A NEW FORMULAE OF VARIABLE STEP 3-POINT BLOCK BDF METHOD FOR SOLVING STIFF ODES

Transcription

1 Joual of Pue ad Appled Maemac: Advace ad Applcao Volume Numbe Page 9-7 A NEW ORMULAE O VARIABLE STEP -POINT BLOCK BD METHOD OR SOLVING STI ODES NAGHMEH ABASI MOHAMED BIN SULEIMAN UDZIAH ISMAIL ZARINA BIBI IBRAHIM HAMISU MUSA ad NEDA ABBASI Iue fo Maemacal Reeac Uve Pua Malaa UPM Sedag Selago Malaa e-mal: agme.abaump@aoo.com Depame of Maemac acul of Scece Uve Pua Malaa UPM Sedag Selago Malaa acul of Maemacal Scece Sad Bee Uve Tea Ia Abac T pape deve a ew vaable ep -po block meod baed o Backwad Dffeeao omula (BD) fo olvg ff Oda Dffeeal Equao (ODE). Te aeg volved e developed meod o cool e ep Maemac Subjec Clafcao: 5L. Kewod ad pae: block backwad dffeeao fomula ff vaable ep ze. Receved Jue Scefc Advace Puble

2 5 NAGHMEH ABASI e al. ze a eac eao o opmze e peco ad poduce ee oluo value mulaeoul a eac ep. Te meod aalzed avg e codo fo zeo abl ad foud o be of ode. Te abl ego of e meod ae alo vegaed ad peeed dc gap. Te popoed meod compaed o MATLAB ue ODE olve amel ode5 ad ode. Numecal eul obaed ae povded o uppo e eaceme of e meod em of accuac.. Ioduco We code block backwad dffeeao fomula (BD) fo e oluo of ff f ode oda dffeeal equao (ODE) of e fom f ( ) ( ) a. b Implc meod o olvg ff ODE ae kow o pefom bee a eplc oe. Solvg ff ODE ug BD meod f wa popoed b wo cem ad e wok ca be foud []. Dalqu [] ed o olve ff ODE ad eplaed e dffcule dffeeal equao olve a ma appea egag ff poblem. Te mplemeao of e BD meod fo olvg ff ODE wa dcued b Gea [] ad e became oe of e well-kow eeace e ud of ff ODE. Te ode ad e accuac of BD meod fo olvg ff ODE wee mpoved b Ca [] oug addg a fuue po e meod wa called eeded block BD. Block mplc meod f wee popoed b Mle [] ad dea ug a Ruge-Kua meod lae wa eeded b Roe []. Covegece ad abl popee of oe-ep mplc block meod ca be followed [ ]. A cla of block mplc meod fo olvg ff ODE ad A-abl popee ca be ee [7]. Block meod o olvg ff ODE va backwad dffeeao fomulae wee developed ece ea ad ca be uded [7]. uemoe vaable ep -po block BD meod fo olvg ff ODE ca be followed [ ]. A fomulao of -po block BD ug vaable ep ze of ode wa obaed b [] wle e meod dd o deeve e codo fo zeo abl eefoe e fomulae ca o be accepable.

3 A NEW ORMULAE O VARIABLE STEP -POINT 5 Te am of pape o oduce a ew fomula of vaable ep -po block backwad dffeeao fomula o appomae ee po cucuel a eac eao. I e followg eco of pape e codo fo zeo abl ad e aal of e abl ego ae lluaed. Te umecal eul obaed of e meod ae compaed w ff ODE olve MATLAB defed a ode5 ad ode. Te advaage of e popoed meod a e oluo ae appomaed a moe a oe po mulaeoul o mpove e accuac of e meod.. Devao of Vaable Sep -Po Block BD Meod I a -po block BD meod ee oluo value ad w ep ze ae compued mulaeoul a block ug fou back value ad w ep ze (gue ). gue. Iepolao po volved e -po vaable ep BBD. I above fgue e ep ao a block. We lm e amou of ep ze ceae o eue zeo abl. I cae we code ad wc coepod o coa ep ze alvg ad 9 ceag e ep ze b a faco of.9. Te movao bed e coce of eac value of ae a follow; f o opmze e oal umbe of ep ad e ecod a eac value ued eue a zeo able fomula. Te epolag polomal P k ( ) of degee k wc epolae e po ( ) ( ) ( ) defed a k Pk ( ) Lk j ( ) ( j ) j

