The Development of Suitable and Well-founded Numerical Methods to Solve Systems of Integro- Differential Equations,
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1 Shiraz Uivrsiy of Tchology From h SlcdWorks of Habibolla Laifizadh Th Dvlopm of Suiabl ad Wll-foudd Numrical Mhods o Solv Sysms of Igro- Diffrial Equaios, Habibolla Laifizadh, Shiraz Uivrsiy of Tchology Availabl a: hps://works.bprss.com/habib_laifizadh//
2 Joural of Mahmaical Scics: Advacs ad Applicaios Volum,, Pags -5 THE DEVELOPMENT OF SUITABLE AND WELL-FOUNDED NUMERICAL METHODS TO SOLVE SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS E. HESAMEDDINI, A. RAHIMI ad HABIBOLLA LATIFIZADEH Dparm of Mahmaics Faculy of Basic Scics Shiraz Uivrsiy of Tchology Shiraz Ira -mail: Absrac This papr has b dvod o apply h rcosrucio of variaioal iraio mhod (RVIM) o hadl sysms of igro-diffrial quaios. RVIM has b iducd wih Laplac rasform from h variaioal iraio mhod (VIM), which was dvlopd from h Iokui mhod. Acually, RVIM ovrcom o shorcomig of VIM mhod o drmi h Lagrag muliplir. So ha RVIM mhod provids rapidly covrg succssiv approimaios o h ac soluio. Th advaag of h RVIM i compariso wih ohr mhods is h simpliciy of h compuaio wihou ay rsriciv assumpios. Numrical ampls ar prsd o illusra h procdur. Compariso wih h homoopy prurbaio mhod has also b poid ou. Mahmaics Subc Classificaio: K, A, 5G5. Kywords ad phrass: rcosrucio of variaioal iraio mhod, sysm of Volrra igro-diffrial quaios. Rcivd Ju, Sciific Advacs Publishrs
3 E. HESAMEDDINI al.. Iroducio Sysms of igral quaios, liar or oliar, appar i sciific applicaios i girig, physics, ad chmisry ad populaios growh modls. Sudis of sysms of igral quaios hav aracd much cocr i applid scics. Volrra sudid h hrdiary iflucs wh h was amiig a populaio growh modl. Th rsarch rsuld i a spcific opic, whr boh diffrial ad igral opraors appard oghr i h sam quaio. This w yp of quaio may rm as Volrra igrodiffrial quaios, giv i h form () i y ( ) f ( ) k(, ) u( ) df, whr k (, ) a fucio of wo variabls ad, is calld h krl. I his papr, w will sudy sysms of Volrra igro-diffrial quaios giv by () i y ( ) f ( y ( ), y ( ),, y ( ) ) g (, y (), y (),, y () ) d,, m m,, m. Th fucios f i ( ) ar giv ral-valud fucios ad ukow fucios y ( ), y( ),, ym ( ) will b drmid. A variy of umrical ad aalyical mhods such as sris soluio mhod [], homoopy prurbaio mhod [-], Adomia dcomposiio mhod [-], ad variaioal iraio mhod [7-9] hav b usd o solv h sysms of igro-diffrial quaios. I is impora o poi ou ha hs mhods will b applid for h sparabl or diffrc krls.
4 THE DEVELOPMENT OF SUITABLE AND I his work, w us h rcosrucio of variaioal iraio mhod for solvig sysm (). This mhod was firs proposd by Hsamddii ad Laifizadh [] ad provids rapidly covrg succssiv approimaios of h ac soluio, if such a closd form soluio iss.. Prlimiaris For h radr s covic, w prs som cssary dfiiios, which ar usd furhr i his papr. Dfiiio. Th Laplac rasform of f ( ) is dfid as follows: s F () s L { f ( ; s) } f ( ) d. O of h mos impora propris of Laplac rasform is h covoluio of fucios f ad g. L h fucios f ( ) ad g () b dfid for, h h covoluio of h fucios f ad g is dod by ( f g)( ), ad is dfid as h followig igral: ( f g)( ) f ( ) g( ) d. L L { f () } F ( s), L{ g( ) } G( s), h L {( f g)( ) } F ( s) G( s). Or quivally, f () g( ) d L F () s G(). s Covrsly, L { F () s G() s } f () g( ) d. Dfiiio. Th Laplac rasform of L [ f ( ; s) ] of h drivaiv is giv by () m i i { () } () ( ik) ( k ; ) L f s s F s s f ( ), F ( s) L f ( ) ; s, k { } i.
