Abstract. W.W. Memudu 1 and O.A. Taiwo, 2

Transcription

1 Theoreicl Mhemics & Applicions, vol. 6, no., 06, 3-50 ISS: prin, online Scienpress d, 06 Eponenilly fied collocion pproimion mehod for he numericl soluions of Higher Order iner Fredholm Inegro-Differenil-Difference Equions W.W. Memudu nd O.A. Tiwo, Asrc This pper is concerned wih he pplicion of eponenilly fied collocion pproimion mehod for he numericl soluions of Higher Order iner Fredholm Inegro-Differenil-Difference Equions. Our pproch enils susiuing n ssumed pproime soluions Cheyshev nd egendre olynomils s ses funcions ino slighly perured form of he given prolem nd hen fied he given mied condiions wih n eponenil, hving one free-u prmeer. Thus, he resuling equion is hen colloced eqully spced inerior poins of given inervls. Thus, resuled ino lgeric liner sysem of equions which re comined wih he eponenilly fied given mied condiions. All ogeher, hese equions nd hen sovled using modificion of MAE 3. The mehod is pplied Kwr Se Fiscl Responsiiliy Commission, Ilorin, igeri. E-mil: wwmemudu@gmil.com Deprmen of Mhemics, Fculy of hysicl Sciemces, Universiy of Ilorin, Ilorin, igeri. E-mil: omoyodeyo@yhoo.com Aricle Info: Received: My 7, 05. Revised: June, 05. ulished online : Jnury 5, 06.

2 3 Eponenilly fied collocion pproimion mehod o wider clss of prolems. The numericl resuls oined for seleced prolems show h he pproime soluion y Cheyshev olynomils s sis funcion performed eer hn h of egendre olynomils s sis funcion in erms ccurncy chieved, compuionl ime nd cos. Keywords: Eponenilly Fied; Cheyshev nd egendre; ses funcion; erured; ccurcy Inroducion The prolem sed in equion ogeher wih he mied condiions given in equion elow hve een recenly considered y As nd Mehdi 00 nd solved equions nd y Tylor Series. n k =0 y k k * r + r y τ = f + K, y τ dτ ; τ 0 r=0 wih he mied condiions n k =0 [ ik y k k k + β y + γ y η] = µ ; i = 0,,..., s ik ik In he mehod, As nd Mehdi, repored h he vrile coefficiens prolems proved difficul o solve y Tylor Series especilly when he vriles re in he form of rnscedenils or eponenils. As nd Mehdi 00, used n pproime soluions y Tylor Series nd rnsform he equion nd he given mied condiion ino mri equion. By solving he sysem of lgeric equions, he Tylor s coefficiens of he soluion funcion re oined. Also, Tylor s mehod hs een eended o solve he Fredholm Inegro-Differenil-Difference Equion. The drwck of he Tylor Series mehod repored in As nd Mehdi 00 is h he higher derivives involved prove imes difficul o oin. Our pproch ssumed n pproime soluion in erms of Cheyshev olynomils nd egendre olynomils s he ses funcions. The ssumed pproime soluion in erms of i

3 W.W.Memudu nd O.A. Tiwo 33 Cheyshev olynomils is susiued ino slighly perured given prolem nd eponenil is fied wih one free u-prmeer o he mied oundry condiions of he given prolem. Thus, fer simplificion, he residul prolem nd he mied oundry condiion, led o lgeric liner sysem of equions which re solved o oin he unknown free u-prmeers nd he unknown consns h ppered in he ssumed soluion. rolem considered In his work, he nh-order iner Fredholm Inegro-Differenil-Difference Equion wih vrile coefficiens given s n k =0 y k k * r + r y τ = f + K, y τ dτ ; τ 0 r=0 wih he mied condiions 3 n k =0 [ ik y k k k + β y + γ y η] = µ ; i = 0,,..., s 4 ik ik i * where, k, r, k, nd f re known smooh funcions. Here, he rel coefficiens α, β, γ,,, η ik ik ik nd i µ re given consns.. Cheyshev polynomils The well known Cheyshev olynomils T n re defined in inervl [-,] s T n = Cos{ ncos }; 5 nd deermined wih he id of he following recurrence formul. T = T ; n+ n Tn n Few erms of Cheyshev polynomils vlid in [-,] re lised T 0 = =,,...

