Research & Reviews: Journal of Statistics and Mathematical Sciences

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1 Reerch & Review: Jourl of Sttitic d Mthemticl Sciece Error Etimtio of Homotopy Perturbtio Method for lier Itegrl d Itegro-Differetil Equtio of the Third id Ehuvtov Z.K.,3 *, Zulri F.S., Ni Log N.M.A.,3 d Mumiov Z. 4 Fculty of Sciece d Techology, Uiveriti Si Ilm Mlyi (USIM), Mlyi Deprtmet of Mthemtic, Fculty of Sciece, Uiveriti Putr Mlyi, Mlyi 3 Ititute for Mthemticl Reerch, Uiveriti Putr Mlyi, Mlyi 4 Mlyi-Jp Itertiol Ititute of Techology, Uiveriti Teologi Mlyi (UTM), KL, Mlyi Reerch Article Received dte: /3/6 Accepted dte: 4/4/6 Publihed dte: 8/4/6 *For Correpodece Ehuvtov ZK, Fculty of Sciece d Techology, Uiveriti Si Ilm Mlyi (USIM), Mlyi ABSTRACT I thi ote, cove Homotopy perturbtio method (HPM) i preeted for the pproimte olutio of the lier Fredholm-Volterr itegrl d itegro-differetil equtio. Covergece d rte of covergece of the HPM re proved for both equtio. Five umericl emple re provided to verify the vlidity d ccurcy of the propoed method. Emple revel tht HPM i very ccurte d imple to implemet for itegrl d itegrodifferetil equtio. E-mil: ziidi@uim.edu.my Keyword: Approimte method, Homotopy perturbtio method, Covergece, Error etimte INTRODUCTION Itegrl equtio occur turlly i my field of ciece d egieerig []. A computtiol pproch to olve itegrl equtio i eetil wor i cietific reerch. Fredholm d Volterr itegrl equtio i oe of the mot importt itegrl equtio. Some itertive method lie Homotopy perturbtio method (HPM) [-] d Adomi decompoitio method (ADM) [-4], Homotopy lyi method [5-] hve bee pplied to olve lier d olier Fredholm d Volterr itegrl equtio d itegro-differetil equtio. HPM h bee ued for wide rge of problem; for fidig the ect d pproimte olutio of olier ordiry differetil equtio(ode) [7], lier d olier itegrl equtio [8], the itegro-differetil equtio [9,] d the Volterr- Fredholm itegrl equtio []. Hetmio et l. [3] hve propoed method which i bed o the homotopy perturbtio method to olve Volterr Fredholm itegrl equtio of the ecod id. The problem of the covergece of the erie cotructed i formulted d proof of the formultio i give. Additiolly, the etimtio of the pproimte olutio i elborted by tig the prtil um of the erie. I 6, Zulri et l. [4] coider Fredholm-Volterro itegro-differetil equtio (FVIDE) of order of the third id d olved it by uig modified homotopy perturbtio method (HPM). It i foud tht MHPM i emilyticl method d ey to pply for FVIDE. Numericl emple re give to preet the efficiecy d relibility of the method.i thi ote, we coider two type of itegrl equtio: b u ( ) ( )= f ( ) + λ K ( tut, ) ( )d t + λ K ( tut, ) ( )d t, [ b, ], () RRJSMS Volume Iue Jue, 6 89

