Journal of Engineering and Applied Sciences. Ultraspherical Integration Method for Solving Beam Bending Boundary Value Problem

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1 Journal of Engneerng and Appled Senes Volue: Edton: Year: 4 Pages: 7 4 Ultraspheral Integraton Method for Solvng Bea Bendng Boundary Value Proble M El-Kady Matheats Departent Faulty of Sene Helwan UnverstyEgypt a_el_ady@yahooo MAAtta * Matheats Departent College Of Sene Al-Zulf Maaah Unversty Saud Araba aatta@uedusa F Ragab Matheats Departent Faulty of Sene Aswan Unversty Egypt u_pants@yahooo Abstrat In ths paper we appled suessve ultraspheral ntegraton atres whh are used for the nueral soluton of fourth order lnear boundary value proble arsng n bendng of a retangular bea on elast foundaton Ths ethod s used to approate for the hghest-order dervatve and generatng approatons to the lower-order dervatves through ntegraton of the hghest-order dervatve We an then use the produed equatons n the for of algebra syste and hene t s onverted to nonlnear prograng ueral eaples llustrated the auray and effeny of the proposed ethod Keywords: : Ultraspheral polynoals; ntegraton atres;bea bendng boundary value proble Artle hstory: Reeved 5 Jan 4 Aepted Feb 4 Introduton Fourth order lnear BVP has been solved by varous nueral ethods Shahd S Sddq et al [9] have used non polynoal splne to solve fourthorder boundary value probles Sra-ul-Isla et al developed a tehnque based on quart nonpolynoal splne funtons for approatons to the soluton of a syste of fourth-order Raz A Usan developed the ethod of the soluton for fourth-order boundary value proble onsderng t to be the proble of bendng a retangular laped bea of length restng on an elast foundaton The a of ths paper s to present ultraspheral spetral ntegraton atres depends on usng ultraspheral polynoals for solvng bea bendng boundary value proble The organzaton of ths paper s as follows: In seton we ntrodue ultraspheral polynoals and soe of ts propertes In seton we presented the proedure of the steps of ultraspheral spetral ntegraton ethod In seton 4 we ntrodue a Desrpton of the used ethod In seton 5 we present nueral results deonstratng the auray of our ethods for soe eaple of bea bendng boundary value proble Seton 6 ontans the onluson of ths paper Ultraspheral Polynoals and Soe Propertes: The ultraspheral or Gegenbauer polynoals wth the real paraeter ( > / ) are a sequene of polynoals = In the fnte doan [ ] ( C ) eah * Correspondng author

2 8 M El-Kady et al/ Journal of Engneerng and Appled Senes () 7 4 degree satsfes the orthogonalty relaton respetvely as follows: ( ) C C d = ( ) ψ () = ( ) Here the noralzaton onstant ψ s defned as Szegö: Γ ( + ) ψ = π () ( + ) Γ Γ ( + ) { } The polynoals ay be generated by the Rodrgue s forula gven as Bell: [ ] ( ) r Γ( r+ ) r = ( ) r= Γ( r!)( r)! C () where [ /] fraton refers to the nteger part of the The general epressons for ultraspheral polynoals an be put n the followng way: Gr where [ ] ( ) r r r= C = G (4) r r Γ( r+ ) G = ( ) (5) r Γ( r!)( r)! A relaton between the oeffents G G s gven by r+ r + ( r )( r ) = Gr 4( r+ )( + r ) In partular we have the speal values G ( ) = G = Γ ( + ) Γ Γ ( + ) and (6) The fundaental reurrent forulae for ultraspheral polynoals are defned as + ( + ) C = ( + ) C ( ) - ( + ) C (7) wth the frst two beng: ( λ ) C = λ Theore (): ( λ ) C = The -th ntegral of the ultraspheral polynoals (4) s epressed n ters of ultraspheral polynoals as follows: t t ( I C ) C ( t ) dt dt dt = where [ ] r = l = γ = G ( ) r+ r r l l( + ) ( l+ ) + ( ) Ε ( l )! γ ( ) r = ; ( ) l= r+ l [ ] ( ) r+ ( ) γr Gr r= Ε = Proof: see M A Ibrah (8) We used an approaton of any ontnuous funton f and an approaton of ther ntegrals by nterpolatng the funton wth ultraspheral polynoals at two sets of nodes: The set of equally spaed ponts: S = { = + = } The set of zeros ponts of the ultraspheral polynoals: { ( : ) + ( ) } S = C = = Theore (): If f ( ) s the ultraspheral approaton to a funton f( ) n fnte epanson e = ( ) f ( ) = a C ( ) (9)

