Nonstandard finite difference scheme associated with harmonic mean averaging for the nonlinear Klein-Gordon equation

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Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology 6 Iss (7) 9-9 Pnrb Aadma Bar Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Jornal ompag: www.aadmabar.com/aras.

1 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology 6 Iss (7) 9-9 Pnrb Aadma Bar Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Jornal ompag: ISSN: Nonsandard fn dffrnc scm assocad w armonc man avragng for nonlnar Kln-Gordon qaon Opn Accss Ern Sryan Saro * Nrl I'zza Oman Mod Agos Salm Nasr Facly of Compr and Mamacal Scncs nvrs Tnolog MARA Slangor 445 Sa Alam Slangor Malaysa ARTICLE INFO Arcl sory: Rcvd Ocobr 6 Rcvd n rvsd form 5 Novmbr 6 Accpd 3 Novmbr 6 Avalabl onln 7 Janary 7 Kywords: Accracy Dnomnaor fncon Harmonc man avragng Kln-Gordon qaon Nonsandard fn dffrnc scm ABSTRACT In s papr w dmonsra a modfd scm for solvng nonlnar Kln- Gordon qaon of PDE yprbolc yps. T Kln-Gordon qaon s a rlavsc wav qaon vrson of Scrodngr qaon wc s wdly sd n qanm mcancs. Addonally nonsandard fn dffrnc scm as bn sd nsvly o solv dffrnal qaons and w av consrcd a modfd scm basd on nonsandard fn dffrnc scm assocad w armonc man avragng for solvng nonlnar nomognos Kln-Gordon qaon wr dnomnaor s rplacd by an nsal fncon. T nmrcal rsls oband av bn compard and sowd o av a good agrmn w rsls aand sng sandard fn dffrnc (CTCS) procdr wc provdd a proposd scm s rlabl. Nmrcal prmns ar sd o valda accracy lvl of scm w analycal rsls. Copyrg 7 PENERBIT AKADEMIA BAR - All rgs rsrvd. Inrodcon In fld of pyscs nonlnar Kln-Gordon qaon plays an mporan rol spcally n applcaons of qanm mcancs and condnsd mar pyscs []. Tr ar many powrfl nmrcal mods a av bn appld n ordr o solv nonlnar Kln-Gordon qaon. T cnqs ncld fn dffrnc mod [3-8] fn lmn mod [9- ] nvrs scarng radal bass fncons (RBF) [3] dffrnal ransform mod (DYM) [45] and omoopy analyss mod (HAM) [67]. * Corrspondng aor. E-mal addrss: rnsryansaro@gmal.com (Ern Sryan Saro) 9

2 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar Nonsandard fn dffrnc (NSFD) mod wc was dvlopd by [] for som class of dffrnal qaons s nson of sandard fn dffrnc mod and as bn sd wdly n nmrcal ngraon of dffrnal qaons. Morovr [3-5] as dnfd cran prncpls for dvlopng bs dffrnal qaons sng nonlocal appromaon by rplacng old dnomnaor of drvavs w a non-ngav fncon φ ( ) a follows crra as nds o zro φ ( ) approacs o zro. Tr s a mnor sdy on nonsandard fn dffrnc mod for Kln-Gordon qaon. In s papr w mplmn nonsandard fn dffrnc mod a s ncorporad w armonc man avragng o approma nown fncon a appars n nomognos Kln- Gordon qaon. In [7] by applyng armonc man appromaon rsls av sowd a nmrcal and appromad solons ar n good agrmn wo mc loss of accracy. T armonc man (HM) as bn sad n [8] as smalls man nc s sabl o b sd n mprovng dgr of accracy. T srcr of s papr s organzd as follows. In Scon w provd w som basc dfnon of Kln-Gordon qaon and fn dffrnc cnq. In Scon 3 w apply proposd mod. In Scon 4 w prsn nmrcal llsraons for drmnng ffcncy and rlably of approac scm and conclson of sdy s gvn n Scon 5.. Kln-Gordon qaon and fn dffrnc cnq.. Nonlnar Kln-Gordon qaon Nonlnar Kln-Gordon qaon as bn sdd nsvly n scnc and ngnrng flds from dffrn prspcvs. T gnral nonlnar Kln-Gordon qaon by Wazwaz [9] n form ( ) ( ) a ( ) F ( ( ) ) ( ) < < L < T sbc o nal condons () ( ) f ( ) ( ) g ( ) > () wr s a fncon of and a s a consan ( ) F ( ( ) ) s a nonlnar fncon of ( ) and f ( ) and ( ).. Fn dffrnc cnq s a nown fncon or fnconal vals g ar gvn fncon. T formlaon of sandard fn dffrnc sng Taylor srs panson n cnral m cnral spac (CTCS) and assoca w for pons of armonc man formla s as follows: a F ( ) {[ 4 ] [ ( ) ( )]} wr dnos as grd sz. By sfng ( ) o ( ) and n smplfyng (3) rsls n (3)