2 b u ( ) ( )= f ( ) + λ K ( tut, ) ( )d t + λ K ( tut, ) ( )d t, [ b, ]. () Eq. ()-() re clled Fredholm-Volterr itegrl equtio (FVIE) d Fredholm-Volterr itegro-differetil equtio (FVIDE) of the third id repectively. For both type of equtio the covergece of pproimte olutio re etblihed d four emple re provided to verify the ccurcy of the propoed method. The pper i orgized follow. I Sectio, we preet the pplictio of the HPM for the problem () d proof of the covergece. I Sectio 3, HPM i modified to olve the Eq. () d preet the rte of covergece of the pproimte olutio. Sectio 4 provide four emple: firt two emple relted to the Eq. () d other two more emple delt with Eq. (). Numericl reult how tht the propoed method re very ccurte d coverge ft. Cocluio i give i Sectio 5. DERIVATION AND ESTIMATION OF THE APPROXIMATE METHOD FOR SOLVING FVIES OF THE THIRD KIND To epli the homotopy perturbtio method (HPM), we coider the geerl itegrl equtio of the form Lu + Nu = f (3) where L i lier opertor d N i olier opertor. A poible remedy, we c defie homotopy Hvp (, ) by Hvp (, )=( pfv ) () + plv ( () + Nv () f ) (4) where Fv () i fuctiol opertor with ow olutio u, which c be obtied eily. It i cler tht, for Hvp (, )= (5) from which we hve Hv (,) = Fv ( ). Hv (,) = Lv+ Nv f (6) Thi how tht Hup (, ) cotiuouly trce implicitly defied curve from trtig poit Hv (,) to olutio Hu (,). The embeddig prmeter p mootoiclly icree from zero to uit the trivil problem Fv ( )= i cotiuouly deformed to the origil problem Lu ( ) + Nu ( ) f =. The embeddig prmeter p (,] c be coidered epdig prmeter []. To obti the pproimte olutio of Eq.(3) we erch olutio i the erie form v ( )= v( p ). (7) = Whe p, Eq.(5) correpod to Eq. (3) d become the pproimte olutio i.e., u ( ) = limv ( ) = v + v +... (8) p The erie (8) i coverget for mot ce, d the rte of covergece deped o Lu ( ) d Nu. ( ) Let u coider the lier Fredholm-Volterr itegrl equtio () where the erel K, K C([ b, ] [ b, ]) d the fuctio, f Cb [, ] re ow, where the fuctio u i to be determied. Defie the opertor Lu ( ) d Nu ( ) for Eq. () follow Lv ( )= v ( ) ( ) b λ Nv ( )= λ K ( tvt,) ()d t K ( tvt,) () dt, v Cb [, ] with the umptio tht ( ) for y [ b., ] Rewritig Eq. (4) i the form H ( v, p)= Lv u + p( u + Nv f ) (9) we olve the problem () follow. For p = the olutio of the opertor equtio Hv (, ) = i equivlet to the olutio of trivil problem v ( ) ( ) u ( )=. For p = the equtio Hv (,) = led to the olutio of Eq. (). By ubtitutig the erie olutio (7) ito the Eq. (5) d ccordig to Abel Theorem the olutio of Eq. () i obtied from the equtio + L pv = u pu ( f) N p v () = = Comprig the me power of prmeter p i both ide of Eq. (), led to the followig reltio v = u () RRJSMS Volume Iue Jue, 6 9

3 v = ( f u Nv) v = Nv =,3, Accordig to the cheme ()-(3) we defie prtil um follow u ( ) ( )=, ( )= vi( ) ( ) i= Theorem thm Let K, K C([ b, ] [ b, ]) d, f Cb [, ] where ( ) for y [ b,, ] be cotiuou fuctio. I dditio, if the followig iequlity α = cb ( )( λ K + λ K ) < (5) where c = i tified d iitil gue u i choe fuctio cotiuou o the itervl [ b, ], the erie olutio (7) i uiformly coverget to the ect olutio u o the itervl [ b, ] for ech p = [,]. Proof. Prove of the Theorem h ot much chge of the Hetmio et l [3]. [Theorem ]. Remr Hetmio et l. [3] [Remr ]. I the thm, the itervl [ b, ] c be replced by ( b, ),( b, ] or [ b, ), where the coditio of cotiuity of K i for i = {,} d f i the pproprite regio Ω d Ω mut be tregtheed by ddig the umptio of boudede of thee fuctio. Moreover, the coditio Ki C([ b, ] [ b, ]) c be replced by ome weer coditio, for emple by the Lebeque itegrbility of erel K o the et [ b, ] [ b, ] d by the iequlity [5] b i i K (, t) dt M ( b ) I the ce of difficultie or impoibility of fidig the um of erie (7) for p =, we my coider pproimte olutio by tig the prtil um of the erie (4). The firt + term of erie (4) i the limit p crete the o-clled th-order pproimte olutio i the form = i ( )= v ( ) (6) The olutio ( ) i (6) c be etimted o the bi of the followig theorem. Theorem i ot much differece give i Hetmio et l. [3] [Theorem ] Theorem The error of th-order pproimte olutio of (6) c be etimted by the iequlity E B α α where E := v ( ) ( ), while v B d α <. Proof. = = v( ) ( ) = v ( ) v ( ) = v ( ) v ( ) v = + = + = + = + Bα = B α α DERIVATION AND ESTIMATION OF THE APPROXIMATE METHOD FOR SOLVING FVIDE OF THE THIRD KIND Let u itroduce the pce of cotiuouly differetible fuctio = ( ) + ( ) C ([ b, ]) equipped with the orm u m u m u (7) b b i.e. u = u + u where i the tdrd orm i Cb [, ]. To pply cove HPM for FVIDE () we rewrite it i the form u ( b ) = ( ( ) + (, ) ( )d + (, ) ( )d ), ( ) = ( ) f λ K tut t λ K tut t u d (8) () (3) (4) RRJSMS Volume Iue Jue, 6 9

