Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Transcription

1 Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-, gldesh MS: 8/ Fe Astrct I this pper, erstei piecewise polyomils re used to solve the itegrl equtios umericlly. A mtri formultio is give for o-sigulr lier Fredholm Itegrl Equtio y the techique of Glerki method. I the Glerki method, the erstei polyomils re eploited s the lier comitio i the pproimtios s sis fuctios. Emples re cosidered to verify the effectiveess of the proposed derivtios, d the umericl solutios gurtee the desired ccurcy. Keywords: Fredholm itegrl equtio; Glerki method; erstei polyomils.. Itroductio I the survey of solutios of itegrl equtios, lrge umer of lyticl ut few pproimte methods for solvig umericlly vrious clsses of itegrl equtios [, ] re ville. Sice the piecewise polyomils re differetile d itegrle, the erstei polyomils [, ] re defied o itervl to form complete sis over the fiite itervl. Moreover, these polyomils re positive d their sum is uity. For these dvtges, erstei polyomils hve ee used to solve secod order lier d olier differetil equtios, which re ville i the literture, e.g. htti d rcke [7]. Very recetly, Mdl d httchry [6] hve ttempted to solve itegrl equtios umericlly usig erstei polyomils, ut they otied the results i terms of fiite series solutios. I cotrst to this, we solve the lier Fredholm itegrl equtio y eploitig very well kow Glerki method [] d erstei polyomils re used s tril fuctios i the sis. For this, we give short itroductio of erstei polyomils first. The we derive mtri formultio y the techique of Glerki method. To verify our formultio we cosider three emples, i which we oti ect solutios for two emples eve usig few d lower order polyomils. O the other hd, the lst emple shows ecellet greemet of ccurcy compred to ect solutio, which cofirms the covergece. All the computtios re performed usig MATHEMATICA. Correspodig uthor:mdshfiqul@yhoo.com

2 . erstei Polyomils The geerl form of the erstei polyomils [-7] of th degree over the itervl [, ] is defied y ( ) i ( i i, (,, i,.,, () i ( ) Note tht ech of these + polyomils hvig degree stisfies the followig properties: i) i, (, if i or i, ii) i, ( ; i iii) i, ( ) i, ( ), i Usig MATHEMATICA code, the first erstei polyomils of degree te over the itervl [, ], re give elow:,,,,,, 8 ( ) ( ) ( ) /( ) 7 ( ) ( ) ( ) /( ) 6 ( ) ( ) ( ) /( ) ( ) ( ) ( ) /( ) ( ) ( ) /( ) ( ) ( ) ( )/( ) 6, 7, 8,,, ( ) ( ) ( ) /( ) ( ) ( )( ) /( ) ( ) ( ) ( ) /( ) ( ) ( ) ( ) /( ) 8 7 ( ) ( ) /( ) 6 Now the first si polyomils over [, ] re show i Fig. (), d the remiig five polyomils re show i Fig. ().

3 . Formultio of Itegrl Equtio i Mtri Form Cosider geerl lier Fredholm itegrl equtio (FIE) of secod kid [, ] is give y ( ( k( t, ( dt f (, () where ( d f ( re give fuctios, k ( t, is the kerel, d ( is the ukow fuctio or ect solutio of (), which is to e determied. Now we use the techique of Glerki method [Lewis, ] to fid pproimte solutio ( ) of (). For this, we ssume tht ( i i, ( () i where i, ( re erstei polyomils (sis) of degree i defied i eq. (), d i re ukow prmeters, to e determied. Sustitutig () ito (), we oti ( i i, ( k( t, i i, ( dt f ( i i or, (, ( k( t,, ( dt i i i f ( () i The the Glerki equtios [Lewis, ] re otied y multiplyig oth sides of () y j, ( d the itegrtig with respect to from to, we hve