4 NAGHMEH ABASI e al. 5 wee j j j L k fo k k j e aocaed polomal fo ca be we a P. Defe ad eplace gve p p

5 A NEW ORMULAE O VARIABLE STEP -POINT 5. Dffeeag w epec o ad ubug ad gve 5 5 f 5 5 ; 9 (5) 9 f ; 9

6 5 f NAGHMEH ABASI e al. 7 5 ( )( )( ) ( 9)( 9 9) ( )( ) 9 ( 9 9) ( 9) ( )( ) 9( ) ( )( ) ( 9) ( ) ( )( ) 9 9. ( )( )( ) (7) Replacg ad o (5) ad (7) epecvel gve e 9 coeffce of e meod a gve below wc ae oed e code. Te value of coe eue e zeo abl of e meod. o f f 75 5 f o f f 5

7 A NEW ORMULAE O VARIABLE STEP -POINT o f. (9) f f f. 5757

8 NAGHMEH ABASI e al. 5. Ode of e Meod T eco deve e ode of e meod coepodg o e equao (9) ad. we code e meod wle. I ca be ewe a f f f Te ma fom of aocaed w f f f Le α α α α α

9 A NEW ORMULAE O VARIABLE STEP -POINT 57 α 5 7 α 7 5 α α 9 5 β 7 β7 β. 9 Te lea dffeece opeao L defed b L[ ( ) ] [ α j ( j) β j ( j)]. j Epadg e fuco ( j) ad devave ( j) a Talo ee aoud ad ubug lead o [ ] ( ) q ; ( q L C ) C C Cq ( ) wee C q ae coa. Te dffeece opeao ad e aocaed meod codeed of ode p f c c c ad c. I cae p p c α j j j j c c ( jα j ) β j ( j α j ) ( β )! j j j j

10 5 NAGHMEH ABASI e al. c c 5 ( j α j ) ( j β j ) 5 5!! j j 5 ( j α j ) ( j β j )! 5! j j c 7 7 ( j α j ) ( j β j ) 7!! j j Teefoe e ode of e meod ad e eo coa deemed b c Applg a mla pocedue o e meod (9) ad ow a e ode of em.. Sabl of e -Po Block BD Meod I eco we dcu abou e codo fo e abl of e meod (9) ad. We a b e followg defo: Defo ([9]). A meod ad o be zeo able f all e oo of f caacec polomal ave modulu le a o equal o u ad oe of modulu u ae mple. Defo ([9]). A meod ad o be aboluel able a ego R fo a gve λ f fo a λ all e oo of abl polomal π( λ) ρ( ) λσ( ) af < k.

11 A NEW ORMULAE O VARIABLE STEP -POINT 59 Defo ([5]). A meod ad o be A-able f e abl ego cove e ee egave lef alf plae. Te abl ego of e meod ae deemed b ubug lea e poblem λ ( λ < λ comple) o Equao (9) ad. We oba e followg fom: AY m BYm CYm (5) wee A B ad C ae e ma coeffce ad ca be pecfed baed o e coeffce (9) ad epecvel. We defe m wee m e block umbe ad e umbe of po e block. Hee ad m. Hece Y m m ( m) ; m Ym ( m) ; m ( m) ad Y m ( m) ( m) ( m) 5. Te abl polomal of e meod R ( ˆ ) wee ĥ λ defed b de( A B C ) wle e abolue abl ego of e meod deemed b olvg de( A B C ) e λ plae. Te followg gve e abolue abl ego of e meod fo e coe ep ze ad epecvel 9

12 NAGHMEH ABASI e al. R( ˆ 5 ) ˆ ˆ ˆ 5 5 ˆ 5 ˆ 7 9 ˆ ˆ ˆ ; ( ) R ( ˆ ) ˆ ˆ 5 57 ˆ ˆ ˆ ˆ ˆ ˆ ; (7) ( ) R( ˆ ) ˆ ˆ ˆ ˆ

13 A NEW ORMULAE O VARIABLE STEP -POINT ˆ ˆ ˆ 5 ˆ. o zeo abl we e ˆ (7) ad o oba e f caacec polomal a ; (9) ( ) ( ) ;

14 NAGHMEH ABASI e al. Te oo obaed fom (9) ad ae led below epecvel ; ( ) ; ( ) A all e oo ave modulu le a o equal o u o e meod zeo able we ad. Te abl ego of e 9 meod we ad ae ow gue ad. 9 gue. Sabl ego we.

