Research Article Cubic B-spline for the Numerical Solution of Parabolic Integro-differential Equation with a Weakly Singular Kernel

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1 Researc Jornal of Appled Scences, Engneerng and Tecnology 7(): 65-7, 4 DOI:.96/afs.7.5 ISS: ; e-iss: Mawell Scenfc Pblcaon Corp. Sbmed: Jne 8, Acceped: Jly 9, Pblsed: Marc 5, 4 Researc Arcle Cbc B-splne for e mercal Solon of Parabolc Inegro-dfferenal Eqaon w a Weakly Snglar Kernel Sad S. Sddq and Sama Arsed Deparmen of Maemacs, Unversy of e Pnab, Laore 5459, Paksan Absrac: Te am of sdy s o solve parabolc negro-dfferenal eqaon w a weakly snglar kernel. Problems nvolvng paral negro-dfferenal eqaons arse n fld dynamcs, vscoelascy, engneerng, maemacal bology, fnancal maemacs and oer areas. Many maemacal formlaons of pyscal penomena conan negro-dfferenal eqaons. Inegro-dfferenal eqaons are sally dffcl o solve analycally so, s reqred o oban an effcen appromae solon. A nmercal meod s developed o solve e paral negro-dfferenal eqaon sng e cbc B-splne collocaon meod. Te meod s based on dscrezng e me dervave sng fne cenral dfference formla and e cbc B-splne collocaon meod for e spaal dervave. Tree eamples are consdered o llsrae e effcency of e meod developed. I s o be observed a e nmercal resls obaned by e proposed meod effcenly appromae e eac solons. Keywords: Cenral dfferences, collocaon meod, cbc B-splne, negro-dfferenal eqaon, weakly snglar kernel ITRODUCTIO Consder e followng paral negro-dfferenal eqaon w a weakly snglar kernel: β ( s) (, s) ds (, ) = f (, ), [ a, b], > Sbec o e nal condon: () (,) = g( ), () and approprae bondary condons: or ( a, ) = f ( ), ( b, ) = f ( ), Drcle condons ( a, ) = r ( ), ( b, ) = r ( ), emann condons () e kernel: α β ( ) =, α ) < α < s a snglar kernel a = and Γdenoes e gamma fncon, g (), f (), f (), r (), r () are known fncons, f (, ) s a gven smoo fncon and e fncon (, ) s nknown. Te negro-dfferenal Eq. () along w e consrans () and () occrs n applcaons sc as ea condcon n maeral w memory (Grn and Ppkn, 968; Mller, 978), compresson of porovscoelasc meda, poplaon dynamcs, nclear reacor dynamcs ec. I can be seen a n Eq. (), e kernel fncon as a weak snglary a e orgn (Tang, 99). Ts s parclar neresng n vscoelascy, becase mg smoo e solon wen e bondary daa s dsconnos (Renardy, 989). Solon of negro-paral dfferenal eqaons as recenly araced mc aenon of researc. Cen e al. (99) sed fne elemen meod for e nmercal solon of a parabolc negro-dfferenal eqaon w a weakly snglar kernel. In Farweaer (994), splne collocaon meods ave been appled o oban e nmercal solon for a class of yperbolc paral negro-dfferenal eqaons. Hang (994) sed me dscrezaon sceme for solvng negrodfferenal eqaons of parabolc ype. X (99a, b and c) sed fne elemen meod o solve parabolc paral negro-dfferenal eqaon. Wlan and X () sed fne cenral dfference/fne elemen appromaons for e nmercal solon of paral negro-dfferenal eqaons. Solman e al. () sed for order fne dfference and collocaon meod for e nmercal solon of paral negrodfferenal eqaon. Correspondng Aor: Sad S. Sddq, Deparmen of Maemacs, Unversy of e Pnab, Laore 5459, Paksan Ts work s lcensed nder a Creave Commons Arbon 4. Inernaonal Lcense (URL: p://creavecommons.org/lcenses/by/4./). 65

