MAT518 Numerical Methods for Differential Equations [Kaedah Berangka Untuk Persamaan Pembezaan]
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1 UNIVERSITI SAINS MALAYSIA Frst Semester Eamnaton 06/07 Academc Sesson December 06 / January 07 MAT58 Numercal Methods for Dfferental Equatons [Kaedah Berangka Untuk Persamaan Pembezaan] Duraton : 3 hours [Masa : 3 jam] Please check that ths eamnaton paper conssts of NINE pages of prnted materal before you begn the eamnaton. [Sla pastkan bahawa kertas peperksaan n mengandung SEMBILAN muka surat yang bercetak sebelum anda memulakan peperksaan n.] Instructons: Answer FOUR (4) questons. [Arahan: Jawab EMPAT (4) soalan]. In the event of any dscrepances, the Englsh verson shall be used. [Sekranya terdapat sebarang percanggahan pada soalan peperksaan, vers Bahasa Inggers hendaklah dguna paka.] /-
2 - - Queston (a) Consder c c 0. t () Wrte a fnte dfference scheme usng forward dfference for c t and backward dfference for c. () Ths scheme has a name assocated wth t. What s the name? (b) Consder c c c 0 t () () () Wrte a fnte dfference scheme usng forward dfference for c t and central dfference for c, c. Obtan the truncaton error of the scheme. Obtan the amplfcaton factor assocated wth a Fourer stablty analyss. Wrte notes (about - pages) on both numercal dsperson and numercal dsspaton. Your notes should touch, amongst others, on physcal dsperson/dsspaton and methods of analyss (n general terms). Soalan [ 00 marks ] (a) Pertmbang c c 0. t () Tuls skema beza terhngga menggunakan beza ke depan untuk c t. dan beza ke belakang untuk c. () Skema n mempunya suatu nama yang bersekutu dengannya. Apakah nama skema n? (b) Pertmbang c c c 0 t () Tuls skema beza terhngga menggunakan beza ke depan untuk c t. dan beza pusat untuk c, c. () () Dapatkan ralat pangkasan untuk skema n. Dapatkan faktor amplfkas yang bersekutu dengan suatu analss kestablan Fourer. Tuls nota-nota (- muka surat) berkatan penyebaran berangka dan pelesapan berangka. Nota-nota anda seharusnya, antara lan, menyentuh penyebaran dan pelesapan fzkal dan kaedah-kaedah analss (secara am). [ 00 markah ] 3/-
3 - 3 - Queston (a) Consder " ' y 3y y 3 y(0), y() 0 By usng N.e. h, wrte the system of equatons that arses when central 3 dfferences are used to dscretze the BVP. There s no need to solve the system. (b) Consder " ' t y y yte t y(0) y'(0) 0 0 t () Wrte as a frst order system. () Obtan the soluton at t 0. usng Eulers method W W hf (t,w ) wth h t 0. Wrte notes (about pages) on stff equatons. Your notes should touch on, amongst others, the characterstcs of the soluton of stff equatons and numercal methods commonly used. [ 00 marks ] Soalan (a) Pertmbang " ' y 3y y 3 y(0), y() 0 Dengan menggunakan N.. h 3, tuls sstem persamaan yang tmbul apabla beza ke pusat dguna untuk mendskretkan MNS. Anda tdak perlu selesakan sstem. (b) Pertmbang " ' t y y yte t y(0) y'(0) 0 0 t () Tuls sebaga sstem perngkat pertama. () Dapatkan penyelesaan d t 0. menggunakan kaedah Euler W W hf (t,w ) dengan h t 0. Tuls nota-nota (sektar muka surat) berkatan persamaan kaku. Nota-nota anda seharusnya menyentuh, antara lan, cr-cr penyelesaan persamaan kaku dan kaedah berangka yang lazmnya dguna. [ 00 markah ] 4/-
4 - 4 - Queston 3 a (a) The lnear system of equatons b a by the teratve method under certan condtons, where a s real, can be solved 0 ( k) a ( k) b. a 0 () For whch values of a s the method convergent when? () For a =0.5, fnd the value of whch mnmzes the spectral radus of the 0 a matr. a 0 r r (b) The coeffcent matr of the lnear equatons Au b s defned by [ 0 marks ] c A c c, c s real and non-zero. c c () Fnd the range of values for c, for whch the Gauss-Sedel teraton method converges. () Wll the Jacob teraton method converge for c? [ 0 marks ] For the equatons , prove that the Jacob teraton method s convergent but Gauss-Sedel method s dvergent. [ 40 marks ] 5/-