3 M El-Kady et al/ Journal of Engneerng and Appled Senes () then ( ) = ( ψ ) a ( ) C f d() Proof: See El-Hawary et al Ultraspheral Integraton Matres Many authors presented spetral ntegraton atres proven suessful n the nueral approaton of any types of dfferental equatons suh as Elbarbary presented spetral suessve ntegraton atr where t an be used to onstrut a Chebyshev epanson ethod for the soluton of boundary value probles We approate the ntegral of a funton f( ) by nterpolatng the funton wth ultraspheral polynoals at the ponts S and S Theore (): If f s approated by ultraspheral polynoals then the -th ntegral of f() t s approated by ultraspheral epanson n the for: t t ( ) f ( t ) dt dt dt = q f () = where the entres of -th ultraspheral ntegraton ( ) atres q ( ) = are gven as follows: S Case : q [ ] ( ) r r ( ) = ψ ( ) = r= l ( ) r+ + = l= l C ( ψ ) Case : θ γ G θ ( ) ( + ) ( + l ) C Ε S! l ( ψ ) [ ] = γr r ψ ϖ = r= q G C ( ) l l r+ ( l+ ) + Ε = l= ( l )! ϖ C Proof: See M A Ibrah 4 Desrpton of the ethod ( ) ( + ) We onsder a general fourth order boundary value proble gven by y + f h( y) = g [ a b] (4) Subet to the boundary ondtons: ya = ν yb = β y ( a) = ν y ( b) = β () () Where ν ν β and β are fnte real onstants and f h( y ) and g( ) are ontnuous ab funtons on the nterval[ ] By applyng ultraspheral ntegraton ethod () the hghest dervatve of y() an be wrtten as y =Φ (4) The low-order dervatves y( ) = = are generated through ntegraton of equaton as follows y = Φ d + a (4) () y ( ) = Φ dd+ ( a) + a a y () (44) ( ) = Φ ddd a a a a ( a ) (45)

4 4 M El-Kady et al/ Journal of Engneerng and Appled Senes () 7 4 y( ) = Φ dddd+ ( a) a a a a 6 + a + ( a ) + 4 (46) The suessve ntegraton of equatons (4) to (46) s approated by ultraspheral ntegraton ethod as follows: () ( ) = Φ ( ) + = y q () () = Φ + + = y ( ) q ( ) ( a) y () = q Φ + ( a) = ( + ( a) + ) = Φ ( ) + ( ) = y q a 6 ( a ) ( a ) Then we an deterne the approaton soluton y by deternng the oeffents = 4 fro the boundary ondtons of the proble of fourth order boundary value proble By Substtutng the approaton soluton n equaton of the proble of fourth boundary value proble (4) and then we obtan the unonstraned optzaton proble whh an be wrtten as Mnze Where F = F ( Φ ) (47) [ ] Φ =Φ Φ Φ The unonstraned optzaton proble (47) an be solved usng partal quadrat nterpolaton ethod (El-Gndy) 5 ueral eaples In ths seton we wll use ultraspheral ntegraton ethod to get an approate soluton n solvng the bea bendng boundary value probles the laped-laped bea whh belongs to the general lass of the boundary value probles n the for: y + f p( y) = g [] () () y() = y() = y () = y () = Eaple 5 Consder the followng boundary value proble whh desrbes the odel of the bendng of a thn bea laped at both ends: 4 y = e [] ( 4 49 ) Subet to the boundary ondton (5) () () y() = y() = y () = y () = (5) The analyt soluton of the above syste soluton s: y = ( ) e We apply ultraspheral ntegraton ethod () the hghest dervatve of y an be wrtten as y =Φ (5) The low-order dervatves y( ) = ; = are generated through ntegraton of equaton (5) as follows y = Φ d + (54) () y = Φ dd + + (55) () y = Φ ddd y = Φ( dddd ) The onstants = 4 (56) (57) an be deterned fro the boundary ondtons these onstants are found to be 4 = (58)

5 M El-Kady et al/ Journal of Engneerng and Appled Senes () = (59) = = q Φ( ) = = 6 q Φ( ) = 6 q Φ( ) + q Φ( ) = (5) (5) Substtutng fro equatons (58) to (5) n equaton (57) = Φ = y q + ( q Φ 6 = 6 q Φ( )) = = + ( 6 q Φ = + q Φ( )) (5) Then by substtutng y( ) n the equaton (5) and we an be wrtten as: Mnze Where F = F ( Φ ) (5) [ () t () t () t ] Φ = Φ Φ Φ It an be solved by usng partal quadrat nterpolaton ethod (El-Gndy) The followng table presents the au absolute error obtaned by usng ultraspheral ntegraton ethods at the ponts gven n S S and Fg 5 ad a oparatve between the approate soluton and the eat soluton for = In S Table (5): The au absolute error by ultraspheral ntegraton Methods at S S S MAE S MAE 4-94E E E-4-5 8E-4 8-6E E E-8 57E E-9 E E-9-49 E-9 Table (5): Coparson between eat soluton wth the result n approaton soluton to Ultraspheral ntegraton ethod for = n S eat soluton Ultraspheral ntegraton ethod error E E E E E E E E E appro soluton eat soluton Fg 5: the approate soluton and the eat soluton for = In S