2 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 In s sdy, e appromae solon of parabolc negro-dfferenal eqaon w weakly snglar kernel s proposed sng cbc B-splne collocaon meod. Te collocaon meod w B-splne bass fncons represens an economcal alernave, snce only reqres e evalaon of e nknown parameers a e grd pons. Haang e al. () sed qnc B- splne collocaon meod for solvng for order paral negro-dfferenal eqaon w a weakly snglar kernel. TEMPORAL DISCRETIZATIO Consder a nform mes Δw e grd pons λ o dscreze e regon Ω = [ a, b] [, T]. Eac λ s e verces of e grd pon (, ) were = a =,,,..., and = k, =,,,..., M, Mk = T. Te qanes and k are e mes szes n e space and me drecons, respecvely. A fne dfference appromaon s sed o dscreze e me dervave nvolved n Eq. () a me pon = as: α α ( s) (, ) (, ( ) (, ) s) s ds = α) s k α) r r α ( s) (, r ) (, r ) ds α) k (, ) (, ) = ds α α). k ( s) r r r (, ) (, ) [( ) α α = α ] α α). k ds (, ) (, ) k. α) ds α ( s) (, r ) (, r) [( r ) α α r α ] α α). k r (, ) (, ) = b Γ ( α ). Γ ( ) (, ) (, ) r r b α r α k α k (4) α α b r = ( r ) r, r =,,,...,. Te dscree dfferenal operaor L α can be defned as: L α (, b α (, ) = ) (, ) (, r ) (, r) α br α ). k α ) k leads o e followng dfference sceme o Eq. (): L α (, ) (, ) f (, ) I can frer be wren as: b (, ) (, ) (, ) (, ) r r (, ) (, ) b r f α α α ) k α ) k Te above eqaon can be rewren as: b ( ) α ) k α ( ) = (, ), br = b ( ) b ( r ( ) r b ( ( ) ( )) α ) k = ( r ) r α α α ( )) f (5) ( ) (6) r,,,,...,. α oe a b = and le a = α ) k, en e rg and sde of Eq. (6) can be reformlaed as: () - a = -b () ( ) -r () - b () (b - b ) () a f (), (7) w e bondary condons: ( a) = f ( ), ( b ) = f ( ) (8) In eac me level, ere s an ordnary dfferenal eqaon n e form of Eq. (7) w e bondary condons Eq. (8), wc s solved by cbc B-splne collocaon meod. Te proposed sceme Eq. (7) s a ree level sceme. In order o apply e proposed sceme, s necessary o ave e vales of a e nodal pons a e zero ( ) and frs ( ) level mes. To compe sbse = (e specal case), n Eq. (5), can be wren as: ( ) a = ( ) a f ( ) (9) Te Eq. (4) can be rewren as: ( s) α ) α ) (, s s ds L α (, ) = (, ) = g ( ) s e vale of a e zero level me (e nal condon). CUBIC B-SPLIE COLLOCATIO METHOD Sbsng L α, ) as an appromaon of: ( α ( s) ) α ) (, s s ds = { a= < < <... < = b} be e Le paron of [a, b]. Le B be B-splne bass fncons w knos a e pons, =,,,. Ts, an appromaon U () o e eac solon U () 66