4 34 Eponenilly fied collocion pproimion mehod T = T = 3 T = T 4 = T = T6 = ec The nlyic form of he Cheyshev polynomils T n of degree n is given y T n = n where [ n n ] denoes he ineger pr of. The orhogonliy condiion is n [ ] i ni n i ni i, i= 0 i! n i! π for i = j = 0 ; Ti Tj π d = for i = j 0 ; 0 for i j.. egendre polynomils egendre polynomils denoed y n in he domin [-,] is defined y nd n + = {n + n nn } n + n d n = n n n! d n ; n = 0,,,...

5 W.W.Memudu nd O.A. Tiwo 35 The firs few erms of he egendre polynomils re given elow for 0 =. =. = 3 3 = = = ec. 3 ierure review The heory of inegro-differenil-difference equions, he mehod used, nd is wide pplicions hve dvnced eyond he dolescen sge o occupy cenrl posiion in pplicle nlysis. In fc, in he ls yers, he proliferion of he sujec hs een winessed y hundreds of reserch ricles, severl monogrphs, mny works hve een done y severl urhors in he heory of inegro-differenil-difference equions. Sefnini nd Bede 008 solved he ove menioned pproch under srongly generlized differeniiliy of inegro-differenil-difference equions.in his cse, he derivive eiss nd he soluion of inegro-differenil-difference equions my hve decresing lengh of he suppor, u he uniqueness is los. Cheyshev finie difference mehod [Dehghn nd Sdmndi 008], egendre Tu mehod [Dehghn nd Sdmndi of equion 00] nd Vriionl Ierion Mehod VIM [Bizr nd Gholmi orshokouhi 00, Bessel mri mehod [Yuzs e l 0. Among hem re Tiwo nd Adeisi 0, Tiwo nd Alimi 04 nd Tiwo nd Rji 04. The

6 36 Eponenilly fied collocion pproimion mehod uhors repored ove hve used collocion pproimion mehod y power series mehod nd cnonicl polynomils pproime soluions. Homoopy Anlysis Mehod HAM ws lso inroduced y io 04 o oin series soluions of vrious liner nd nonliner prolems of his ype. 4 Descripion of eponenilly fied Tu-mehod In his secion, we discuss he eponenilly fied collocion u-mehod for he soluions of iner Fredholm Inegro-Differenil-Difference Equions. In his mehod, we ssumed n pproime soluion of he form y = rr ; 6 r=0 where r r 0 re unknown consns o e deermined nd r re he egendre polynomils defined ove. Thus, equion 3 is susiued ino slighly perured equion o oin, where H S k * i k r r + i r r τ r=0 k=0 r= i=0 r r τ r=0 = f + K, d + H, τ 0 is he perurion erm given s 7 H = τ T + τ T + τ T nd τ, τ, τ,... τ re n free u prmeers o e deermined long wih 3 4 n n 0. n We hen fied n eponenil wih one free u-prmeer ino equion o oin k k k k,, ikrr βikrr γ ik rr η τ ne η + + ` + = µ i r=0 k =0 i = 0,,..., n ; 8

7 W.W.Memudu nd O.A. Tiwo 37 Equion 4 is furher simplified o oin 0 * 0 { τ K, 0 τ d} 0 * { + τ K, τ d} + * { + τ K, τ d} n + n * + { + K, τ d} = f H τ 9 According o Oriz 969, he numers of τ i inrodurced re equivlen o he degrees or orders of he prolems considered. In order o sisfy Oriz 969, he remining τ -free prmeer is hen fied o he mied ondry condiions s shown in equion 5. Hence, equion 6 is hus colloced poin i = o oin k 0 * 0 { τ, 0 k k k k K τ d} 0 + k * { τ, k k k k K τ d k } * { τ, k k k k τ k } n * + { n +, k k k τ τ } K d + + τ T = f i i l i= K d k k where for some ovious prcicl reson, we hve chosen he collocion poins o e k k = + ; k =,,..., + + Thus, we hve + collocion equions in + n + 0 unknowns,,..., conns o e deermined long wih n free-u prmeers. n 0 er equions re oined y equion 5. Alogeher, we hve ol of + n + lgeric liner sysem of equions in + n + unknown consns. The + n + liner lgeric sysem of equions re hen solved y Gussin

