Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations

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1 World Acdemy of Scece Egeerg d Tecology No-Polyoml Sple etod for te Solto of Prolems Clcls of Vrtos. Zre. Hosyr. Sedgt Astrct I ts pper mercl solto sed o opolyoml cc sple fctos s sed for fdg te solto of odry vle prolems wc rse from te prolems of clcls of vrtos. Ts pproxmto redce te prolems to explct system of lgerc eqtos. Some mercl exmples re lso gve to llstrte te ccrcy d pplclty of te preseted metod. Keywords Clcls of vrto; No-polyoml sple fctos; Nmercl metod INTRODUCTION HE clcls of vrtos d ts extesos re devoted to T fdg te optmm fcto tt gves te est vle of te ecoomc model d stsfes te costrts of system. Te eed for optmm fcto rter t optml pot rses meros prolems from wde rge of felds egeerg d pyscs wc clde optml cotrol trsport peome optcs elstcty vrtos sttcs d dymcs of sold odes d vgto[]. I compter vso te clcls of vrtos s ee ppled to sc prolems s estmtg optcl flow[] d spe from sdg []. Severl mercl metods for pproxmtg te solto of prolems te clcls of vrtos re kow. Glerk metod s sed for solvg vrtol prolems [4]. Te Rtz metod [] slly sed o te sspces of kemtclly dmssle complete fctos s te most commoly sed pproc drect metods of solvg vrtol prolems. Ce d Hso [6] trodced te Wls seres metod to vrtol prolems. De to te tre of te Wls fctos te solto oted ws pecewse costt. Some ortogol polyomls re ppled o vrtol prolems to fd te cotos soltos for tese prolems [7-9]. A smple lgortm for solvg vrtol prolems v Berste ortoorml polyomls of degree sx s proposed y Dxt et l. []. Rzzg et l. [] ppled drect metod for solvg vrtol prolems sg Legedre wvelets. He s vrtol terto metod s ee employed for solvg some prolems clcls of vrtos []. Sple fctos re specl fctos te spce of wc pproxmte soltos of ordry dfferetl eqtos. I oter words sple fcto s pecewse polyoml. Zre s wt te Deprtmet of temtcs Uversty of ogeg Ardl P. O. Box. 79 Ardl Ir e-ml: zre@m.c.r.hosyr s wt te Deprtmet of temtcs Uversty of ogeg Ardl P. O. Box. 79 Ardl Ir e-ml: osyrmrym@yoo.com.sedgt s wt te Deprtmet of temtcs Uversty of ogeg Ardl P. O. Box. 79 Ardl Ir e-ml: sedgt64@yoo.com stsfyg cert codtos of cotty of te fcto d ts dervtves. Te pplctos of sple s pproxmtg terpoltg d crve fttg fctos ve ee very sccessfl[-6]. Qdrtc d cc polyoml d opolyoml sple fctos sed metods ve ee preseted to fd pproxmte soltos to secod order odry vle prolems[7]. K [8] sed prmetrc cc sple fcto to develop mercl metod wc s fort order for specfc coce of te prmeter. Te m prpose of te preset pper s to se o-polyoml cc sple metod for mercl solto of odry vle prolems wc rse from prolems of clcls of vrtos. Te metod cossts of redcg te prolem to set of lgerc eqtos. Te otle of te pper s s follows. Frst Secto we trodce te prolems clcls of vrtos d expl ter reltos wt odry vle prolems. Secto otles o-polyoml cc sple d sc eqtos tt re ecessry for te formlto of te dscrete system. Also ts secto we report or mercl reslts d demostrte te effcecy d ccrcy of te proposed mercl sceme y cosderg two mercl exmples. II. STATEENT OF THE PROBLE Te gerl form of vrtol prolem s fdg extremm of te J[ t) t)... t)] G t t) t)... t) t) t)... t)) dt. ) To fd te extreme vle of J te odry codtos of te dmssle crves re kow te followg form: ) γ... ) ) δ.... ) Te ecessry codto for t)... to extremze J[ t) t)... t)] s to stsfy te Eler-Lgrge eqtos tt s oted y pplyg te well kow procedre te clcls of vrto [] G d G )... 4) dt sect to te odry codtos gve y Eqs. )-). 986