5 E. HESAMEDDINI al.. Sysm of Igro-Diffrial Equaios ad h Rcosrucio of Variaioal Iraio Mhod (RVIM) I his sudy, w cosidr h followig sysm of igro-diffrial quaios: () i y ( ) f (, y ( ), y( ),, ym ( ) ) g (, y (), y(),, y ()), m () i y ( ) f (, y ( ), y( ),, ym ( ) ) g(, y (), y(),, y ()), m () i y m ( ) fm (, y ( ), y( ),, ym ( ) ) g (, (), (),, () ), m y y ym () whr g s ar liar/oliar fucios of, y, y,, ym ad h drivaiv of y of ordr i, subc o h iiial codiios: () i y is ( k) y ck, m. () W summariz sysm (), i h form of () i y ( ) N ( y ( ), y ( ),, y ( ) ),,,, (), m m wih h zro arificial iiial codiios. By akig Laplac rasform o boh sids of Equaio (), i h usual way ad usig h arificial iiial codiios, h followig rsul is obaid: i s L { y ( ) } L{ N ( y ( ), y ( ),, y ( ) )},,,. () Thrfor, w ca coclud ha, m m L { y ( ) } L{ N (, y ( ), y( ),, ym ( ) )},,, m. (5) i s Suppos ha H(), s s i h by usig h covoluio horm, o obai
6 THE DEVELOPMENT OF SUITABLE AND 5 L { y ( ) } H() s L{ N ( y ( ), y ( ),, y ( ) )} L{ ( h N },, m,, m, () whr L { H () s } h( ). Takig h ivrs Laplac rasform o boh sids of Equaio (), h iraio formula of Equaio (), ca b giv as y ( ) h( ) N ( y (), y (),, y () ) d,,, m. (7), Now, w mus impos h acual iiial codiios o obai h soluio of Equaio (). Thus, w hav h followig iraio formulaio: y ( ) y ( ) h( ) N (, y ( ), y ( ),, y ( )) d, m m,, m. (8) Th valus y ( ), y ( ), ym ( ) ar giv by, ( y ) ( ) y ( ) y ( ). (9)! y Thrfor, accordig o h rcosrucio of variaioal iraio mhod y ( ) is obaid as follows: y ( ) lim y ( ),,,, () whr y ( ) idicas -h approimaio of ( ). y. Numrical Rsuls To dmosra h ffcivss of h mhod, w cosidr som sysms of liar ad oliar fracioal diffrial quaios:
7 E. HESAMEDDINI al. Eampl. Cosidr h followig sysm of Volrra igrodiffrial quaios: u ( ) si ( u( ) ν( ) ) d, ν ( ) si cos ( u() ν() ) d, () subc o h iiial codiios u ( ), u ( ), ν( ), ν ( ). () Applyig h RVIM o Equaio (), h rsul is as follows: L { u( ) } L si ( () ()), u ν d s L { ν( ) } L si cos ( () ()). u ν d () s By applyig h ivrs Laplac rasform o boh sids of Equaio (), rsul i u( ) ( ) si ( u( ) ν ( ) ) d d, ν ( ) ( ) si cos ( u() ν() ) d d. () Cosidrig h iiial codiios of Equaio (), iraiv rlaios ar obaid as u ν d ν ( ) u( ) ( ) si ( u ( ) ( )) d, ν ( ) ν( ) ( ) si cos ( u ( ) ( )) d d, (5) whr u ( ), ν ( ) ad u ( ), ν ( ) idica h -h approimaio of u ( ) ad ν ( ), rspcivly.
8 THE DEVELOPMENT OF SUITABLE AND 7 Accordig o rlaios (5), afr som simplificaio ad subsiuio, h followig ss of quaios ar rsuld: u ( ) si,!! ν ( ) si cos,!! ( ),!!! u ( ).! 5! 7! ν Thus h closd form soluios ar as follows: u( ) lim u ( ) cos ad ν ( ) lim ν ( ) si. Now, w will solv his ampl by h homoopy prurbaio mhod (HPM). Thy cosruc a cov homoopy of h form H( u, p) u ( ) si p ( u() ν() ) d, 5 H( ν, p) ν ( ) si cos p ( u( ) ν( ) ) d. () Th mbddig paramr p moooically icrass from o. Th homoopy prurbaio mhod admis h us of h pasio u( ) p u ( ), ν( ) p ν ( ), (7) 7 whr u ( ) ad ν ( ), ar h compos of u ( ) ad ν ( ) ha will b lgaly drmid i h rcursiv mar. Subsiuig (7), i o Equaio (), ad quaig h rms wih lik powrs, h followig ss of rlaios ar rsuld:
9 8 E. HESAMEDDINI al. u ( ) si, ν ( ) si cos, 7 u ( ) cos si, ν ( ) si cos, u ( ) si, ν ( ) si cos, u ( ) cos si, ν ( ) cos si, ad so o. Thrfor, h soluios by h homoopy prurbaio mhod wih hr rms will b drmid as u( ) cos, ν ( ) si. Figur shows compariso bw h approima soluios by HPM ad h ac soluio ha is obaid by RVIM mhod for Eampl, ad also shows h absolu rror bw h boh mhods.