4 where u ( ) i the firt order derivtive of u ( ) with iitil coditio u ( )= d d itegrte both ide of Eq. (??) to yield b t u( ) = d + ( f( t) + λ (, ) ( )d + (, ) ( )d )d () K t τ u τ τ λ K t τ u τ τ t (9) t Write Eq. (9) i opertor form Lv = g+ Nv () where () ( )= ( ), ( )= + f t Lv v g d dt t () b t Nv( ) = ( λ (, ) ( )d + (, ) ( )d )d, [, ] () K t τ v τ τ λ K t τ v τ τ t v C b () t Uig the bove defiitio we obti the homotopy opertor for Eq.(??) ( ) H ( v, p)= Lv u + p u g N ( v ) () where p [,] i homotopy prmeter d u ( ) defie iitil gue for Eq.(??). I imilr wy, ubtitutig the erie olutio (7) ito equtio H ( v, p)= led + L pv = u + pg ( u) N p v (3) = = Comprig the epreio with the me power of prmeter p, we obti v = u (4) v= g u Nv (5) v = Nv, =,3, (6) Net, for the derivtive of v we defie f( ) Lv ( )= v ( ), g( )= ( ) Nv b ( ) = ( λ (, ) ( )d + (, ) ( )d ) ( ) K tvt t λ K tvt t (7) By uig the homotopy perturbtio method, the olutio of opertor equtio H ( v, p)= Lv u + pu ( + Nv g )= (8) i erched i the form of power erie v ( )= pv ( ) (9) = If the erie (9) poee rdiu of covergece ot greter th, the the erie i bolutely coverget. By puttig (9) ito the Eq. (8) d comprig the epreio with the me power of prmeter p led to the reltio v = u (3) v = g u Nv (3) v = Nv, =,3, (3) The covergece of the pproimte olutio (7) of Eq. () i give i the Theorem 3.. Theorem 3 Let K, K C([ b, ] [ b, ]),, f Cb [, ] be cotiuou fuctio o the repective domi. I dditio, if the iequlity β = cb ( ) ( λ K + λ K ) < (33) where c =, hold d the iitil vlue u i choe cotiuou fuctio, the the erie i Eq. (7) i uiformly coverget to the ect olutio u ( ) i the ee of the C orm o the itervl [ b, ] for ech p [,]. Proof. Sice the erel K, K d the fuctio f( ), ( ) with ( ), [ b,, ] re cotiuou i the repective cloed domi the RRJSMS Volume Iue Jue, 6 9

5 K( t, ) K, K( t, ) K, f( ) f, =,, [, ] ( ) c t b Let u be choe cotiuou fuctio d u = N The from the reltio (4)-(6) it follow tht v = u = N ( λ λ ) v d + u + c( b ) f + ( b ) v ( K + K ) d + N + c( b ) f + ( b ) N α := B For the geerl ce, from (6) we hve the followig etimtio: v Bb ( ) α Bβ (34) where β =( b ) α. From the erie (7), for p [,] it follow tht = = = v ( ) pv( ) v = v + v N + Bβ = B = N + (35) β Sice the the etimte erie (??) i the geometric erie with the commo rtio β d therefore it i coverget if commo rtio β <. It implie tht v ( ) i uiformly coverget o [ b, ] for ech p [,]. The umptio for reltio (3)-(3) implie v = u = N v u + c( f + ( b ) v ( λ K + λ K )) N + c f + ( b N ) α = B v c(( b ) v ( λ K + λ K ) v α. Subtitutig (??) ito the etimtio of v ( ) yield: v Bβ α, B =. ( b ) β For the erie i (9), we get = = = v ( ) pv ( ) v v + v, + ( b ) = B = N +. ( b ) β The erie β i the coverget geometric erie poeig the commo rtio β <. Hece, the olutio v ( ) i = differetible. So tht both erie (7) d (9) re coverget if β <. The ccordig (7) we obti covergece of the erie (7) to the ect olutio u ( ) i the ee of the C orm. The prtil um ( ) i Eq. (6) d it derivtive c be etimted follow. Theorem 4 The error E of th-order pproimte olutio of (9) d it error of th-order pproimte derivtive defied by (8) c be etimted by (36) E ' RRJSMS Volume Iue Jue, 6 93

