Amerc Jorl of Egeerg Reserch (AJER) 04 Amerc Jorl of Egeerg Reserch (AJER) e-iss : 30-0847 p-iss : 30-0936 Volme-03, Isse-08, pp-43-47 www.jer.org 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
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, 0 3 0 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... 5 0 0 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... 0 0 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
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 0.000000000.000000000.000000000.000000000.000000000 0..00000000.09987500.09987508.0985560.0985396 0..00000000.099500000.09950007.097448.09743853 0.3.70000000.6956500.6956557.6798934.679630 0.4.560000000.559963000.550000098.55853049.558578 0.5.000000000.999737500.995937649.94409508.984063738 0.6.60000000.69800000.6760007.59594590.60690780 0.7 3.450000000 3.4498500 3.4478774 3.4374999 3.34774003 0.8 4.50000000 4.58700000 4.509700350 4.5094993 4.48467 0.9 5.860000000 5.858787500 5.849787937 5.6786744 5.759646.0 7.500000000 7.497500000 7.489800353 7.3956043 7.39996085 Emple : - e e t Sject to the coos 0 0 0 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 0.000000000.000000000.000000000 0.999999998 0.999999973 0..057098.057098.057037.0570906.0570586 0..40758.40759.39366.40664.4038 0.3.349858808.349858808.349796664.349858494.349853557 0.4.4984698.4984698.49779456.4984493.49838 0.5.64877.648770.648687634.64870056.64868450 0.6.88800.88800.80948.87485.8097068 0.7.0375707.0375706.0373657.0375484.03735387 0.8.554098.554099.5004.55406.5585 0.9.459603.45960309.4594085.4596003.45959944.0.78888.78888.7865764.78304575.77964004 w w w. j e r. o r g Pge 45
Emple 3: v Sject to The Amerc Jorl of Egeerg Reserch (AJER) 04-4 0 0, 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 0 0 0 0.5388 0 -.53837 0-0. 0.0953080 0.095309970 0.0953088 0.09499653 0.094900490 0. 0.83557 0.8309 0.885697 0.89578 0.8969997 0.3 0.636465 0.6346 0.63995 0.609395 0.603595 0.4 0.3364737 0.336448896 0.336437637 0.3369984 0.3363787 0.5 0.40546508 0.405437993 0.405430058 0.4096354 0.4090838 0.6 0.47000369 0.46997553 0.46996799 0.46750896 0.46660689 0.7 0.530685 0.53059544 0.5300904 0.584937 0.58386 0.8 0.587786665 0.58760864 0.58737405 0.5863339 0.585405 0.9 0.64853886 0.6468434 0.645036 0.6403677 0.6397676.0 0.693478 0.69967 0.6987487 0.69938947 0.696 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 0 0 0 0 0 0..50 0-5.49 0-5.4484 0-4.46704 0-4 0. 5.000 0-4 4.99973 0-4.55879 0-3.56477 0-3 0.3 4.375 0-4 4.37443 0-4.00676 0-3.036880 0-3 0.4 3.700 0-4 9.99990 0-3.46975 0-3.4879 0-3 0.5.65 0-4 4.0635 0-3 5.590498 0 -.59366 0-0.6.000 0-4.399793 0-3.408540 0 -.30980 0-0.7.875 0-4.876 0-3.58007 0 -.057997 0-0.8.300 0-3.099650 0 -.0758007 0 -.0757833 0-0.9.5 0-3.0063 0 -.838856 0 -.00388538 0 -.0.5000 0-3.0996468 0 -.04839857 0 -.0003795 0 - Tle 5: Errors for emple Power seres Cheyshev seres X Stdrd Collocto C.G.L Collocto Stdrd Collocto C.G.L Collocto 0 0 0.0 0-9.7 0-8 0. 0 6.0 0-7. 0-8 3.3 0-7 0..0 0-9 9.4 0-6 9.4 0-8 4.3 0-7 0.3 0 6.44 0-5 3.4 0-7 5.5 0-6 0.4 0 4.54 0-5.05 0-7.378 0-6 0.5.0 0-9 3.3637 0-5.35 0-6 3.98 0-5 0.6 0.737 0-5.97 0-6.73 0-5 0.7.0 0-9.6450 0-5.3 0-6.730 0-5 0.8.0 0-9 4.30887 0-4.78 0-6.73 0-5 0.9.0 0-9.067 0-4.079 0-6.67 0-5.0 0.6064 0-4.747 0-5 3.78 0-4 w w w. j e r. o r g Pge 46
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 0 0 0.588 0 -.53837 0-0.. 0-7.898 0-6 3.8057 0-4 4.0969 0-4 0..365 0-6 3.586 0-5 3.68839 0-4 3.556 0-4 0.3.653 0-5 3.434 0-5 3.44870 0-4 3.5067 0-4 0.4.334 0-5 3.460 0-5 3.453 0-4 3.4045 0-4 0.5.75 0-5 3.505 0-5 4.68754 0-3 3.55678 0-3 0.6.837 0-5 3.633 0-5.494667 0-3 3.4094 0-3 0.7 3.837 0-5 5.37 0-4.34530 0-3.48963 0-3 0.8.7805 0-4 4.64 0-4.55346 0-3.38544 0-3 0.9.69544 0-4 3.557 0-4.48675 0-3.377 0-3.0.34564 0-4.73 0-4.0834 0-3.0456 0-3 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 90 95. [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 657 668. [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 578 583. [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 55 60. [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 39 3303. [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 005 0. [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 556 570. [] 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 869 878. [] Rshed. M.T (004): mercl Solto of Fctol Dfferece, Itegrl d Itegro Dfferetl Eqtos. Jorl of Appled Mthemtcs d Comptto 56. Pp 485 49. w w w. j e r. o r g Pge 47