Numerical Solution of Higher Order Linear Fredholm Integro Differential Equations.

Transcription

1 Amerc Jorl of Egeerg Reserch (AJER) 04 Amerc Jorl of Egeerg Reserch (AJER) e-iss : p-iss : Volme-03, Isse-08, pp Reserch Pper Ope Access mercl Solto of Hgher Order Ler Fredholm Itegro Dfferetl Eqtos., Gegele. O.A,, Evs. O.P, 3, Akoh,D,,3, Deprtmet of Mthemtcs d Sttstcs, Federl Polytechcs, Bd. ger Stte ABSTRCT : Power seres d Cheyshev seres ppromto methods were sed to solve hgher order ler Fredholm tegro dfferetl eqtos v two collocto pots: Stdrd collocto pot d Cheyshev Gss Lotto collocto pot. mercl emples were gve to demostrte the effectveess of the methods. KEY WORDS: Fredholm tegro dfferetl eqto, Power seres, Cheyshev seres, Stdrd collocto pot, Cheyshev Gss Lotto collocto pot. I. ITRODUCTIO: I recet yers cosderle work hs ee doe oth the developmet of the techqe, ts theoretcl lyss d mercl pplcto the tretmet of Itegro Dfferetl eqtos, ecse of t wde rge of pplctos scetfc feld sch s fld dymcs, sold stte physcs, plsm physcs d mthemtcl ology []. Itegro dfferetl eqtos re clssfed to vros types mog whch Fredholm tegro dfferetl eqto, the focs of ths pper. Geerlly, Fredholm tegro dfferetl eqto s of the form j F k, t t k 0, 0 k - Where k j re the th dervtves, F, k, t re gve cotos smooth fctos, s the kow fcto to e determed d,, re costts. Becse the reslt of () come the k dfferetl d tegrl opertors, the t s ecessry to defe tl coos s (). The Fredholm tegro dfferetl eqto of the secod kd pper vrety of scetfc pplcto sch s the theory of sgl processg d erl etworks []. Becse of the mportce of Fredholm tegro dfferetl eqto scetfc reserch, severl mercl methods were sed to solve oth ler d o ler Fredholm tegro dfferetl eqto sch s T opertol method [8], Hr wvelets method [5], Lgrge terpolto method [] d Dfferetl trsformto method [3], jst to meto t few.[] focsed o the se of Cheyshev terpolto to solve med ler tegro dfferetl eqto wth pecewse tervl. Also [], Lgrge d Cheyshev terpolto ws ppled o fctol tegrl eqto. The se of verse Fzzy trsforms sed o fzzy prtto wth comto collocto techqes hs ee vestgted (see [4]). Reserch hs ee codcted o the se of Legedre mlt-wvelets to solve wekly sglr Fredholm tegro dfferetl eqtos [7]. Power seres method ws se y [9] to solve system of ler d o-ler tegro dfferetl eqtos d ot close form solto f the ect soltos re polyoml otherwse prodces ther Tylor seres solto. Cheyshev seres hs ee sed to solve Fredholm tegrl eqtos t three dfferet collocto pots [6].I ths pper we cosder the se of power seres d Cheyshev seres ppromto methods to solve hgher order Fredholm tegro dfferetl eqtos sg two collocto pots. w w w. j e r. o r g Pge 43

2 Amerc Jorl of Egeerg Reserch (AJER) 04 II. POWER SERIES APPROXIMATIO METHOD I ths secto we cosder the se of power seres ppromto solto of the form, Where the coeffcets re kow to e determed. Pttg (3) () gves, j F 0 (4) c e wrtte smpler form s k, t 4 0 j... F k, t t t We tegrte the rght hd sde of (5) d fter smplfcto, the resltg eqto s the collocted sg the followg collocto pots ()Stdrd collocto pot defed s - p * p, p,,..., -. 6 ()Cheyshev Gss Lotto collocto pot defed s q cos * q, q,,..., -. 7 Ech of the two collocto pots descre ove together wth the tl coo gve () reslted ( + ) ler lgerc eqtos ( + ) kow costts whch re the solved to ot the kow costts tht re ssttted (3) to get the mercl solto of (). III. CHEBYSHEV SERIES APPROXIMATIO SOLUTIO: I ths secto we cosder the se of Cheyshev seres ppromto solto of the form T, 0 8 j 0 Where T s Cheyshev polyoml defed s T - cos cos, -, 9 d t stsfed the recrrece relto T T - T, - 0 Pttg (0) (), we ot j T j 0 () c e wrtte smpler form s F k, t T t j 0 j T T... F k, t T t T Usg the sme procedre s.0 ove d sg shft Cheyshev polyoml where pplcle, () together wth () gves ( + ) ler lgerc eqtos ( + ) kow costts. These eqtos re solved sg mple 3 to ot the kow costts s whch re the ssttted to (8) to get the mercl solto of (). IV. UMERICAL EXAMPLES AD RESULTS: I ths secto we cosder the followg emples o ler Fredholm - tegro dfferetl eqtos. These emples hve ee chose from []. w w w. j e r. o r g Pge 44