4 i ( i ( k( t, i ( dt j ( d,,, j, ( f ( d, j,,, i Sice i ech equtio, there re three itegrls. The ier itegrd of the left side is fuctio of d t, d is itegrted with respect to t from to. As result the outer itegrd ecomes fuctio of oly d itegrtio with respect to yields costt. Thus for ech j (,,, ) we hve lier equtio with ukows i ( i,,, ). Filly () represets the system of lier equtios i ukows, re give y i Ci, j Fj, j,,,, i where (), ( ), ( ) C i j (, ), ( ), ( ), i k t i t dt j d i, j,,,,. () Fj i, ( f ( d, j,,,, (c) Now the ukow prmeters i re determied y solvig the system of equtios (), d sustitutig these vlues of prmeters i (), we get the pproimte solutio ( ) of the itegrl equtio (). The solute error E for this formultio is defied y ( ( E. (. Numericl Emples I this sectio, we epli three itegrl equtios which re ville i the eistig litertures [,, 6]. For ech emple we fid the pproimte solutios usig differet umer of erstei polyomils. Emple : We cosider the FIE of d kid give y [6] ( ( t t ) ( dt, (6)

5 hvig the ect solutio, ( Usig the formultio descried i the previous sectio, the equtios () led us, respectively, i i Ci, j F j, j,,,,, (7) Ci, j i ( j ( d ( t t ) i ( dt,,, j, ( d, i, j,,, (7) Fj j, ( d, j,,,, (7c) Solvig the system (7) for, the vlues of the prmeters re: 7 7,,, 7 7 Sustitutig ito () d for, the pproimte solutio is, ( which is the ect solutio. Emple : Now we cosider other FIE of d kid give y [6] ( ( t ) ( dt, (8) hvig the ect solutio ( Proceedig s the emple, the system of equtios ecomes s i Ci, j F j, j,,, i where, () C i, j i, ( i, ( d ( t ) i, ( dt j, ( d, i, j,,, ()

6 F j j, ( d, j,,, (c) Now solvig the system () for, the vlues of the prmeters, i re:,,,., is ( which is the ect solutio. d the pproimte solutio, for Emple : Cosider other FIE of d kid give y [, pp ] ( ( t t ) ( dt, () 8 8 hvig the ect solutio ( Proceedig s the previous emples, the system of equtios ecomes: Ci, j i, ( j, ( d ( t t ) i, ( dt j, ( d, i, j,,, () F j j, ( d, j,,, () For, solvig system (), the vlues of the prmeters ( i ) re: 6 8,,, 8 8 d the pproimte solutio is (, which is the ect solutio. Emple : Cosider other FIE of d kid give y [, pp ] t ( e e ( dt e,, () hvig the ect solutio e (. e Sice the equtios () d (c) re of the form 6

7 t Ci, j i, ( j, ( d e e i, ( dt j, ( d i, j,,,, () F j e j, ( d, j,,, () For,,, d 6, the pproimte solutios re, respectively ( ( ( ( Plot of solute differece, the error E etwee ect d pproimte solutios, is depicted i Fig. for vrious vlues of. Oserve tht the miimum order of ccurcies re -, -6, -7, d -8, respectively, with,, 6 d 7 erstei polyomils. This cofirms us tht we if icrese the umer of polyomils, the ccurcy lso icreses. Now the pproimte solutios, ect solutios, d the error E, etwee ect d the pproimte solutios t vrious poits of the domi re displyed i Tle. 7

8 Tle. Numericl solutios t vrious poits d correspodig solute errors of the emple Ect Solutios Approimte Solutios Asolute Reltive Error, E Approimte Solutios Polyomils used Polyomils used Asolute Reltive Error, E Polyomils used 6 Polyomils used

9 . Coclusio We hve cosidered the itegrl equtios to solve umericlly. We hve otied the pproimte solutio of the ukow fuctio y the well kow Glerki method usig erstei polyomils s tril fuctios. We hve verified the derived formuls with the pproprite umericl emples. I this cotet we my ote tht the umericl solutios coicide with the ect solutios eve few of the polyomils re used i the pproimtio. Refereces Ref formt/style ot OK. Adul J. Jerri, Itroductio to Itegrl Equtios with Applictios (Joh Wiley & Sos Ic. ).. Shti Swrup, Itegrl Equtios, Krish Prksh Medi (P) Ltd ( th Editio, 7).. P. E. Lewis d J. P. Wrd, The Fiite Elemet Method, Priciples d Applictios (Addiso- Wesley, ).. J. Reikehof, It. J. Numer. Methods Egrg., 67 (86).. E. Kreyszig, It. J. Numer. Methods Egrg., (7). 6.. N. Mdl d S. httchry, Appl. Mth.Comput., 77 (7). 7. M.I. htti d P. rcke, J. Comput. Appl. Mth., 7 (7).