3 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Table : Coeffcen of cbc B-splne and s dervaves a knos - - else B () 4 () B () / -/ () B () 6/ -/ 6/ A me level, can be epressed n erms of e cbc B-splne bass fncons B () as: = U c B = ( ) ( ) ( ) () c are nknown me dependen qanes o be deermned from e bondary condons and collocaon form of e negro-dfferenal eqaon. Te cbc B-splne B (), =,,..., can be defned as nder: ( ), [, ], ( ) ( ) ( ), [, ], B ( ) = ( ) ( ) ( ), [, ], ( ), [, ],, oerwse Te vales of sccessve dervaves ( r B ) ( ), =,..., ; r =,, a nodes, are lsed n Table. Le, U () sasfes e bondary condons: U ( a) = f ( ), U ( b ) = f ( ) and e collocaon eqaons: U U ( ) a b U ( ) ( b b ) U ( ) r = r r b U ( ) ( b b ) U ( ) a f ( ),, =,,,..., Te above eqaon can be rewren, by omng e dependence of U () on as: Smplfyng e above relaon leads o e followng sysem of () lnear eqaons n () nknowns c, c, c,..., c, c. 6 6 a c 4 a c a c = F,, =,,,..., r r r = r r F b ( c 4 c c ) ( b b )( c 4 c c ) b ( c 4 c c ) ( b b )( c 4 c c ) a f () To oban e nqe solon of e sysem (), wo addonal consrans are reqred. Tese consrans are obaned from e bondary condons. Imposon of e bondary condons enables s o elmnae e parameers c - and c from e sysem (). Frs e Drcle bondary condons are sed n order o elmnae c - and c, as: ( a, ) = ( c 4 c c ) = f ( ) ( b, ) = ( c 4 c c ) = f ( ) c = 4 c c f ( ) c = 4 c c f ( ) Afer elmnang c - and c, e sysem () s redced o a r-dagonal sysem of () lnear eqaons n () nknowns. Ts sysem can be rewren n mar form as: A C =F, =,,,... C = [ c, c,..., c ], =,,,... Te coeffcen mar A s gven as nder: (4) U U a b U b b U r = ( r r ) b U ( b b ) U a f, =,,,..., () Sbsng Eq. () no Eq. (), can be wren as: 6 ( c 4 c c ) a ( c c c ) r r r r r = b ( c 4 c c ) ( b b )( c 4 c c ) b ( c 4 c c ) ( b b )( c 4 c c ) a f () 67 6 a α β α α β α A = O O O α β α 6 a α a 6, β = 4 a =

4 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Te emann bondary condons can also be appled n order o elmnae c - and c, as: Te vale of C s obaned, solvng Eq. (9) sng cbc B-splne collocaon meod, as: ( a, ) = ( c c) = r ( ) ( b, ) = ( c c ) = r ( ) c = c r ( ) c = c r ( ) Afer elmnang c - and c, e sysem () s redced o a r-dagonal sysem of () lnear eqaons n () nknowns. Ts sysem can be rewren n mar form as: A C =F, =,,,... C = [ c, c,..., c ], =,,,... Te coeffcen mar A s gven as nder: (5) 4 a a γ δ γ γ δ γ A= O O O γ δ γ a 4 a 6 γ = a, δ = 4 a 6 a c 4 a c a c = F, =,,,..., (7) F = ( c 4 c c ) a f Te above Eq. (7) s a sysem of () lnear eqaons n () nknownsc, c, c,..., c, c. To oban e nqe solon of s sysem, elmnae c - and c, sng Drcle and emann bondary condons. Te me evolon of e appromae solon U s deermned by e me evolon of e vecor C. Ts s fond by repeaedly solvng e recrrence relaonsp, once e nal vecor C = [ c, c,..., c ] T, as been comped from e nal condon. Te recrrence relaonsp s r-dagonal and so can be solved sng Tomas algorm. Te nal sae vecor: Te nal sae vecor C can be deermned from e nal condon (,) = ( ) = g ( ) wc gves () eqaons n () nknowns. For deermnng ese nknowns e followng relaons a e knos are sed: U ( a,) = (,), (,) = ( ), =,,,..., U ( b,) = (, ) U wc gves a r-dagonal sysem of eqaons n e followng mar: G C =E Usng e sysem (), for =, followng s e sysem of () lnear eqaons n () nknowns c, c, c,..., c, c. 6 6 a c 4 a c a c = F, =,,,..., (6) F = b ( c 4 c c ) ( b b )( c 4 c c ) a f In order o fnd e vale of C = [ c, c,..., c ] T, s frs needed o fnd e vale of C = [ c, c,..., c ] T G = 4 O O O 4 4 UMERICAL RESULTS Te proposed meod s esed on e followng ree problems. Le:

