Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials

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1 Nuercl Solutos of Fredhol Itegrl Equto of Secod Kd Usg Pecewse eroull Polyols Afroz Shr, Md. Shfqul Isl* Isttute of Nturl Sceces, Uted Itertol Uversty, Dh-9, gldesh. El: Deprtet of Mthetcs, Uversty of Dh, Dh, gldesh *Correspodg uthor, El: Astrct: The of ths pper s to solve the tegrl equtos uerclly usg pecewse eroull polyols. The eroull polyols re derved eplctly over the ut tervl. A tr forulto for o-sgulr ler Fredhol tegrl equto of the secod d s derved y the techque of Gler ethod. I the Gler ethod, the eroull polyols re eploted s the ler coto the pproto s ss fuctos. Nuercl eples re cosdered to verfy the effectveess of the proposed dervtos. Keywords: Fredhol tegrl equto, Gler ethod, eroull polyols, Nuercl solutos. I. Itroducto I the survey of solutos of tegrl equtos, lrge uer of lytcl ut few pprote ethods re vlle for solvg uerclly vrous clsses of tegrl equtos [,, 7, 8 ]. Sce the pecewse polyols re dfferetle d tegrle, the erste polyols [5 8] hve ee used for solvg dfferetl d tegrl equtos uerclly. Recetly, tegrl equtos hve ee solved y the well ow vrtol terto ethod [9]. I the lterture [7], Mdl d httchry hve ttepted to solve tegrl equtos uerclly usg erste polyols, ut they oted the results ters of fte seres solutos I cotrst to ths, we solve the ler Fredhol tegrl equto of the secod d y eplotg very well ow Gler ethod, d eroull polyols [] re used s trl fuctos. For ths, we gve short troducto of eroull polyols frst. The we derve tr forulto y the techque of Gler ethod. To verfy our forulto we cosder three eples, whch we ot ect solutos for two eples eve usg few d lower order polyols. O the other hd, the lst eple shows ecellet greeet of ccurcy copred to ect soluto, whch cofrs the covergece. All the coputtos re perfored usg MATHEMATICA.

2 II. eroull Polyols: The eroull polyols [Atso, ] upto degree c e defed over the tervl [,] plctly y where re eroull uers gve y d d. These eroull polyols y e defed eplctly s ( ) ( ) ( ), () The frst 6 eroull polyols ( 5) re gve ellow for usg ths pper: Note tht eroull polyols hve specl property t d, respectvely, r ( ), d ( ),. 5 III. Forulto of Itegrl Equto Mtr For Cosder geerl ler Fredhol tegrl equto (FIE) of secod d [, ] s gve y ( ( dt f, () where ( d f ( re gve fuctos, ( s the erel, d ( fucto or ect soluto of (), whch s to e detered. s the uow Now we use the techque of Gler ethod [Lews, ] to fd pprote soluto ( ) of (). For ths, we ssue tht () where ( re eroull polyols (ss) of degree defed eq. (), d re uow preters, to e detered. Susttutg () to (), we ot

3 ( ( dt f or, ( ( dt f or, ( ( dt f () The the Gler equtos [Lews, ] re oted y ultplyg oth sdes of () y ( d the tegrtg wth respect to fro to, we hve or, ( ( dt d f d ( ( dt d f d,,,, (5) I ech equto, there re three tegrls. The er tegrd of the left sde s fucto of d t d s tegrted wth respect to t fro to. As result the outer tegrd ecoes fucto of oly d tegrto wth respect to yelds costt. Thus for ech (,,, ) we hve ler equto wth uows (,,, ). Flly (5) represets the syste of ler equtos uows. Equvletly, C, F,,,,,, (5) where C, ( ( dt d,,,,,. (5c) F f d,,,,, (5d) Now the uow preters re detered y solvg the syste of equtos (5), d susttutg these vlues of preters (), we get the pprote soluto ( ) of the tegrl equto (). The solute error E for ths forulto s defed y ( E. (