8 38 Eponenilly fied collocion pproimion mehod eliminion mehod o oin he unknown consns 0 i i which re hen susiued ck ino he pproime soluion in equion 3. Thus, equions 6 nd 7 re hen pu in Mri form of he form B X A = where, = = m m m m m T T S A X B f f f f f τ τ τ + + = d, = * τ τ + d, = 0 * 0 τ τ + d, = 0 * 0 3 τ τ + d K S S, = * τ τ + d, = * τ τ + d, = 0 * 0 τ τ + d, = 0 * 0 3 τ τ + d S S, = * τ τ +

9 W.W.Memudu nd O.A. Tiwo 39 The mri equion is hen solved y Gussin Eliminion Mehod o oin he unknown consns, which re hen susiued ino he pproime soluion given in equion 6. 5 Illusrive emples Eponenilly Fied Collocion Tu-Mehod By Cheyshev olynomils Funcion In his secion, severl numericl emples re given o illusre he ccurcy nd effeciveness properies of he mehod nd MAE 3 pckge is used o crry-ou he clculion. The solue errors used is defined s y y ; umericl Emple Consider he firs-order liner Fredholm inegro-differenil-difference equion [As nd Mehdi 00] y y + y + y = + y d, 3 wih he mied condiion y y0 + y = 0. 4 The ec soluion of he prolem is y = umericl Emple Consider he second order liner Fredholm inegro-differenil-difference equion = 4 y" 4 y y y = ln ln3 5ln5 + y d + 3 wih condiions 5 y 0 = ln4, y 0 =, 6 4

10 40 Eponenilly fied collocion pproimion mehod nd he ec soluion of his prolem is y = ln + 4. umericl Emple 3 In his emple, we consider hird-order liner Fredholm inegro-differenil-difference equion wih vrile coefficiens given s y y + y y = = + sin + cos cos + y d wih condiions y 0 = 0, y 0 =, y 0 = 0, 8 nd he ec soluion is y = sin. 7 6 Tle of resuls In his secion, we uled he resuls oined for vrious vlues of nd he ec soluion when evlued eqully spced inerior inervls of considerion.

11 W.W.Memudu nd O.A. Tiwo 4 Tle : Asolue Error for Emple Cse =6 As nd Mehdi Eponenilly Eponenilly Tylor 00 Fied Fied Series Shifed egendre y Cheyshev y egendre [9] Tu Mehod olynomil olynomil

12 4 Eponenilly fied collocion pproimion mehod Tle : Asolue Error for Emple Cse =7 Tylor Series [9] As nd Mehdi 00 Shifed egendre Tu Mehod Eponenilly Fied y Cheyshev olynomil Eponenilly Fied y egendre olynomil

13 W.W.Memudu nd O.A. Tiwo 43 Tle 3: Emple : Eponenilly Fied Collocion Tu Mehod for Cse =6 Ec soluion Ylcins & Akky EFC EF CM [4] Ylcins & Akky EFC EF y= In+4 =6 y y y E E E E * * * * * * * * * * * * * * * * * * * * *

14 44 Eponenilly fied collocion pproimion mehod * * * * * * * * *

15 W.W.Memudu nd O.A. Tiwo 45 Tle 4: Emple : Eponenilly Fied Collocion Tu Mehod for Cse = 7 Ec soluion Ylcins & Akky EFC EF CM [4] Ylcins & Akky EFC EF y= In+4 =7 y y y E E E E * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,

16 46 Eponenilly fied collocion pproimion mehod oe: i Eponenilly Fied y Cheyshev olynomil EFC ii Eponenilly Fied y edendre olynomil EF

17 W.W.Memudu nd O.A. Tiwo 47 Tle 5: Asolue Error for Emple 3 Cse =6 As nd Mehdi Eponenilly Eponenilly Tylor 00 Fied Fied Series [9] Shifed egendre Tu Mehod y Cheyshev olynomil y egendre olynomil

18 48 Eponenilly fied collocion pproimion mehod Tle 6: Asolue Error for Emple 3 Cse =7 Tylor Series [9] As nd Mehdi 00 Shifed egendre Tu Mehod Eponenilly Fied y Cheyshev olynomil Eponenilly Fied y egendre olynomil Conclusion nd discussion of resuls In his work, we solved Higher Order iner Fredholm Inegro-Differenil-Difference Equions y eponenilly fied using wo ses funcions nmely, Cheyshev nd egendre olynomils. The resuls oined y he proposed mehod for he soluion of Higher Order iner Fredholm Inegro-Differenil-Difference Equions re compred wih oher eising works in lierure such s Tylor Series Approch nd Shifed egendre Tu Mehods repored y As nd Mehdi 00. We oserved from he les of resuls presened h using Cheyshev

