MAT 518 Numerical Methods for Differential Equations [Kaedah Berangka untuk Persamaan Pembezaan]

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1 UNIVERSITI SAINS MALAYSIA Frs Semeser Examao 03/04 Academc Sesso December 03 / Jauary 04 MAT 58 Numercal Mehods for Dffereal Equaos [Kaedah Beragka uuk Persamaa Pembezaa] Durao : 3 hours [Masa : 3 jam] Please check ha hs examao paper cosss of FIVE pages of pred maerals before you beg he examao. [Sla paska bahawa keras peperksaa megadug LIMA muka sura yag berceak sebelum ada memulaka peperksaa.] Isrucos: [Araha: Aswer all four [4] quesos. Jawab semua empa [4] soala.] I he eve of ay dscrepaces, he Eglsh verso shall be used. [Sekraya erdapa sebarag percaggaha pada soala peperksaa, vers Bahasa Iggers hedaklah dgua paka]. /-

2 -- [MAT 58]. osder he lear adveco equao + u = 0 x where s he depede varable, x ad are he depede varables. u s a cosa. (a) By cosderg = u x ad akg paral dervaves wh respec o me o boh sdes, show ha = uxx (b) By cosderg he Taylor expaso of ( x, ) (c) + ad makg use of, + = uxx show ha = 0.5 u xx osruc a scheme for he lear adveco equao usg he above expresso for ad ceral dfferece for all spaal dervaves.. Permbagka persamaa adveks lear + u = 0 x dega pembolehubah bersadar, x da pembolehubah ak bersadar. u alah pemalar. (a) Dega mempermbagka = u x da megambl erba separa erhadap masa pada kedua-dua belah, ujukka = uxx (b) Dega mempermbagka perkembaga Taylor bag ( x, ) + da meguaka, + = uxx ujukka = 0.5 u xx (c) Baguka suau skema uuk persamaa adveks lear dega megguaka ugkapa d aas uuk da beza pusa uuk semua erba ruag. 3/-

3 -3- [MAT 58]. osder he equao = σ σ > 0 s a posve cosa. The DuFor-Frakel scheme for he above equao replaces u he ceral + dfferece approxmao of by u + u ad also uses he ceral dfferece approxmao for. (a) Wre he DuFor -Frakel scheme for = σ (b) alculae he rucao error of he scheme. Permbagka persamaa = σ σ > 0 alah pemalar posf. Skema DuFor-Frakel uuk persamaa d aas meggaka u dalam + hampra beza pusa dega u + u da juga megguaka beza pusa uuk. (a) Tuls skema DuFor-Frakel uuk = σ (b) Kra rala pagkasa skema 4/-

4 -4- [MAT 58] 3. (a) osder he problem of orso. The orso fuco φ sasfes he Posso's equao φ + = 0 wh φ = 0 o he boudary. Fd φ over he cross-seco of a recagular bar bouded by x= 0, y = 0, x= 3, y = 4. Takg he mesh sze of u boh he drecos, solve he Posso's equaos, usg Successve Over Relaxao (S.O.R) mehod ul eraos. () Fd ρ ( ) for hs sysem L ωb () alculae he rae of he covergece R ( L ωb ) () Esmae he opmal relaxao facor ω b for S.O.R ad (v) ompare he umber of eraos bewee S.O.R mehod ad 7 Gauss Sedel mehod for hs sysem.( ε = 0 ) (b) Gve B as a m m marx wh m dsc egevalues; α, α,, αm ad correspodg egevecors β, β,, β m. Suppose A s a block B I I B I I B I rdagoal marx of he form A =. I B I I B Fd he egevalues of A ad he correspodg egevecors. 3. (a) Permbagka masalah klasa berku. Fugs klasa φ memeuh persamaa Posso au φ + = 0 dmaa φ = 0 pada sempada. Dapaka φ pada keraa reas sau baag segempa epa dsempada oleh x= 0, y = 0, x= 3, y = 4. Dega megambl saz mesh u pada kedua-dua arah, selesaka persamaa Posso megguaka kaedah Pegedura Berlebha Beruru-uru (P.B.B) sehgga lelara. () Dapaka ρ ( ) bag ssem L ωb () Kra kadar peumpua R ( L ωb ) bag kaedah () Aggar fakor pegedura opmal ω b uuk kaedah P.B.B da (v) Badgka blaga lelara aara kaedah P.B.B da kaedah 7 Gauss Sedel bag ssem ( ε = 0 ). 5/-

5 -5- [MAT 58] (b) Dber B sebaga marks m m dega m la ege yag berbeza; α, α,, αm da vekor ege yag sepada β, β,, β m. Adaka A dalah marks blok ga B I I B I I B I pepejuru dalam beuk A =. I B I I B Dapaka la ege A da vekor ege yag sepada. 4. (a) alculae he egevalues of he S.O.R erao marx correspodg o 4 0 marx A = (b) 4 Gve he block sysem AU j j = ; =,,3,4. Wre he block Jacob j= erave mehod ad he correspodg block S.O.R erave mehod solvg hs sysem. (c) Show ha he weghed Jacob mehod defed by ( + ) ( ) ( ) Dx = ( ω) Dx + ω( b+ ( L+ U) x ) has he erao marx G = [( ω) I + ωd ( L+ U)]. 4. (a) Kraka la ege bag lelara P.B.B uuk marks (b) berku. Dberka ssem blok 4 j= 4 0 A = AU j j = ; =,,3,4. Tulska kaedah lelara blok Jacob da kaedah lelara blok P.B.B yag sepada dalam meyelesaka masalah. (c) Tujukka kaedah Jacob berpembera yag ddefska oleh ( + ) ( ) ( ) Dx = ( ω) Dx + ω( b+ ( L+ U) x ) mempuya marks lelara G = [( ω) I + ωd ( L+ U)]. - ooo O ooo -