3 Amerc Jorl of Egeerg Reserch (AJER) 04 Emple : 3 - t t Sject to the coos - 0, 0 The lytcl solto s gve s 3 * 5 * 3 Tle : mercl solto of emple for = 0 Ect solto Power seres solto Cheyshev seres solto Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto Emple : - e e t Sject to the coos The lytcl solto s gve e 0 s Tle : mercl solto of emple for = 0 Ect solto Power seres solto Cheyshev seres solto Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto w w w. j e r. o r g Pge 45

4 Emple 3: v Sject to The Amerc Jorl of Egeerg Reserch (AJER) , 0, 0 -, 0 lytcl the I. coos solto s 6 * - gve s 4 - t t 0 Tle 3: mercl solto of emple 3 for = 0 Ect solto Power seres solto Cheyshev seres solto Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto Tles of Errors: Tle 4: Errors for emple Power seres Cheyshev seres X Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto Tle 5: Errors for emple Power seres Cheyshev seres X Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto w w w. j e r. o r g Pge 46

5 Amerc Jorl of Egeerg Reserch (AJER) 04 Tle 6: Errors for emple 3 Power seres Cheyshev seres X Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto V. COCLUSIO: Most tegro dfferetl eqtos re dffclt to solve lytclly, my cses t reqre to ot the ppromte soltos, for ths prpose we preset the solto of hgher order ler Fredholm tegro dfferetl eqtos. Or methods re sed o Power seres d Cheyshev seres whch redces ler Fredholm tegro dfferetl eqto to set of ler lgerc eqtos tht c e esly solved y compter. The reslt oted show tht the two methods sed c hdle those prolems effectvely s c e see the tles of errors. REFERECES [] Adl Mjd.W (0): Ler d o ler tegrl eqtos, Methods d Applctos. Sprger Hedelerg Dordrecht Lodo ew York. [] Bels. S.A (007): A ew method for optml cotrol of Volterr tegrl eqtos. Jorl of Mthemtcs d Comptto 89 Pp [3] Dr. P d Ed. A (007): A method for the mercl Solto of tegro dfferetl eqtos. Jorl of Appled Mthemtcs d Comptto 88. Pp [4] Ezzt. R d Mokhtr. F (0): mercl Solto of Fredholm Itegrl Eqtos of the Secod kd y sg Fzzy Trsforms. Itertol Jorl of Physcl sceces 7(0). Pp [5] Fyyz. M d Azrm. M (03): ew Algorthms for mercl Solto of o ler Itergro Dfferetl Eqtos of Thrd Order Usg Hr Wvelets. Jorl of Sceces Itertol (Lhore). 5(). Pp 6. [6] Ishol. C.Y d Aolr. O.E (009): Soltos of Ler Fredholm Itegrl Eqtos Usg Cheyshev Seres Method. ger Jorl of Art, Sceces d Techology (IJASAT). 5(). Pp [7] Mehrdd. L, Behzd..S d Mehd. D (0): mercl Solto for Wekly Sglr Fredholm Itegro dfferetl Eqtos Usg Legedre Mlt Wvelets. Jorl of compttol d Mthemtcs 35. Pp [8] Mohmmd. S.H d Shhmord. S (005): mercl Pecewse Appromte solto of Fredholm Itegro Dfferetl Eqtos y the t Method. Jorl of Appled Mthemtcl Modelg 9. Pp [9] Mortz. G (009): mercl Scheme to Solve Itegro Dfferetl Eqtos System. Jorl of Advced Reserch Scetfc Comptg (). Pp. [0] Mstf. G d Ylc. O (0): O the mercl Solto of Ler Fredholm Volterr Itegro Dfferetl Eqtos wth Pecewse Itervl. Itertol Jorl of Applcto d Appled Mthemtcs 7(). Pp [] Rshed. M.T (004): Lgrge Iterpolto to Compte the mercl Solto of Dfferece, Itegrl d Itegro Dfferetl Eqtos. Jorl of Appled Mthemtcs d Comptto 5. Pp [] Rshed. M.T (004): mercl Solto of Fctol Dfferece, Itegrl d Itegro Dfferetl Eqtos. Jorl of Appled Mthemtcs d Comptto 56. Pp w w w. j e r. o r g Pge 47