5 - 5 - (d) Prove that the truncaton error of the fve-pont fnte dfference formula appromatng Laplace's equaton at the pont, y j, for a square mesh of sde h, can be wrtten as 4 4 h U(, y ) (, ), 4 j U 4 y where h h, y h y h, and t s assumed that the frst-, j j second-, thrd- and fourth-order partal dervatves of U wth respect to and y are contnuous throughout these ntervals respectvely. [ 0 marks ] Soalan 3 a (a) Sstem persamaan lnear b dmana a adalah nyata, boleh a dselesakan dengan kaedah lelaran d bawah syarat tertentu 0 ( k) a ( k) b. a 0 () Bag nla a yang manakah kaedah n akan menumpu untuk? () Bag a =0.5, dapatkan nla yang memnmumkan jejar spektrum matrks 0 a. a 0 r r (b) Matrks koefsen bag persamaan lnear Au b d defnskan sebaga c A c c, c adalah nombor nyata dan bukan sfar. c c [ 0 markah ] () Dapatkan julat nla c d mana lelaran Gauss-Sedel menumpu. () Adakah lelaran Jacob menumpu bag nla c? [ 0markah ] 6/-
6 - 6 - Bag persamaan berkut , buktkan bahawa lelaran Jacob menumpu manakala lelaran Gauss-Sedel mencapah [ 40 markah ] (d) Buktkan bahawa ralat pangkasan bag rumus pembezaan lma-ttk untuk menganggarkan persamaan Laplace pada ttk, y j pada seg empat mesy berss h boleh dtuls sebaga (, ) (, ), y 4 4 h U y 4 j U 4 d mana h h, y h y h, dan dandakan perngkat pertama, j j kedua, ketga dan keempat persamaan pembezaan U terhadap dan y adalah masng-masng selanjar pada semua selang. [ 0 markah ] Queston 4 (a) Gven B as a m m matr wth m dstnct egenvalues;,, K, m and correspondng egenvectors,, K, m. Suppose A s a n n block trdagonal matr of the form B I I B I I B I A. O O O I B I I B () Fnd the egenvalues of A and the correspondng egenvectors. [ 0 marks ] 7/-
7 - 7 - (b) Consder the system Au b where 9 A, b. Generate teratons usng precondtoned Conjugate Gradent method wth 9 0 precondtoner matr M. 0 [ 0 marks ] Consder the problem of torson. The torson functon satsfes the Posson's equaton 0 wth 0 on the boundary. Fnd over the cross-secton of a rectangular bar bounded by 0, y 0, 3, y 4. Takng the mesh sze of unt n both the drectons, solve the Posson's equatons, usng Successve Over Relaaton method untl teratons. () Fnd ( L b ) for ths system () Calculate the rate of the convergence R ( L b ) () Estmate the optmal relaaton factor b for S.O.R and (v) Compare the number of teratons between S.O.R method and Gauss Sedel 7 method for ths system.( 0 ) [ 40 marks ] r r (d) In solvng the system A b usng the second order Rchardson's method, we use the formula r ( k ) r ( k ) r ( ) ( ) ( ) ( k r b A ) ( k r k ) where and are constants chosen to provde mamum convergence to the soluton. If are the egenvalues of A and are the egenvalues of the matr assocated wth the teratve process, show that and are related by the followng equaton: ( ) ( ) 4 [ 0 marks ] 8/-
8 - 8 - Soalan 4 (a) Dber B sebaga matrks m m dengan m nla egen yang berbeza;,, K, m dan vektor egen yang sepadan,, K, m. Andakan A adalah n n matrks blok tga pepenjuru dalam bentuk B I I B I I B I A. O O O I B I I B () Dapatkan nla egen A dan vektor egen yang sepadan. [ 0 markah ] (b) Pertmbangkan sstem Au b d mana 9 A, b. Janakan lelaran dengan menggunakan kaedah Kecerunan Konjugat 9 0 berprasyarat dengan matrks prasyarat M. 0 [ 0 markah ] Pertmbangkan masalah klasan berkut. Fungs klasan memenuh persamaan Posson atu 0 dmana 0 pada sempadan. Dapatkan pada keratan rentas satu batang segempat tepat dsempadan oleh 0, y 0, 3, y 4. Dengan mengambl saz mesy unt pada kedua-dua arah, selesakan persamaan Posson n menggunakan kaedah Pengenduran Berlebhan Berturut-turut (P.B.B) sehngga lelaran. () Dapatkan ( L b ) bag sstem n () Kra kadar penumpuan R ( L b ) bag kaedah n () Anggarkan faktor pengenduran optmal b untuk kaedah P.B.B n dan (v) Bandngkan blangan lelaran antara kaedah P.B.B dan kaedah Gauss 7 Sedel bag sstem n dengan 0. [ 40 markah ] 9/-
9 - 9 - r r (d) Dalam menyelesakan sstem A b menggunakan kaedah Rchardson perngkat kedua, kta menggunakan formula berkut r ( k ) r ( k ) r ( ) ( ) ( ) ( k r b A ) ( k r k ) d mana dan adalah pemalar yang dplh untuk mendapatkan penumpuan maksmum kepada penyelesaan. Jka adalah nla-nla egen bag A dan adalah nla-nla egen bag matrks yang bersepadan dengan proses lelaran, tunjukkan bahawa dan adalah dkatkan melalu persamaan berkut: ( ) ( ) 4 [ 0 markah ] - ooo O ooo -
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