15 A NEW ORMULAE O VARIABLE STEP -POINT gue. Sabl ego we. gue. Sabl ego we. 9

16 NAGHMEH ABASI e al. Te abl ego wle gue cove e wole egave lef alf plae o e meod codeed a A-able. Te abl ego we ad wc ae ow gue 9 ad gue epecvel almo cove e ee egave lef alf plae ece e meod ffl able. To oba dffee value of ave bee eed. Zeo 9 abl ad abolue abl of e meod fo e eed ae led Table. Table. Te value of eed fo e -po Sau of e meod o zeo able 9.9 o zeo able 5. o zeo able.5 zeo able o aboluel able 5 7. zeo able o aboluel able. zeo able o aboluel able 5. zeo able o aboluel able 9.9 zeo able o aboluel able zeo able o aboluel able 9.9 zeo able ad aboluel able

17 A NEW ORMULAE O VARIABLE STEP -POINT 5 5 om Table ee a we ad e meod 9 o zeo able a doe o af e Defo. Teefoe e -po block BD of vaable ep ze fomulaed b [] o accepable a e meod o zeo able we. Coog 5 5 ad gve a zeo able meod o aboluel able. o eample f we e ˆ. e abl polomal obaed fo ad olve fo gve a lea oe 97 oo bgge a oe. Teefoe b e Defo e meod o aboluel able fo. Seg dffee amou of ĥ fo ad gve a mla eul wc ow a 9 7 e meod o aboluel able e coe. gue 5- ae povded o ow e abl ego we 5 5 ad

18 NAGHMEH ABASI e al. gue 5. Sabl ego we. 97 gue. Sabl ego we. 9

19 A NEW ORMULAE O VARIABLE STEP -POINT 7 5 gue 7. Sabl ego we. gue. Sabl ego we.

20 NAGHMEH ABASI e al. 5 gue 9. Sabl ego we. 7 gue. Sabl ego we.

21 A NEW ORMULAE O VARIABLE STEP -POINT 9 5. Implemeao of e -Po Block BD Meod Newo eao appled fo e mplemeao of e meod. we defe e eo e eao a ( ) ( ) ( ) eo eac appoma e ad e mamum eo gve b MAXE ma TNS ( ( eo ) ). Te abbevao TNS gve e oal umbe of ep. Le defe ( ) j j deoe e ( ) eave value of j ( ) ( ) ( ) e j j j j. Te -po block BD meod ca be we a αf β γ η αf β γ η αf β γ η wee α β ad γ epee e coeffce we ad epecvel. η 9 η ad η epee e back value. Le β γ αf η β γ αf η β γ αf η. Newo eao defed a ( ) j j j [ ] j j j.

22 NAGHMEH ABASI e al. 7 Hece Newo eao ca be defed e fom.. Equao equvale o.. ma Jacoba e e e (5) Le J deoe e Jacoba ma (5) e. α α γ α β α γ α α β α γ α β α f f f f f f f f f J Teefoe e value of j e j ca be appomaed ad e oluo j j ae compued fom

23 A NEW ORMULAE O VARIABLE STEP -POINT 7 ( ) ( ) ( ) j j e j j we ad. 9 Coog e ep ze Coog e ep ze a mpoa faco e educo of e umbe of eao. Te ep ze eleco fall o ee aege. Ug a pecbed oleace value (TOL) a al ep ze deemed. A e coduced o compae e local ucao eo (LTE) w TOL wee LTE ( k ) ( k ) k. If e LTE < TOL e ep codeed a ucceful (IST). A ep e pevou ep ze maaed (coepodg o ug ) ad e followg e wll be coduced: TOL ew c old k LTE wee c e afe faco k e ode of e meod ad equal o. ew ad old ae e ep ze fo e cue ad pevou block epecvel. Hee c.5. If ew > (.9) old e ew (. 9) old. T coepod o ug e fomula. O e oe 9 ad f LTE > TOL e ep ze alved ad we egad ep a a faled ep (IST) (coepodg o e fomula we ).. Numecal Reul I eco e umecal eul of e -po block BD meod of ode o a e of ff poblem fo oleace ad ae abulaed ad compaed w MATLAB ff umecal olve fo ODE a ode5 ad ode [5]. Te mamum global eo ad e oal umbe of ep fo eac poblem ae gve. Te e poblem ad e oluo ae led a

24 7 NAGHMEH ABASI e al. Poblem ([]).. Eac oluo ( ) e. 5 5 Poblem ([ ]). ( ). Eac oluo ( ) e. Egevalue: λ. Poblem ([]). ( ). Eac oluo e e. Egevalue: λ ad λ. Poblem ([]) Eac oluo e e e e.