5 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Table : Te errors e M and e M wen = 6 and k =. for eample M e M e M 4.48 E E E-5.77 E-6.7 E-5.9 E E E E-6.76 E-7 Table : Te errors e M and e M wen = 6 and k =. for eample M e M e M E E E E E E E E E E-5 Table 4: Te errors e M and e M wen M = and k =. for eample e M e M.76 E E E E-5.5 E-4.74 E E E E E-6 Table 5: Te errors e M and e M wen = 6 and k =. for eample M e M e M 9.4 E E-6 9. E E E E E E E E-6 Table 6: Te errors e M and e M wen = 6 and k =. for eample M e M e M 9.94 E E E E E E E E E E-5 Table 7: Te errors e M and e M wen M = and k =. for eample e M e M E E-4.68 E E E E-6 4. E-6.6 E-6 = k, =,,,..., M, =, were M denoes e fnal me level M and s e nmber of nodes. In order o ceck e accracy of e proposed meod, e mamm norm errors and L norm errors beween nmercal and eac solon are gven w e followng defnons: Eample : Followng s e second order parabolc negro-dfferenal eqaon: ( s) α) α (, s) ds (, ) = f (, ), [,], >, α=.5 w e nal condon: (,) = sn π, [,] and bondary condons: (, ) = = (, ), Te eac solon of e problem s: (, ) = ( )snπ (8) Te nmercal solons a = 6, k =. and k =., w dfferen me levels M, are presened n Table and respecvely. Te nmercal solons a M = and k =. for dfferen vales of are ablaed n Table 4. In Table o 4, e me ncremen k, e space ncremen = and me level M are vared o es e accracy of e proposed meod, wc ndcaes a e proposed meod s sbsanally effcen. In order o ndcae e effec of e proposed meod for larger M, e eac solon and e nmercal solon are ploed sng =, M = 5 and k =. as sown n Fg.. Wen =, k =. and M = e eac solon and e nmercal solon a e M me level are sown n Fg.. I can be observed from e Table o 4 and Fg. and, a e proposed meod appromaes e eac solon very effcenly. Eample : Followng s e parabolc negrodfferenal eqaon: ( s) α) α (, s) ds (, ) = f (, ), [,], >, α=.5 w e nal condon: (, ) = cos π, [,] and Drcle bondary condons: (9) Mamm norm error: e = ma (, ) U M M M L e U M norm error: M = (, ) M = (, ) = ( ), (, ) = ( )cos( π ), Te eac solon of e problem s: 69

6 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Fg. : Te resls a M = 5 for eample.5 Te M- eac solon Te M- nmercal solon Fg. : Te eac and nmercal solons a M = (, ) = ( ) cosπ Te nmercal solons a = 6, k =. and k =., w dfferen me levels M, are presened n Table 5 and 6 respecvely. Te nmercal solons a M = and k =. for dfferen vales of are ablaed n Table 7. In Table 5 o 7, e me ncremen k, e space ncremen = and me level M are vared o es e accracy of e proposed meod, wc ndcaes a e proposed meod s sbsanally effcen. In order o ndcae e effec of e proposed meod for larger M, e eac solon and e 7 nmercal solon are ploed sng =, M = 5 and k =. as sown n Fg.. Wen =, k =. and M = e eac solon and e nmercal solon a e M me level are sown n Fg. 4. I can be observed from e Table 5 o 7 and Fg. and 4, a e proposed meod appromaes e eac solon very effcenly. Eample : Followng s e parabolc negrodfferenal eqaon: ( s) α) α (, s) ds (, ) = f (, ), [,], >, α=.5 ()