10 THE DEVELOPMENT OF SUITABLE AND 9 Figur. Compariso bw h soluios obaid by RVIM ad HPM.
11 5 E. HESAMEDDINI al. By cosidrig h abov rsuls, w coclud ha h RVIM ca giv ac soluios i a fw iraios. Eampl. Now, w cosidr h followig sysm of hr Volrra igro-diffrial quaios: u ( ) ( ν ( ) w( ) ) d, ν ( ) ( w() u() ) d, w ( ) ( u( ) ν( ) ) d, (8) wih h iiial codiios: u( ), ν ( ), w( ). (9) By applyig h RVIM o Equaio (8), h rsul is as follows: L { u( ) } L ( ν() w() ) d, s L{ ν ( ) } L ( w() u() ) d, s L { w( ) } L ( u() ν() ) d. () s Similarly, bfiig h ivrs Laplac rasform o boh sids of Equaio (), h followig RVIM formula ca b giv as: u( ) ( ν () w() ) d d, ν ( ) ( w( ) u( ) ) d d, w( ) ( u( ) ν ( ) ) d d. ()
12 THE DEVELOPMENT OF SUITABLE AND 5 To obai h approima soluio of Equaio (8), h followig iraiv rlaio is cosidrd: ( ) ( ) ( ( ) ( )), d d w u u ν ( ) ( ) ( ( ) ( )), d d u w ν ν ( ) ( ) ( ( ) ( )), d d u w w ν () whr h iiial approimaio mus b saisfid i h followig quaios: ( ) ( ) ( ).,, w u ν For h sysm of Equaio (8), by mas of RVIM chiqu, h succssiv approima soluios ca b obaid ( ), 7 u ( ), ν ( ), w ( ),!!! u ( ) ( ) ( ) ( ),!!! ν ( ) ( ) ( ) ( ).!!! w
13 5 E. HESAMEDDINI al. Thrfor, h ac soluios ar giv by u ( ) lim u ( ), ν ( ) lim ν ( ), w ( ) lim w ( ). Now, w will us h homoopy prurbaio mhod (HPM) for solvig sysm (8). Th followig ss of quaios ar rsuld: u ( ), ν ( ), w ( ), u ( ) 9, ν ( ), w ( ), ad so o. Thrfor, w g h approima soluio h soluios ( ), ν ( ) w ( ) as h fourh rm approimaio of h soluios of u, sysm (8), by h homoopy prurbaio mhod u( )
14 THE DEVELOPMENT OF SUITABLE AND ν ( ) , , w ( )
15 5 E. HESAMEDDINI al. Figur shows ha h RVIM is mor ffici ha HPM. Figur. Compariso bw h RVIM ad HPM for sysm (7).
16 THE DEVELOPMENT OF SUITABLE AND Coclusio I his work, w applid h RVIM for solvig sysms of Volrra igro-diffrial quaios. Th w mhod rquirs o kowldg of variaioal hory o driv a dd variaioal iraio mhod. I is impora o poi ou ha som ohr mhods will b applid for sysms wih sparabl or diffrc krls. Whras, h RVIM ca b usd for solvig sysms of Volrra igro-diffrial quaios wih ay kid of krls. By comparig h rsuls of ohr umrical mhod such as homoopy prurbaio mhod, w coclud ha h RVIM is mor accura, fas, ad rliabl. Bsids, RVIM dos o rquir small paramrs, hus h limiaios of h radiioal prurbaio mhods ca b limiad, ad h calculaios ar also simpl ad sraighforward. This has b cofirmd i h prs papr by mployig wo ampls. Thrfor, his mhod is a vry ffciv ool for calculaig h ac soluios of sysms of igro-diffrial quaios. Rfrcs [] K. Malkad ad Y. Mahmoudi, Taylor polyomial soluio of high-ordr oliar Volrra-Frdholm igro-diffrial quaios, Appl. Mah. Compu. 5 (), -5. [] J. H. H, Homoopy prurbaio chiqu, Compu. Mh. Appl. Mch. Eg. 78 (999), 57-. [] A. Golbabai ad M. Javidi, Applicaio of H s homoopy prurbaio mhod for -h ordr igro-diffrial quaios, Appl. Mah. Compu. 9 (7), 9-. [] A. Brasos, M. Ehrhard ad Th. Famlis, A discr Adomia dcomposiio mhod for discr oliar Schrodigr quaios, Appl. Mah. Compu. 97 (8), 9-5. [5] A. M. Wazwaz, A rliabl modificaio of Adomia s dcomposiio mhod, Appl. Mah. Compu. (999), [] J. Biazar, E. Babolia ad R. Islam, Soluio of h sysm of Volrra igral quaios of h firs kid by Adomia dcomposiio mhod, Appl. Mah. Compu. 9 (), 9-58.
17 5 E. HESAMEDDINI al. [7] S. Abbasbady ad E. Shivaia, Applicaio of variaioal iraio mhod for -h ordr igro-diffrial quaios, Vrlag dr Zischriffuraur for Schug a (9), 9-. [8] S. Q. Wag ad J. H. H, Variaioal iraio mhod for solvig igrodiffrial quaios, Phys. L. A 7 (7), [9] A. J. Jrri, Iroducio o Igral Equaios wih Applicaios, Scod Ediio, Wily Irscic, 999. [] E. Hsamddii ad H. Laifizadh, Rcosrucio of variaioal iraio algorihm usig h Laplac rasform, Iraioal Joural of Noliar Scics ad Numrical Simulaio (-) (9), g
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