25 A NEW ORMULAE O VARIABLE STEP -POINT 7 Egevalue: λ ad λ. Te oao ued e able ae led below: BBD TNS TOL -po vaable ep block BD meod. e oal umbe of ep. oleace value. MAXE mamum global eo 9. Table. Compao fo Poblem TOL Meod TNS MAXE ode5 9.7e- ode 9.e- BBD 97.7e- ode5.7e- ode.7e- BBD.979e- ode5 9.5e- ode.9e-5 BBD 5.9e- Table. Compao fo Poblem TOL Meod TNS MAXE ode5.e- ode 9.5e- BBD 5.5e-5 ode5.e- ode.5e- BBD.e-7 ode5.75e- ode.5e-5 BBD 5.57e-9

26 7 NAGHMEH ABASI e al. Table. Compao fo Poblem TOL Meod TNS MAXE ode5 9 5.e- ode 5.e- BBD 9.79e-7 ode e-5 ode.9e-5 BBD 7.97e-9 ode e- ode.e- BBD 9.7e- Table 5. Compao fo Poblem TOL Meod TNS MAXE ode5.7e- ode 7.e- BBD.7e- ode e- ode.7e- BBD.e- ode5.959e- ode.79e-5 BBD 7.e- om Table -5 ca be obeved a e mamum global eo fo eac gve oleace a deceaed e -po block BD meod wc ow a e meod covege fae fo all e poblem eed compao w ode5 ad ode.

27 A NEW ORMULAE O VARIABLE STEP -POINT Cocluo A fomulao of a block BD a compue ee po cocuel fo e oluo of ff ODE codeed pape. Te meod aalzed ad foud o be A-able we ad ffl able we ad. Te eul obaed dcae 9 a e code developed a bee olve fo ff poblem educg e eo compao w ff MATLAB olve. I fac e meod oupefomed e ode5 ad ode em of accuac. Ackowledgeme Te auo would lke o ak Iue fo Maemacal Reeac Uve Pua Malaa fo uppog eeac. Refeece [] K. H. K. Aua K. I. Oma. Iak Z. B. Ibam ad Z. Majd Developg paog evalwe block meod fo olvg oda dffeeal equao I Poceedg of e Wold Academ of Scece Egeeg ad Tecolog -5. [] J. Ca O e egao of ff em of ODE ug eeded backwad dffeeao fomulae Numecal Maemak (9) 5-. [] C.. Cu ad J. O. Hfelde Iegao of ff equao I Poceedg of e Naoal Academ of Scece of e Ued Sae of Ameca (95) 5-. [] G. Dalqu A pecal abl poblem fo lea mulep meod BIT Numecal Maemac (9) 7-. [5] S. O. aula Block meod fo ecod ode ODE Ie. J. Compue Ma. (99) 55-. [] C. W. Gea Numecal al value poblem oda dffeeal equao COMM. ACM (97) 5-9. [7] Z. B. Ibam Block Mulep Meod fo Solvg Oda Dffeeal Equao PD Te Uve Pua Malaa. [] Z. B. Ibam K. I. Oma ad M. B. Sulema ed coeffce block backwad dffeeao fomula fo e umecal oluo of ff oda dffeeal equao Euopea Joual of Scefc Reeac ) 5-5.

28 7 NAGHMEH ABASI e al. [9] J. Lambe Compuaoal Meod Oda Dffeeal Equao Jo Wle ad So Ic. New Yok 97. [] W. E. Mle Numecal Soluo of Dffeeal Equao Jo Wle New Yok 95. [] H. Mua M. B. Sulema. Imal N. Seu ad Z. B. Ibam A accuae block olve fo ff al value poblem ISRN Appled Maemac. [] J. B. Roe A Ruge-Kua fo all eao SIAM Revew 9 (97) 7-5. [] L.. Sampe ad H. A. Wa Block mplc oe-ep meod Maemac of Compuao (99) 7-7. [] M. B. Sulema H. Mua. Imal ad N. Seu A ew vaable ep ze block backwad dffeeao fomula fo olvg ff IVP Ieaoal Joual of Compue Maemac. [5] L.. Sampe ad M. K. Recel Te MATLAB ODE ue SIAM Joual of Scefc Compug (997) -. [] H. A. Wa ad L.. Sampe A-able block mplc oe-ep meod BIT (97) 5-. [7] J. Wllam ad. dehoog A cla of a-able advaced mulep meod Maemac of Compuao (97) -. [] S. A. M. Yam Z. B. Ibam K. I. Oma ad M. B. Sulema A quaave compao of umecal meod fo olvg ff oda dffeeal equao Maemacal Poblem Egeeg -; Acle ID 99. g