6 β B β E B, E' β b β where E := v ( ) ( ), E ' := v ( ) ( ), while v B d β <. Proof. v ( ) ( ) = v( ) v d = + Bβ = + = + = B β. β v ( ) ( ) = v '( ) v = + = + = + B B β β =. b b β EXAMPLES Let the error etimte E d reltive error δ be defied by E = u = m u d d b d ud = % Emple Coider the Fredholm-Volterr lier itegrl equtio 49 3 u( )= u() t dt+ u() t dt (37) Solutio. The ect olutio i u ( )=. We begi by verifyig the coditio i Theorem. Sice the erel K, K d fuctio f( ), ( ) re cotiuou we eed to chec the iequlity (5). For Eq. (37) we hve λ =, λ =, = =, = =, = = 3 8 M K M K c Hece 4 α = cb ( )( λ M+ λ M) = < 5 coditio of Theorem hold, therefore HPM i coverget for y choice of iitil gue. Let u chooe u ( )=, the clcultig the ucceive fuctio v + determied by reltio ()-(3) we obti v = ( + ) u ( ) =, 47 v = ( + ) +, 6 47 v = ( ), Tble how tht the error decree very ft whe umber of term icree. Tble. Error of the pproimte olutio ( ) defied by (6) for Eq. (37). E = u δ (%) E = u (%) d d δ Emple Coider the Fredholm-Volterr itegrl equtio t + t e u( )=6 e e e u() t dt e u() t dt (38) RRJSMS Volume Iue Jue, 6 94

7 with the ect olutio u ( )= d e. Solutio. Let iitil gue u = be give. From Eq. (38) it follow tht e α = cb ( )( λ M+ λ M) = < 8 Prmeter α i Eq. (38) tifie the iequlity (5). From the Tble it c be ee tht the bolute d reltive error re decreed very ft whe umber of Tble. Error of the pproimte olutio ( ) defied by (6) for Eq. (38) E = u δ (%) E = u (%) d d δ term icree. Emple 3 Coider the Fredholm-Volterr itegro-differetil equtio of third id 8 ( + ) u ( ) = + ( )d ( )d tut t ut t (39) with the ect olutio ud = d iitil coditio u () =. Solutio. From Eq. (39) we hve λ =, λ =, M = K =, M = K =, c = =,( b )= 3 Hece, 5 β = cb ( ) ( λ M+ λ M) = < 6 tifie the coditio of Theorem 3. By chooig the iitil fuctio u ( )= we obti (Tble 3): Tble 3. Error of the pproimtio olutio (6) for Eq. (39) E = ud δ (%) E ˆ ud v δ (%) From thee reult we c coclude tht the pproimte olutio coverge very ft to the ect olutio whe icree. Emple 4 Coider the Fredholm-Volterr itegro-differetil equtio of third id u ( )= + + ( ) ( ) ( ) ( ), ()= tut dt 3 t tut dt u Solutio. Ect olutio of Eq. (4) i ud =+ 3. It i ot difficult to ee tht β = cb ( ) ( λ M+ λ M) = < 4 tifie the coditio of Theorem 3. The reult i Tble 4, how tht the bolute d reltive error of Eq. (4) decree very ft with iitil gue u ( )= +. Tble 4. Error of the pproimtio olutio (6) for Eq. (4) E = ˆ ud v δ (%) E = ud δ (%) Emple 5 Coider the Fredholm-Volterr itegrl equtio of the third id i the form (4) u( )= + u() t dt+ ( t) u() t dt (4) RRJSMS Volume Iue Jue, 6 95

