Akpan s Algorithm to Determine State Transition Matrix and Solution to Differential Equations with Mixed Initial and Boundary Conditions

Transcription

1 IOSR Joural o Elcrcal ad Elcrocs Egrg IOSR-JEEE -ISSN: ,p-ISSN: 3-333, Volu, Issu 5 Vr. III Sp - Oc 6, PP kpa s lgorh o Dr Sa Traso Marx ad Soluo o Dral Euaos wh Mxd Ial ad Boudary odos kpa, E.. Dpar o Elcrcal/Elcroc/opur Egrg, Uvrsy o UYO, Uyo, Ngra bsrac: Solvg a sys o lar cosa coc dral uaos wh xd al ad boudary valus whou h us o a copur s o a rval davor h ordr o h sys s hghr ha wo. Ths arcl dlas algorhs or copug h sa raso arx STM ad h soluo o syss o cosa coc lar dral uaos o ay ordr. Ths algorhs obva h shorcogs hr Lvrrr s algorh, Sylvsr s xpaso hor, ayly-halo s hor, ad Puzr s algorh. Furhror, hs algorhs do o rur sybolc sowar sc h STM ad dral uao soluo ca b copud usg rgular Malab or ++. Kywords: lgorh, sa raso arx, cosa coc, dral uaos I. Iroduco odl o a sys o cosa coc lar ordary dral uaos s gv as x x wh al codo x x, whr, x s a colu vcor wh -rs ad ach ry s a ral ubr; ad s a cosa arx wh ral or coplx rs. I hs arcl h sa raso arx STM s dod by. Sc h STM s a pora rda rsul h soluo o., svral rsarchrs hav proposd algorhs ad hors o copu h STM. Noabl aog h ar h Lvrrr s algorh [], Sylvsr s hor [], ayly-halo s hor [3], ad Puzr s algorh [4]. I Lvrrr s algorh, a rsolv arx s s oud or., xprssd as s si Th h STM o h sys s h vrs Laplac rasor o s. I s larg, akg h vrs Laplac rasor o s rurs sybolc sowar. Thus, h Lvrrr s algorh dos o yld h STM drcly. Sylvsr s xpaso hor ca b usd o copu h STM or h cas whr h arx has dsc gvalus as ollows. whr,, F,,..., I F, ar h gvalus o h arx, ad Whl F ca b copud urcally, hs algorh s urcally usabl bcaus volvs dvso by h drc o gvalus whch ay approach zro so cass. ayly-halo s hor ca b adapd o d h STM or h cas o ral ad dsc gvalus. I hs cas h sys o uaos o solv or h vcor s:,,,..., 3 Th, I 4 DOI:.979/ Pag

2 kpa s lgorh o Dr Sa Traso Marx ad Soluo o Dral Euaos wh.. Th drawback o hs approach s ha dg s o aabl o urcal hod. Fally, h Puzr algorh ca b usd o dr h STM v or h cas o rpad gvalus as ollows. r P whr, P I, ad P k I,,,..., k Th vcor r r r r s a soluo o a spara sys o ordary dral uaos ro. ; ad r s oud aually or by usg sowar capabl o sybolc apulaos. Edd kpa s algorh, roducd hs arcl, s asly pld by spl copur progras Malab, ++, ad ohr copug sowar; ad dos o hav h drawbacks hr h ohr algorhs, such as urcal sably ad a d o possss xpsv sybolc sowar. Ths arcl s orgazd as ollows: lgorhs or copug h sa raso arx or al ad xd al ad boudary valu probls ar gv sco. algorh or copug h soluo o syss o dral uaos as a uco o gvalu xpoals s gv sco 3. xapl ha dosras h applcao o h proposd algorhs appars sco 4. Sco 5 s h cocluso.. Edd kpa s lgorh or Fdg h Sa Traso Marx or Ial Valu Probls L do h sa raso arx STM o h auooous lar sys gv.. Suppos h arx has dsc ral or coplx gvalus,,, xpoal as E T. 5. D a vcor o gvalus Furhr, l P b h arx whos colus ar h gvcors o. ddoally, l h vcor or xd ad k, b gv as P P,,,..., k ad, k k M,, k,,...,. 6 Th Edd kpa s orula or copug h STM s gv as E,, k,,...,. 7 k Takg E ousd h brack o h rgh had sd o. 7, h sa raso arx s wr as M E 8 whr, h sybol "" pls l-by-l do produc o k h rs o M wh E. k,. Edd kpa s lgorh or Drg h Sa Traso Marx or Mxd Ial ad Boudary Valu Probls I so applcaos, such as opzao probls, h opal soluos ca occur a h boudars o h doa o soluos. Wh hs happs, hods o soluo o al valu probls ar adua bcaus xd al ad boudary valus ar volvd. Ths yps o probls cssa h us o Lagrag ulplrs []. osuly, or a h -ordr sys, h sz o h sys o uaos o solv s. L x b h -dsoal sa vcor or h sys., ad z s a vcor o Lagrag ulplrs wh h sa dso as x. Th h sys wh xd al ad boudary codos s wr as DOI:.979/ Pag