19 W.W.Memudu nd O.A. Tiwo 49 olynomil s he sis funcion gve eer resuls when compred wih h of egendre olynomil. We lso oserved h where he soluions were known in As nd Mehdi 00, he resuls of he proposed mehod re in good greemens. The proposed mehod does no require rigorous clculion nd compuion cos in erms of eecuion of resuls when compred wih he work of Ylcins nd Akky 0. References [] As, S. nd Mehdi, D., umericl Soluion of he Higher Order iner Fredholm Inegro-Differenil-Difference Equions wih vrile Co efficiens, Deprmen of Applied Mhemics, Fculy of Mhemics nd Compuer Science, Amirkir Universiy of Technology, 44, Hfez Ave. Tehrn, Irn, 00. [] Bhrwy, A. H. Tohidi, E. nd Soleymni, F., A ew Bernoulli Mri Mehod For Solving High Order iner And onliner Fredholm Inegro- Differenil Equions Wih iecewise Inervl, Applied Mhemics nd Compuions, 9, 0, [3] Bi-zr, J. nd Gholmi orshokouhi, M., Applicion Of Vriionl Ierion Mehod For iner And onliner Inegro-Differenil-Difference Equions, Inernionl Mhemicl Forum, 565, 00, [4] Dehghn, M. nd Sdmndi, A., Cheyshev Finie Difference Mehod For Fredholm Inegro-Differenil-Difference Equions, Inernionl Journl of Compuer Mhemics, 85, 008, [5] Dehghn, M. nd Sdmndi, A., umericl Soluion of The Higher Order iner Fredholm Inegro-Differenil-Difference Equions Wih Vrile Coefficiens, Compuer And Mhemics Wih Applicions, 598, 00, [6] Gulsu, M. nd Sezer, M., Approimions To The Soluion Of iner Fredholm

20 50 Eponenilly fied collocion pproimion mehod Inegro-Differenil-Difference Equions Of High Order, Journl Of Frnklin Insiue, 343, 006, [7] io, S.J., Beyond erurion: Inroducion To The Homoopy Anlysis Mehod, Vol. of Modern Mechnics And Mhemics, C.R.C ress, Boc Ron, Fl, USA, 04. [8] Oriz, E.., The Tu Mehod, SIAM J. umer. Anl., 6, 969, [9] Sefnini,. nd Bede, B. Generlized Hukuhr Differeniiliy Of Inervl Vlued Funcions And Inervl Differenil Equions, Working ppers, 080, Universiy Of Urino Crlo Bo, Deprmen Of Economics, Sociey And oliics-scienific Commiee, 008. [0] Tiwo, O.A. nd Adeisi, A.F., Muliple erured Collocion Mehod For Specil Clss Of Higher Order iner Fredholm And Voler Inegro-Differenil Equions, Deprmen of Mhemics, Fculy of hysicl Sciences, Universiy of Ilorin, igeri, 0. [] Tiwo, O. A., Alimi, A. T. nd Aknmu, M.A., umericl Mehods For Solving iner Fredholm Inegro-Differenil-Difference Equions By Collocion Mehods, Journl of Educionl olicy nd Enerpreneurl Reserch JEER,, 04, [] Tiwo, O. A. nd Rji, M.T., umericl Soluions Of Second Order Inegro-Differenil-Difference Equions IDEs Wih Differen Four olynomil Bses Funcions, Inernionl Journl of Engineering Reserch And Technology, 3, Feury, 04. [3] Ylcins, S. nd Akky, T., A umericl Approch For Solving iner Inegro-Differenil-Difference Equions Wih Bouker olynomil Bses, Deprmen of Mhemics, Fculy of Science And Ar, Cell Byr Universiy, Mnis Turkey, 0. [4] Yuzs, S., Shin,. nd Sezer, M., A Bessel olynomil Approch For Solving Generl iner Fredholm Inegro-Differenil-Difference Equions, Inernionl Journl of Compuer Mhemics, 884, 0,