2 World Acdemy of Scece Egeerg d Tecology I ts pper we cosder te specl form of te vrtol prolem) s Jt [ )] Gtt ) t)) dt ) wt odry codtos ) γ ) δ 6) d J[ t) t)] G t t) t) t) t)) dt 7) sect to odry codtos ) γ ) δ 8) ) γ ). 9) δ Ts for solvg te vrtol prolems ) we cosder te secod order dfferetl eqto G d G ) dt ) wt te odry codto 6). Ad lso for solvg te vrtol prolems 7) we fd te solto of te system of secod-order dfferetl eqtos G d G ) dt ) wt te odry codtos 8)-9). Terefore y pplyg o-polyoml cc sple metod for te Eler-Lgrge eqtos ) d ) we c ot pproxmte solto to te vrtol prolems ) d 7). III. No-polyoml Cc sple metod Cosder te prtto Δ { t t t... t} of [ ] R. Let S k Δ) deote te set of pecewse polyomls of degree k o stervl I [ t t ] of prtto Δ. I ts work we cosder o-polyoml cc sple metod for fdg pproxmte solto of vrtol prolems. Cosder te grd pots t o te tervl [ ] s follows: t < t < t <... < t < t ) t t... ) 4) were s postve teger. Let t) e te exct solto of te Eq.) d S t) e pproxmto to t ) oted y te segmet P t). Ec opolyoml sple segmet P t) s te form: P t) s k t t) cos k t t) c t t) d... ) were c d d re costts d k s te freqecy of te trgoometrc fctos wc wll e sed to rse te ccrcy of te metod d Eq. ) redce to cc polyoml sple fcto [ ] we k. We cosder te followg reltos: Pt ) Pt ) D P t ).. 6) We c ot te vles of c d d v strgtforwrd clclto s follows: cos c 7) s D D cos ) cos ) s s 8) d 9) were k d.... Usg te cotty codtos ) ) P x ) P x ) we get te followg reltos for... : cos ) D ) D ) k k s d cos cos cos s s ) k s k ) 987

3 World Acdemy of Scece Egeerg d Tecology D D D D ) s cos cos ) cos k s k k s k t cos cos cos ) cos s ) s k k t k s k s cos ) ). k ks ks ) By redcg te dces of Eqs. ) d ) y oe we get te followg eqtos: cos D ) ) D k k s cos cos ) k s k ) d lso D D D D ) s cos cos ) cos cos ) k s k k s k t k s cos cos ) s cos s ) k t k k s k cos ). k s k s ) D re elmted from Eq. ) y sg Eq. ). As reslt we get te followg sceme: [ α β α ]... 4) were cos α β ) s s I order to llstrte te performce of te o-polyoml cc sple metod we preset two exmples. Exmple. We frst cosder te followg vrtol t prolem wt te exct solto t) e []: t m J t) t) 4 e ) dt 6) sect to odry codtos ) ) e. 7) Cosderg te Eq. 6) te Eler-Lgrge eqto of ts prolem c e wrtte te followg form: t ) t ) t 8e. 8) Te solto of te secod-order dfferetl eqto 8) wt odry codtos 7) s pproxmted y te preseted sple metod. For or prpose We cosder te odry vle prolem 8) geerl form s follows: ) t g) tt ) f ) t 9) t Were g t) d f t) 8e. Te exct solto of ts t prolem s t) e. For mercl solto of te odry-vle prolem 9) te tervl [ ] s dvded to set of grd pots wt step sze. Settg t t Eq. 9) we ot g t ) f t ) ) y sg te ssmpto P t ) ) we ve g t ) f t ). ) Replcg or β g t s Eq. ) Eq.4) we get ) [ α g t f t ) )) α g t f t ) )) f t ))]... ) 988

4 World Acdemy of Scece Egeerg d Tecology α α αf t g t g t )) ) ) ) ) β.... g t αf t ) ) ) βf t Usg Tylor s seres for Eq. ) we c ot locl trcto error s follows: ) ) ) t α β ) α β ) 4 4) ) α β) α β) ) 7 α β ) O ) ) Te ler system ) cossts of ) eqto wt kows.... To ot qe solto we eed two eqtos. For ts prpose we c se te followg eqtos tt re fod y sg metod of determed coeffcet ) ) ). 6) Te locl trcto errors t... ssocte wt te sceme ) ) d 6) c e oted s follows: 9 6 6) 7 O ) 6 6) 7 t O ) ) 7 O ) wt α β. Te errors re reported o te set of form grd pots S { t... t... t } t t.... 7) Te mxmm error o te form grd pots S s E ) mx t ) t ) 8) were t ) s te exct solto of te gve exmple d s te compted solto y te o-polyoml cc sple metod. Te mxmm solte errors mercl solto of te Exmple re tlted Tle I. Tese reslts sow te effcecy d pplclty of te preseted metod. TABLE I RESULTS FOR EXAPLE E ) Exmple. I ts exmple cosder te followg prolem of fdg te extremls of te fctol[]: J [ t) t)] π ) t ) t ) t )) t dt wt odry codtos ) ) π ) 4) π ) ) 4) wc s te exct solto gve y t) t)) s t) s t)). For ts prolem te correspodg Eler-Lgrge eqtos re ) t ) t ) t ) t 4) 989