8 Solutio. The ect olutio i u ( )= with the iitil gue u ( )=. I thi emple, the coditio of the Theorem fil. Sice + λ =, λ 4 =, M = K =, M = K =, c = = 8 5 hece 6 α = cb ( )( λ M+ λ M) = > 5 From the Tble 5, we c coclude tht pproimte olutio for Eq. (4) would coverge to the ect olutio but it might deped o the choice of the iitil gue. Tble 5. Error of the pproimtio olutio (6) for Eq. (4) E = u vˆ (%) E = u (%) d d δ CONCLUSION I thi ote, we hve lyzed HPM for olvig lier FVIE d FVIDE of the third id. From the Theorem it follow tht HPM for FVIE coverge uiformly if α <. Mewhile, HPM for FVIDE i uiformly coverget d differetible if β <. Emple - correpodig Tble d verify tht HPM i very ccurte d tble for the FVIE. Mewhile Emple 3-4 how tht HPM i coverget to the ect olutio very ft whe umber of poit icree. Emple 5, how tht cove HPM coverge to the ect olutio by the uitble choice of iitil gue eve though coditio of Theorem i ot tified. From the emple bove, we c coclude tht HPM method for olvig FVIE d FVIDE coverge very ft whe umber of poit icree d give fuctio d erel re tified the certi coditio i thm d thm3. ACKNOWLEDGEMENT Thi wor w upported by Ger Putr GP-I/4/9443, Uiverity Putr Mlyi (UPM). We re very grteful for the upport of Reerch Mgemet Ceter (RMC), UPM. REFERENCES [] Wzwz AM. Lier d olier itegrl equtio: Method d pplictio. Spriger, New Yor, NY, USA,. [] He JH. Homotopy perturbtio techique. Comp Method Appl Mech Egrg. 999;78:57-6. [3] He JH. A couplig method of homotopy techique d perturbtio techique for o-lier problem, It. J. Nolier Mech. ;35: [4] Kh Y d Wu Q. Homotopy perturbtio trform method for olier equtio uig He polyomil. Comp Mth Appl. ;6: [5] Mdi M, et l. O couplig of the homotopy perturbtio method d Lplce trformtio. Mth Comp Modellig. ; 53: [6] Kh Y, Abrzde M, Krgr A. Couplig of homotopy d the vritiol pproch for coervtive ocilltor with trog odd-olierity. Sci Ir. ;9:47-4. [7] Rmo JI. Piecewie homotopy method for olier ordiry differetil equtio. Appl Mth Comput. 8;98:9-6. [8] Jfri H, Alipour M, Tdodi H. Covergece of homotopy perturbtio method for olvig itegrl equtio. Thi J Mth. ;8:5-5. [9] Golbbi A d Jvidi M. Applictio of He homotopy perturbtio method for th-order itegro-differetil equtio. Appl Mth Comput. 7;9: [] Dehgh M d Sheri F. Solutio of itegro-differetil equtio riig i ocilltig mgetic field uig He Homotopy Perturbtio Method. Progre i Electromgetic Reerch PIER 78. 8;36:-376. [] Ghemi M, Ki MT, Dvri A. Numericl olutio of the olier Volterr-Fredholm itegrl equtio by uig Homotopy Perturbtio Method. Appl Mth Comput. 7;88: [] Abbbdy S. Numericl olutio of the itegrl equtio: homotopy perturbtio method d Adomi decompoitio method. AppliedMthemtic d Computtio. 6;73: RRJSMS Volume Iue Jue, 6 96

9 [3] Abbbdy S d Shivi E. A ew lyticl techique to olve Fredholm itegrl equtio. Numericl Algorithm.;56:7 43. [4] Bboli E, Bizr J, Vhidi AR. The decompoitio method pplied to ytem of Fredholm itegrl equtio of the ecod id. Applied Mthemtic d Computtio. 4;48: [5] Lio SJ. The propoed homotopy lyi techique for the olutio of olier problem, PhD thei. Shghi Jio Tog Uiverity. 99. [6] Lio SJ. Beyod perturbtio: itroductio to the homotopy lyi method. Boc Rto: Chpm d Hll/CRC Pre; 3. [7] Lio SJ. O the lytic olutio of mgetohydrodymic flow of o-newtoi fluid over tretchig heet. J Fluid Mech 3;488:89. [8] Lio SJ. O the homotopy lyi method for olier problem. Appl Mth Comput 4;47: [9] Mutf Ic. O ect olutio of Lplce equtio with Dirichlet d Neum boudry coditio by the homotopy lyi method. Phy Lett. 7;A 365:4 45. [] Abbbdy S, Mgyri E, Shivi E. The homotopy lyi method for multiple olutio of olier boudry vlue problem. Commu Nolier Sci Numer Simult. 9;4: [] Bizr J d Ghzvii H. Covergece of the Homotopy Perturbtio Method for prtil differetil equtio. Rel World Appl. 9;: [] Bizr J d Ghzvii H. Study of covergece of Homotopy Perturbtio Method for ytem of prtil differetil equtio. Comput Mth Appl. 9;58:-3. [3] Hetmio E, et l. A tudy of the covergece of d error etimtio for the homotopy perturbtio method for the Volterr-Fredholm itegrl equtio, Appl Mth Lett. 3;6: [4] Zulri FS, et l. Modified homotopy perturbtio method for Fredholm-Volterr itegro-differetil equtio. Recet Adv Mth Sci. 6: [5] Kythe PK d Puri P. Computtiol Method for Lier Itegrl Equtio, Birhäuer, Boto,. RRJSMS Volume Iue Jue, 6 97