6 4 M El-Kady et al/ Journal of Engneerng and Appled Senes () 7 4 Eaple 5 Consder the followng nonlnear fourth-order BVPs: + = y y + ( ) (54) [] Subet to the boundary ondtons: y() = y() = y () () () = y () = Ths has the analyt soluton gven by: y = ( ) We apply ultraspheral ntegraton ethod () the hghest dervatve of y an be wrtten as y =Φ (55) The low-order dervatves y( ) = = are generated through ntegraton of equaton (55) as follows y = Φ d + (56) () y = Φ dd + + (57) () y = Φ ddd y = Φ dddd The onstants = 4 (58) (59) an be deterned fro the boundary ondtons these onstants are found to be 4 = (5) = (5) = = q Φ( ) 6 q Φ( ) = = = q Φ( ) 6 q Φ( ) = (5) (5) Substtutng fro equatons (5) to (5) n equaton (59) we get = Φ = y q + ( q Φ 6 = 6 q Φ( )) = = + ( q Φ = 6 q Φ( )) (54) Then by substtutng y( ) n the equaton (54)) and we an be wrtten as: Mnze Where F = F ( Φ ) (55) [ () t () t () t ] Φ = Φ Φ Φ It solved by usng partal quadrat nterpolaton ethod (El-Gndy) The followng table presents the au absolute error obtaned by usng ultraspheral ntegraton ethod at the ponts gven n S S and Fg (5) ad a oparatve between the approate soluton and the eat soluton for = In S

7 M El-Kady et al/ Journal of Engneerng and Appled Senes () Table (5): The au absolute error by ultraspheral ntegraton ethods at S S S S MAE MAE 4 E- - 66E- 6-6E-8-7 E E-8 - E-8 4E E E-8 - E E-8-5E-8 Table (54): Coparson between eat soluton wth the result n approaton soluton to Ultraspheral ntegraton ethod for = ns eat soluton 6 Conluson Ultraspheral ntegraton ethod error E E E E E E E E E-9 In ths study ultraspheral ntegraton ethod has been appled to obtan the nueral solutons for solvng bea bendng boundary value proble at the set of equally spaed ponts or the set of zeros ponts The nueral results deonstrated the effeny and auray of the proposed shee of the ethod The nueral results obtaned by the proposed ethod are n a good agreeent wth the eat solutons avalable n the lterature By onsderng that the auray of our ethod depends on spefed value of the paraeter ultraspheral Fg 5: the approate soluton and the eat soluton for = In S Referenes appro soluton eat soluton Bell W W Speal funtons for sentsts and engneers D Van ostrand Copany (969) E M E Elbarbary Pseudospetral ntegraton atr and boundary value probles Intern J of Cop Math (7) El-Gndy T M ueral Studes n Optal Control Theory PhD Thess Unversty of Wales (977) El-Hawary H M Sal M S and Hussen H S Ultraspheral Integral Method for Optal Control Probles Governed by Ordnary Dfferental Equatons J of Optzaton 5 8 M A Ibrah Ultraspheral Approatons and Ther Applatons for Solvng Optal Control Probles PhD Thess Qena Unversty (9) RA Usan SA Wars Sooth splne solutons for boundary value probles n plate defleton theory Cop Math Appl 6 (98) 5 Raz A Usan Dsrete ethods for boundary value probles wth applatons n plate defleton theory ZAMP (979) Raz A Usan Dsrete varable ethods for a boundary value proble wth engneerng applatons Matheats of Coputaton (44) (978) S S Sddq Ghazala Ara Soluton of the syste of fourth order boundary value probles usng non-polynoal splne tehnque Appled Matheats and Coputaton 85 (7) 8-5 Sra-ul-Isla Ira A Trz Fazal Haq Shahruh K Taseer Faly of nueral ethods based on non-polynoal splnes for soluton of ontat probles Counaton n onlnear sene and ueral Sulaton (8) Szegö G Orthogonal Polynoals A Math So Colloq Pub (985)