5 World Acdemy of Scece Egeerg d Tecology wt odry codtos 4) d 4). I smlr mer d pplyg 4) we ssme tt fctos ) d t ) defed over te tervl [ π ] re pproxmted y P t) s k t t ) cos k t t ) c t t ) d... 4) P s k t t ) cos k t t ) c t t ) d... 44) Were c t d d re costts d k s te freqecy of te trgoometrc fctos. Smlrly we c ot te followg reslts: [ α β α ]... [ α β α ]... 4) were α d β re defed ). Now cosder te system 4) d ssttte t t ts we c wrte: 46) coseqetly we ve: 47) By sg reltos 4) -47) we get: [ α β α ]... [ α β α ]... 48) Te system 48) cots ) eqtos wt kow coeffcets.... To. ot qe solto for more eqtos re eeded. Tese eqtos re fod y sg metod of determed coeffcets d re gve elow: ) 4 ) 4 49) d ) 4 ). 4 ) Te Eqs. 48)-) prodce ler system tt cots eqtos wt kow coeffcets. Solvg ts ler system we c ot te pproxmte solto of te system of secod-order odry vle prolems4). Sppose ) d E ) e te mxmm E solte errors. We solved Exmple for dfferet vles of. Te mxmm of solte errors o te form grd pots 7) re tlted Tle II. TABLE II RESULTS FOR EXAPLE ) E E ) IV. CONCLUSION I ts pper o-polyoml cc sple metod employed for fdg te extremm of fctol over te specfed dom. Te m prpose s to fd te solto of odry vle prolems wc rse from te vrtol prolems. Te o-polyoml cc sple metod redce te comptto of odry vle prolems to some lgerc eqtos. Te proposed sceme s smple d compttolly ttrctve. Applctos redemostrted trog llstrtve exmples 99

6 World Acdemy of Scece Egeerg d Tecology REFERENCES [] R. Westock Clcls of Vrtos: Wt Applctos to Pyscs d Egeerg Dover 974. [] B. Hor B. Scck Determg optcl flow Artfcl Itellgece vol. 7 o. -) pp [] K. Ikec B. Hor Nmercl spe from sdg d occldg odres. Artfcl Itellgecevol. 7o. -) pp [4] L. Elsgolts Dfferetl Eqtos d Clcls of Vrtos r oscow 977 trslted from te Rss y G. Ykovsky). [] I.. Gelfd S.V. Fom Clcls of Vrtos Pretce-Hll Eglewood Clffs NJ 96. [6] C.F. Ce C.H. Hso A wls seres drect metod for solvg vrtol prolems J. Frkl Ist.vol. pp [7] R.Y. Cg.L.Wg Sfted Legedre drect metod for vrtol prolems J. Optm. Teory Appl.vol. 9 pp [8] I.R. Horg J.H. Co Sfted Ceysev drect metod for solvg vrtol prolems Itert. J. Systems Sc. vol. 6 pp [9] C. Hwg Y.P. S Lgerre seres drect metod for vrtol prolems J. Optm. Teory Appl. Vol. 9 o. pp [] S. Dxt V.K. Sg A.K. Sg O.P. Sg Berste Drect etod for Solvg Vrtol Prolems Itertol temtcl Formvol. -7. []. Rzzg S. Yosef Legedre wvelets drect metod for vrtol prolems temtcs d Compters Smlto vol. pp []. Ttr. Deg Solto of prolems clcls of vrtos v He s vrtol terto metod Pyscs Letters A vol. 6 pp [] J.H. Alerg E.N. Nlso J.L. Wls Te Teory of Sples d Ter Applctos Acdemc Press New York 967. [4] T.N.E. Grevlle Itrodcto to sple fctos : Teory d Applcto of Sple Fctos Acdemc Press New York 969. [] P.. Preter Sples d Vrtol etods Jo Wley & Sos INC. 97 [6] G. cl Sd cl Hd Book of Sples Klwer Acdemc Plser s 999. [7].A. Rmd I.F. Lse W.K. Zr Polyoml d opolyoml sple pproces to te mercl solto of secod order odry vle prolems Appled temtcs d Comptto vol. 84 pp [8] A. K Prmetrc cc sple solto of two pot odry vle prolems Appled temtcs d Compttovol. 4 pp