7 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Fg. : Te resls a M = 5 for eample. Te M- eac solon Te M- nmercal solon Fg. 4: Te eac and nmercal solons a M = w e nal condon: (,) = sn π, [,] and emann bondary condons: Table 8: Mamm norm errors e M for = 4 for eample M k =. e M k =.5 e M E-4.8 E- 4. E E E-4.6 E E-.988 E- 5.4 E-.875 E- (, ) = π( ) cos π, π π (, ) = ( ) cos( ), Te eac solon of e problem s: π (, ) = ( ) sn Te nmercal solons a = 4, k =. and k =.5, w dfferen me levels M, are presened n Table 8 and 9 respecvely. In Table 8 and 9, e me 7 Table 9: L norm errors e M for = 4 for eample M k =. e M k =.5 em E-5.57 E E E-5.7 E E E E E-4.44 E-4 ncremen k and me level M are vared o es e accracy of e proposed meod, wc ndcaes a e proposed meod s sbsanally effcen. In order o ndcae e effec of e proposed meod for larger M, e eac solon and e

8 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Fg. 5: Te resls a M = 5 for eample Te M- eac solon Te M- nmercal solon Fg. 6: Te eac and nmercal solons a M = nmercal solon are ploed sng =, M = 5 and k =. as sown n Fg. 5. Wen =, k =. and M = e eac solon and e nmercal solon a e M me level are sown n Fg. 6. I can be observed from e Table 8 and 9 and Fg. 5 and 6, a e proposed meod appromaes e eac solon very effcenly. COCLUSIO Te nmercal solon of parabolc negrodfferenal eqaon w a weakly snglar kernel s sded sng cbc B-splne collocaon meod. Te parabolc negro-dfferenal eqaon s dscrezed by e fne cenral dfference formla n e me drecon and e cbc B-splne collocaon meod for spaal dervave. Te parameers, k and M are vared n order o es e accracy of e proposed meod. I s observed from e nmercal epermens, a e proposed meod possesses g degree of effcency and accracy. Moreover, e nmercal resls are n 7 good agreemen w e eac solons. Te nmercal solons of non-lnear parabolc negro-dfferenal eqaons are n progress. REFERECES Cen, C., V. Tomée and L.B. Walbn, 99. Fne elemen appromaon of a parabolc negrodfferenal eqaon w a weakly snglar kernel. Ma Comp., 58: Farweaer, G., 994. Splne collocaon meods for a class of yperbolc paral negro-dfferenal eqaons. SIAM J. mer. Anal., : Grn, M.E. and A.C. Ppkn, 968. A general eory of ea condcon w fne wave speed. Arc. Raon. Mec. An., : -6. Haang, Z., H. Xl and Y. Xea,. Qnc B- splne collocaon meod for for order paral negro-dfferenal eqaons w a weakly snglar kernel. Appl. Ma. Comp., 9:

9 Res. J. Appl. Sc. Eng. Tecnol., 7(): 65-7, 4 Hang, Y.Q., 994. Tme dscrezaon sceme for an negro-dfferenal eqaon of parabolc ype. J. Comp. Ma., : Mller, R.K., 978. An negro-dfferenal eqaon for rgd ea condcors w memory. J. Ma. Anal. Appl., 66: -. Renardy, M., 989. Maemacal analyss of vscoelasc flows. Ann. Rev. Fld Mec., : -6. Solman, A.F., A.M.A. EL-Asyed and M.S. El-Azab,. On e nmercal solon of paral negrodfferenal eqaons. Ma. Sc. Le., : 7-8. Tang, T., 99. A fne dfference sceme for paral negro-dfferenal eqaons w a weakly snglar kernel. Appl. mer. Ma., : 9-9. Wlan, L. and D. X,. Fne cenral dfference/fne elemen appromaons for parabolc negro-dfferenal eqaons. Compng, 9: 89-. X, D., 99a. On e dscrezaon n me for a parabolc negro-dfferenal eqaon w a weakly snglar kernel I: Smoo nal daa. Appl. Ma. Comp., 58: -7. X, D., 99b. On e dscrezaon n me for a parabolc negrodfferenal eqaon w a weakly snglar kernel II: onsmoo nal daa. Appl. Ma. Comp., 58: 9-6. X, D., 99c. Fne elemen meods for e nonlnear negro-dfferenal eqaons. Appl. Ma. Comp., 58: