso Equatos Joural of Computa- toal ad Appled Matema tcs Vol 3 No 9 pp 6-66 do:6/cam8 5 [3] A Moebb ad M Dega Hg-Order Compact Soluto of te Oe-Dmesoal Heat ad Advecto- Dffuso Equatos Appled Matematcal Modellg V ol 34 No pp do:6/apm3 [4] M Sar G Gürasla ad A Zeytoglu Hg-Order Fte Dfferece Scemes for Solvg te Advecto-Dffuso Equato Computers & Matematcs wt Applcatos Vol 5 No 3 pp [5] S Cadra Sekara Rao ad M Kumar Expoetal B- Sple Collocato Metod for Self-Adot Sgularly Perturbed Boudary Value Problems Appled Numercal Matematcs Vol 58 No 8 pp do:6/apum798 [6] S Cadra Sekara Rao ad M Kumar Parameter-Uformly Coverget Expoetal Sple Dfferece Sceme for Sgularly Perturbed Semlear Reacto-Dffuso Problems Nolear Aalyss Vol 7 9 pp do:6/a9 [7] B J Mc Cart Teory of Expoetal Sples Joural of Approxmato Teory Vol 66 No 99 pp -3 do:6/-945(9)95 -K [8] S Pruess Propertes of Sples Teso Joural of Approxmato Teory Vol 7 No 976 pp do:6/-945(76)93- [9] B J Mc Cart Teory Computato ad Applcato of Expoetal Sples Courat Matematcs ad Computg Laboratory Researc ad Developmet Report DOE/ER/ Copyrgt 3 ScRes