3 kpa s lgorh o Dr Sa Traso Marx ad Soluo o Dral Euaos wh.. x z x z wh h accopayg al ad boudary codos x x ad x x, rspcvly. Th arcs ad ar cosa ar couplg rs. L x ad z arcs wh ral or coplx rs; ad h arcs. Th h sys o uaos wh xd al ad boudary codos s gv by DOI:.979/ Pag 9 ad I. h sz o h -arx s. Th STM or hs cas s E,, k,,..., k whr, k P P,,,..., k ad T E., Edd kpa s lgorh or Drg h Soluo o Syss o Lar Dral Euaos wh Ial ad Mxd Ial ad Boudary Valus L a vcor b gv as T whr, or al valu probls,,,...,,,..., 3, ad or xd al ad boudary valu probls,. For h al valu probl., h soluo o h sys o uaos s gv by x 4 Th vcor s a uco o al codos, ad sass x 5 I, x x 6 For xd al ad boudary valu probls, s a h al ad al. I ha cas, l ~ ~ x x arx, ad ~ do h rs -rows o ad, rspcvly, rprs. Th vcor sass 7 Th h soluo o s 8 wh h vcor gv by. 7.

4 kpa s lgorh o Dr Sa Traso Marx ad Soluo o Dral Euaos wh.. lhough h soluos o s. ad ca b wr h or o s. 4 ad 8, s prrabl o xprss x adr succcly as ucos o h gvalus xpoal [5]. Edd kpa s orula or h soluo o syss o cosa coc lar ordary dral uaos s as ollows: For xd ad k, l w k k 9 whr, s gv as. 6 or, ad s spcd as. 5 or 7. lso, l w k W,, k,,..., Th,, or x W E h cas o al valu probls, s Dosrao Exapl or a Mxd Ial ad Boudary Valus Probl For a xapl ha dosras h applcao o h algorhs urad hs arcl, cosdr a scalar sys gv []: x x.5z Th al ad boudary valus ar x ad x, rspcvly. Th varabl z s a Lagrag ulplr havg h dyacs z x z 3 Hc, h sys o uaos o solv s:.5 4 Th gvalus o h -arrx ar. Thror, E T P.84.96, ad P M k,, k =,; so by. 6, , hus, copud slarly. Hc, M Th STM s: , ad ; ad, ad ar Usg h al ad boudary codos, ~ ad ~. DOI:.979/ Pag

5 kpa s lgorh o Dr Sa Traso Marx ad Soluo o Dral Euaos wh By. 9, w. 63; w, w ad w ar copud slarly. osuly, x z II. ocluso Ec algorhs or copug h sa raso arx STM ad soluo o syss o cosa coc lar dral uaos wh al or xd al ad boudary valus hav b prsd. Th STM algorh ulzs h gvalus ad gvcors o h sys; ad dos o hav h urcal sably probl or d or sybolc sowar whch s h cas wh xsg algorhs, such as h Lvrrr s algorh, Sylvsr s xpaso hor, ayly-halo s hor, ad Puzr s algorh. Th soluo o h syss o uaos s xprssd rs o h gvalus xpoal. Ths algorhs ca b pld usg rgular copug sowar such as Malab or ++. Fally, a xd al ad boudary valu xapl ha hghlghs h us o hs algorhs has b xhbd. Rrcs []. Kalah, T., Lar Syss, Prc Hall, Nw Jrsy, 98. []. Schulz D. G. ad Mlsa, J. L., Sa Fucos ad Lar orol Syss, McGraw-Hll, Nw York, 967. [3]. Jacob B., Lar lgbra, W. H. Fra ad opay, Nw York, 99. [4]. Wala P., Scod ours Elary Dral Euaos, cadc Prss, Orlado, 986. [5]. Brau M., Dral Euaos ad Thr pplcaos, 4 h d., Sprgr-Vrlag, Nw York, 993. DOI:.979/ Pag