4 IV. Nuercl Eples I ths secto, we epl three tegrl equtos whch re vlle the estg ltertures [, 7]. For ech eple we fd the pprote solutos usg eroull polyols. Eple : We cosder the FIE of d d gve y [7] ( t t ) ( dt,, (6) hvg the ect soluto. 9 Usg the forulto descred the prevous secto, the equtos (5) led us, respectvely, C, d ( t t ) dt d,,,,,, (7) F d,,,,, (7) Solvg the syste (7) for, the vlues of the preters re:,,, 9 9 d the pprote soluto s 9 whch s the ect soluto. Eple : Now we cosder other FIE of d d gve y [Mdl, 7] ( t ) ( dt, (8) hvg the ect soluto Proceedg s the eple, the syste of equtos ecoes s where, C, d ( t ) dt d,,,,,, (9)

5 F d,,,,, (9) For, solvg syste (9), the vlues of the preters ( ) re:,,,, d the pprote soluto s whch s the ect soluto. Eple : Cosder other FIE of d d gve y [pp (), wth, Jerry, ] ( t t ) ( dt, () 8 8 hvg the ect soluto 9 9 Proceedg s the prevous eples, the syste of equtos ecoes s where, C, F,,,,,, () C F, ),,,,,, d ( t t ) dt ( d d, (),,,, (c) For, solvg syste (), the vlues of the preters ( ) re: 6 8,,,, 9 9 d the pprote soluto s whch s the ect soluto. Eple : Cosder other FIE of d d gve y [pp (v), wth, Sht, ] t e e ( dt e,, () e hvg the ect soluto. e Sce the equtos (5) d (5c) re of the for 5

6 t C, d e e dt d,,,,,, () F e d,,,,,, () Solvg the syste (5) usg () d () sted of (5) d (5c) respectvely, we hve the followg results: For, the pprote soluto s For, the pprote soluto s For 5, the pprote soluto s For 6, the pprote soluto s Plot of reltve error, E s depcted Fg. for vrous vlues of. Now the pprote solutos, ect solutos, d the reltve error E t vrous pots of the do re dsplyed Tle. The ccurcy s ovous. E E Fg.. Reltve error E usg polyols Fg.. Reltve error E usg 5 polyols 6

7 E E Fg.c. Reltve error E usg 6 polyols Fg.d. Reltve error E usg 7 polyols 7

8 Tle-: Nuercl solutos t vrous pots d correspodg solute errors of the eple Ect Solutos Approte Solutos Asolute Reltve Error, E Approte Solutos Polyols used Polyols used Asolute Reltve Error, E Polyols used 5 Polyols used V. Cocluso We hve solved uerclly Fredhol tegrl equtos of secod d. We hve oted the pprote soluto of the uow fucto y the well ow Gler ethod usg eroull polyols s trl fuctos. The uthors cocludg rer s tht the uercl solutos cocde wth the ect solutos eve few of the polyols re used the pproto. 8

9 Refereces. Adul J. Jerr, 999 Itroducto to Itegrl Equtos wth Applctos, Joh Wley & Sos Ic.. Sht Swrup, 7 Itegrl Equtos, Krsh Prsh Med (P) Ltd, Ffteeth Edto.. Lews, P. E., J. P. Wrd,99 The Fte Eleet Method, Prcples d Applctos, Addso-Wesley.. Kedl E Atso,989 A Itroducto to Nuercl Alyss, Joh Wley d Sos, d edto, (989), pp Reehof, J., 977 Dfferetto d tegrto usg erste s polyols, It. J. Nuer. Methods Egrg,, Kreyszg, E., 979 erste polyols d uercl tegrto, It. J. Nuer. Methods Egrg,, Mdl,. N., S. httchry, 7 Nuercl soluto of soe clsses of tegrl equtos usg erste polyols, Appled Mthetcs d Coputto, 9, htt, M.I., P. rce, 7 Solutos of dfferetl equtos erste polyol ss, J. Coputtol d Appled Mthetcs, 5, Wg, S.Q., J.H. He, 7 Vrtol ethod for solvg tegro-dfferetl equtos, Physcs Letters A, 67,