The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

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1 Journl of Mtemtcs ttstcs ():-9, 9 IN 9-9 cence Publctons Te Estence, Unqueness Error Bounds of Appromton plnes Interpolton for olvng econd-order Intl Vlue Problems Abbs Y Al Byt, Rostm K eed Frdun K Hm-l Deprtment Mtemtcs, Mosul Unversty, Mosul, Irq Deprtment of Mtemtcs, lddn Unversty, Erbl, Irq Deprtment of Mtemtcs, ulmn Unversty, ulmn, Irq Abstrct: Problem sttement: Te lcunry nterpolton problem, wc we d nvestgted n ts study, conssted n fndng te s degree splne () of defcency four, nterpoltng dt gven on te functon vlue trd fft order n te ntervl [, Also, n etr ntl condton ws prescrbed on te frst dervtve Oter purpose of ts constructon ws to solve te second order dfferentl equtons by two emples sowed tt te splne functon beng nterpolted very well Te convergence nlyss te stblty of te ppromton soluton were nvestgted compred wt te ect soluton to demonstrte te prescrbed lcunry splne (,, ) functon nterpolton Approc: An ppromton soluton wt splne nterpolton functons of degree s defcency four ws derved for solvng ntl vlue problems, wt prescrbed nonlner endpont condtons Under sutble ssumptons, te estences; unqueness te error bounds of te splne (,, ) functon d been nvestgted; lso te upper bounds of errors were obtned Results: Numercl emples, sowed tt te presented splne functon proved ter effectveness n solvng te second order ntl vlue problems Also, we noted tt, te better error bounds were obtned for smll step sze Concluson: In ts study we treted for frst tme lcunry dt (,,) by constructng splne functon of degree s wc nterpolted te lcunry dt (,,) te constructed splne functon ppled to solve te second order ntl vlue problems Key words: Estence unqueness, splne functon, mtemtcl model, nd order dfferentl equtons INTRODUCTION Te ntl vlue problems ply n mportnt role n mtemtcl pyscs, becuse mny problems n scence tecnology re formulted mtemtclly n boundry vlue problems s n et trnsfer deflecton n cbles everl numercl metods ve been nvestgted for clcultng te solutons of suc problems Among tese, fnte dfference tecnques te sootng metod ply n mportnt role Tese metods provde te vlue of te unknown of some grd knots If we wnt te soluton t te ponts, tere s need of nterpoltng tese vlues Also, splne functons of Hermte types s been used by mny utors for solvng tese problems [,9- Te lterture on te numercl solutons of ntl vlue problems by usng lcunry splne functons s not too muc Gyovr [ solved Cucy problem by sng modfed lcunry splne functon wc nterpoltng te lcunry dt (,, ) en [7 used defcent lcunry splne for solvng Cucy problem lso en Venturno [8 used two-pont boundry vlue problem by usng lcunry splne functon wc nterpoltes te lcunry dt (, ) Amed et l [ found te ppromton soluton of te fourt order lcunry splne functons In ts study, we try to solve te ntl vlue problem: y f (, y,y ), y( ) y,y ( ) y () n by usng tt f C ([, R ), n tt t stsfes te Lpsctz contnuous: Correspondng Autor: Abbs Y Al Byt, Deprtment Mtemtcs, Mosul Unversty, Mosul, Irq { } (q) f (,y,y ) f (,y, y ) L y y y y q,,, n- for ll [, ll rel y, y, y, y Tese condtons ensure te estence of te unque soluton of te problem [

2 Ts study s orgnzed s follows: Frst consder te splne functon of degree s s presented wc nterpoltes te lcunry dt (,, ) ome teoretcl results bout estence unqueness of te splne functon of degree s re ntroduced lso convergence nlyss s studed To demonstrte te convergence of te prescrbed lcunry splne functon, numercl emples presented, fnlly, we prescrbe te concluson dscusson of te result MATERIAL AND METHOD Descrptons of te metod: We present for te frst tme ccordng to our knowledge s degree splne (,, ) nterpolton for one dmensonl gven suffcently smoot functon f() defned on [, n : < n Denote te unform prtton of I wt knots, were,,,, n- We denote by n, te clss of s degree splnes () suc tt: s () y y, y (), y, J Mt & tt, ():-9, 9 () on te ntervl [, were,j, j,, re unknowns to be determned [, Let us emne now ntervls [,,,,, n- By tkng nto ccount te nterpoltng condtons, we cn wrte te epresson, for () n te followng form: s () y,, y (), y, () were,, j, ()(n ), j,,, re unknowns we need to determne t On te lst ntervl [ n-, n we defne n () s follows: s () y ) ) n n n n, n n, n) y n ( n ) n, n ) y () n ( n ) n, () were, n-,j, j,,, re unknowns to be determned Te estence unqueness teorem for splne functon of degree s wc nterpolte te lcunry dt (,, ) re presented emned Teorem : Estence unqueness: Gven te rel numbers y( ), y () ( ) y () ( ) for,,,, n, ten tere est unque splne of degree s s gven n te Eq - suc tt: ( ) y( ) ( y (, r, for,,,n ( y ( Proof: Let s defne splne functon () s follows: () wen [, () () wen [, ;,,,n n () wen [ n, n () were te coeffcents of tese polynomls re to be determned by te followng condtons: ( ) ( ) y ( ) ( ) y, r, ;,,,n ' ' ( ) ( ) n n n n n n () ( ) y, ( ) y ; r, (7) To fnd unquely te coeffcents n () of Eq by usng te condton () were, we obtn te followng: y y y () () y y y y y y () () (), 7 y y () () From te boundry condton () we ve: () (), y y y (8) olvng tese equtons to obtn te followng:

3 J Mt & tt, ():-9, 9 () (), [y y y [y y [y () () y 7 (9) () () () (), [y y [y y ( () (), [y y () 7 ubsttutng tese vlues of,,,, we get: () (), [y y y [y y [y () () y () We sll fnd te coeffcents of () for,,,, n- Here we ve: y y y y () () (), y y y 7 y y () (), () () y y () (),, Te coeffcent mtr of te system of Eq n te unknown,,,,,n- s non-sngulr mtr ence te coeffcents,,,,,n- re determned unquely so re, terefore te coeffcents,,,, Fnlly, for fndng te coeffcents of n- (), we ve: y y y n, n, n, n n n () () yn yn y y y y 7 y y () () () n, n, n n n () () n, n n () () n, [yn y n y n [yn y n [y () () n y n 7 olvng tese equtons, we see tt te coeffcents n-,j ;,, re unquely determned Hence te proof of Teorem s completed Te error bound of te splne functon () wc s soluton of te problem () s obtned for te unform prtton I by te followng teorem: olvng te frst tree equtons, we obtn te followng: () (),, [y y [y y [y () () y 7 () () () () (), [y y [y y () () (), [y y () 7 ubsttutng te vlues of,,,, n te fourt equton, we obtn te followng relton between,,, were () for,,,n-: () (),, [y y [y y [y () () y () Teorem : Let y C [, () be unque splne functon of degree s wc soluton of te problem () Ten for [, ;,,, n-: () y () 7 r W () for r,,,,,n r W (), for r,,,,n W (), for r ( ) W (), for r,,,n W () for r ( ) W () for r,,,n were, W () denotes te modules of contnuty of y (), W () m W () ; defned by { }

4 J Mt & tt, ():-9, 9 To prove ts teorem we need te followng lemm: Lemm : Let,,, n- Were: y C [, Ten, e W () for, e,, y (7) W () denotes te modules of contnuty of y () Proof of lemm : If y C [, ten usng Tylor s epnson formul, we ve: y() y( y ( y ( () y ( θ 7 were, < θ smlr epressons for te dervtves of y() cn be used Now from Eq usng (7) we obtn: () () e, e, y ( θ, ) y ( θ, () () y ( θ, y ( θ, 7 (8) were < θ s, for,,, n-; s,,, : () () e, y ( θ, y ( θ, () () y ( θ, y ( θ, 7 (9) were, < θ,, θ,, θ,, θ, We see tt te system of Eq 8 9 s te unknowns e,,,,,n- s te unque soluton: e m m ( ) m, () () y ( θ, ) y ( θ, m e () () θ, θ, y ( ) y ( ) 7 w () Hence: W (), Wc completes te proof of te Lemm Proof of Teorem : Let [, were,,, n- We ve from Eq by pplyng Tylor s epnson formul we ve: () () 7 k, ( Usng ( (), we ve: () 7 () () k, y () () y () w () () () y () y () From Eq we ve: () y 7 () (), from wc we obtn: () y () y y () 7 () () () () (), From Eq we get: () y () W () () () by usng Eq usng Tylor seres epnson on y () () () y bout () () From (), we ve ( y (, from wc we obtn: Were: () () () m y ( θ, ) y ( θ, y ( θ, 7 y () ( θ, ) It s cler tt: () () () () To fnd equton: ( ) () y () (t) y (t) dt w ()dt w () () y (), we need te followng () ()

5 J Mt & tt, ():-9, 9 y () y ( ) )y ( ) () () () () () y ( y ( () () () () () y () y, y () y ( β ) y ( ) )y ( ) () () () () () y ( y ( β) () () (), y ( y ( β) y ( β) () were, < β, β From ( usng Tylor seres epnson, we get: () () (), y ( y ( α) y ( α) W () () were, < α, α From () usng () we get tt: () () () y () W () () () () By (), ( ) y ( ), from wc we obtn: Ten: () y () y ( ) y ( ),, 9 (), y ( ) () 7, y ( () Usng Eq, pply Tylor s epnson formul, we cn sow tt:, y ( W () () () 9, y ( W () (7) From (), usng (), (7) Lemm we cn get: () y () ( ) W () Crryng on smlr rguments we esly fnd tt: ( ) () y () (t) y (t) dt () () () () (t) y (t) dt () () W ()dt W () To fnd () y () we need te followng equtons: () () y () y ( )y ( y ( () () y ( y ( y () ( α ) from Eq tt: ) () y ( ) ( ) () y (, (),,, 7 () y() ( ) W () Ts proves Teorem for [,,,,, n For [,, we ve from (): () y () 7 y () () () (), () y () W () () () y () y () w () () () Crryng on smlr steps s for te cse [,,,,,n-, we fnd te followng nequltes: W () () y () W (), () () () y () W () () y () () () But for te frst dervtve of (), we ve te followng nequlty:

6 J Mt & tt, ():-9, 9 () y () y ( ), () (), y ( 7, y ( W () W () W () W () 7 Also for (), we get: () y(), y ( () (), y ( 7, y ( 7 W () W () W () W () 7 7 Ts proves Teorem for [, Hence, te proof of Teorem s completed REULT AND DICUION We present numercl results to demonstrte te convergence of te splne (,, ) functon of degree s wc constructed before to te second order ntl vlue problem Problem : we consder tt te second order ntl vlue problem y (y y) were [, y( y`( wt te ect soluton y() e [ Problem : we consder tt te second order ntl vlue problem y -y were [, y( y`( From Eq t s esy to verfy tt: ( ) y y y ( ),, y (), [y y y [y y y () () () () y [y y 7 () () [y y () y y () () [y y () () [y y y () 7 Also t s esy from Eq to verfy tt: ( ) y for,,, n-: (), [y y () () () () [y y [y y () () () () () () [y y [y y () () () () [y y From () we ve: ( ) y ( ) y () () () () From, wt usng te vlues of,j,,,, n- j,, gven n te Eq 9-, we get: () (),, [y y [y y [y () () y It turns out tt te s degree splne wc presented n ts study, yeld ppromte soluton tt s O( ) s stted n Teorem Te results re sown n te Tble for dfferent step szes Tble : An bsolute mmum error for () t s dervtve s for problem s() y() s () y () s () y () s () y () () () s () y () () () s () y () 8 () () s () y ()

7 J Mt & tt, ():-9, 9 Tble : An bsolute mmum error for () t s dervtve s for problem s() y() s () y () s () y () s () y () () () s () y () () () s () y () () () s () y () CONCLUION In ts study we tret for frst tme lcunry dt (,,) by constructng splne functon of degree s wc nterpoltes te lcunry dt (,,) te constructed splne functon ppled to solve te second order ntl vlue problems Numercl emples, sowed tt te presented splne functon proved ter effectveness n solvng te second order ntl vlue problems Also, we note tt, te better error bounds re obtned for smll step sze REFERENCE Amed, M, M Eml, T Fwzy H Elmosel, 99 Defcent splne functon ppromton to fourt order dfferentl equtons Appled Mt Modell, 8: 8- DOI: /7-9X(9)99- Gyovr, J, 98 Cucy problem modfed lcunry splne functons Construct Teor Funct, 8: 9-9 Howell, G AK Vrm, 989 Best error bounds for qurtc splne nterpolton Appromt Teor, 8: 8-7 ttp://portlcmorg/cttoncfm?d99 &coll&dlacm Knt, A, VR V Bttcry, Cubc splne for clss on non-lner sngulr boundry vlue problems rsng n pysology Appled Mt Comput, 7: ttp://ctnstfr/?modeleffcen&cpsdt 777 Kn, A T Azz, Te numercl soluton of trd-order boundry vlue problems usng quntc splne Appled Mt Comput, 7: - ttp://portlcmorg/cttoncfm?d 8 llm, MA Hussen, 98 Defcent splne functon ppromton to second-order dfferentl equtons Appled Mt Modell, : 8- ttp://ctnstfr/?modeleffcen &cpsdt99 7 en, A, 987 oluton of cucy's problem by defcent lcunry splne nterpoltons tud Bbes-Boly Mt, : -7 8 en, A E Venturno, 99 olvng twopont boundry vlue problems by mens of defcent qurtc splnes Appled Mt Comput, : - ttp://portlcmorg/cttoncfm?d 8 9 ddq, G Akrm, Quntc splne solutons of fourt order boundry vlue problem, rxv Mt Ntl, : - ttp://rvorg/p_cce/mt/pdf//7v pdf ddq, G Akrm, 7 olutons of tent-order boundry vlue problems usng eleven degree splnes Appled Mt Comput,8: -7 ttp://ctnstfr/?modeleffcen&cpsdt 88 ddq,, G Akrm Nzeem, 7 Quntc splne soluton of lner st-order boundry vlue problems; Appled Mt Comput, 89: ttp://ctnstfr/?modeleffcen&cpsdt ddq,, G Akrm A El, 7 Quntc splne soluton of lner fft-order boundry vlue problems Appled Mt Comput, 89: DOI: /jmc78 9