3 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar ( ) ] [ { [ ( ) ( )]} F a 4 (4) Trfor fnal form of gnral CTCS scm assocad w armonc man avragng can b wrn as (4). Sds on s of fn dffrnc scms wc lz alrnavs o or man avragng mod as bn rpord n [3] for lnar Kln-Gordon qaon. 3. Nonsandard fn dffrnc armonc man scm Nonsandard fn dffrnc mods wr nrodcd by Mcns n 98s [] as sopscad nmrcal cnqs wc approma drvavs and dffrnal qaons by sng nonlocal dscr rprsnaons. In s papr w analys applcaon of a nonsandard fn dffrnc mod a s assocad w armonc man avragng by sng as dnomnaor fncon for nonlnar nomognos Kln-Gordon qaon. Ts dnomnaor fncon sasfs propry as ( ) φ [3]. T fnal form for or nonsandard fn dffrnc scm s as follows: ] [ { [ ( ) ( )]} F a 4 (5) 4. Nmrcal llsraons To drmn ffcncy of modfd scm dscrbd n prvos scon w dmonsra som ampls. 4.. Eampl W frs consdr nonlnar nomognos Kln-Gordon qaon n [93] ( ) < < < < (6) w followng nal condons: ( ) ( ) > (7) T analycal solon of Eampl s ( ) ( ) a can b fond n [9]. Hr by sng scm (5) w acqr scm of Problm blow: [ { ] ( ) [ ( ) ( ) ( ( ) ( ) ) ( ) ( )]} 4 (8)

4 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar W crad compr programs for applcaon of sandard CTCS scm and scm (8) for Eampl. T prsnd nmrcal rsls and graps n Fg. and Fg. sow rspcv approma solon and rlav rrors a slcd ms pons w svral grd szs. Eac Solon for Nonlnar Problm.8 z-as y-as..4 -as.6.8 Fg.. T Eac Solon of Eampl n grapcal form a.5 Fg.. T Solon of Eampl n grapcal form sng scm (8) a.5 T abov grapcal prsnaons sow a grap for approma solon n Fg. loos mrly sam as grap for ac solon n Fg. d o occrrnc of smallr rrors. Tabl Rlav rrors for CTCS scm a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..)

5 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar Tabl Rlavs rrors for scm (5) a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..) Tabl 3 Comparson of avrag rlav rrors bwn CTCS scm and scm (8) Scm CTCS scm Scm (8) Tabl Tabl and Tabl 3 ndca rlav rrors and avrag rlav rrors for rspcv CTCS scm and approma scm (8) for Eampl a slcd grd szs. T rlav rrors of approma solons approac zro as grd sz rdcs. On or and avrag rlav rrors n Tabl 3 bcom smallr wn grd sz approacs o zro. Ts vdncs ndca bo scms convrg. In addon as grd szs bcom fnr nmrcal approma solons convrg o ac solon. Hnc bo scms ar conssn and sabl as grd szs nd o zro. Howvr s llsras a scm (5) s mor accra an CTCS scm. 4.. Eampl W n consdr nonlnar nomognos Kln-Gordon qaon n [3] 4 sn < < < < (9) w nal condons ( ) ( ) > () T analycal solon of Problm s ( ) sn a can b fond n []. Hr by sng scm (5) w oband nw approma scm accordng o Eampl as follow: 3

Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 4 Pnrb Aadma Bar sn sn sn sn sn sn sn sn sn sn sn sn 4 4 () W dvlopd compr programs for applcaon of CTCS scm and scm ()

6 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) Pnrb Aadma Bar sn sn sn sn sn sn sn sn sn sn sn sn 4 4 () W dvlopd compr programs for applcaon of CTCS scm and scm () for Eampl. T prsnd nmrcal rsls and graps n Fg. 3 and Fg. 4 dmonsra rspcv approma solon and rlav rrors a slcd ms pons w svral grd szs. Fg. 3. T Eac solon of Eampl n grapcal form a.5 Fg. 4. T Solon of Eampl n grapcal form sng scm () a.5 T abov grapcal prsnaons sow a grap for approma solon n Fg. 4 loos sam as grap for ac solon n Fg. 3 d o occrrnc of smallr rrors.

7 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar Tabl 4 Rlav rrors for CTCS scm a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..) Tabl 5 Rlav rrors for scm () a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..) Tabl 6 Comparson of avrag rlav rrors bwn CTCS scm and scm () Scm CTCS scm Scm () Tabl 4 Tabl 5 and Tabl 6 dmonsras rlav rrors and avrag rlav rrors for CTCS scm and approma scm for Eampl a slcd grd szs rspcvly. T rlav rrors of approma solon approac zro as grd sz rdcs. On or and avrag rlav rrors n Tabl 6 bcom smallr wn grd sz approacs zro. Ts vdncs ndca bo scms ar convrgn. In addon as grd szs bcom smallr nmrcal approma solons convrg o ac solon. Hnc bo scms ar sabl and conssn as grd szs srn o zro. Howvr s sows a scm () s mor accra an CTCS scm Eampl 3 As a fnal problm w consdrd nonlnar nomognos Kln-Gordon qaon n [5-4333] ( ) < < < < () w nal condons ( ) ( ) > (3) a can b fond n [3]. Hr by sng scm (5) w oban nw approma scm accordng o Eampl 3 as follow: T analycal solon of Eampl 3 s ( ) ( ) 5

8 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar ( ) ( ) ( ) {[ 4( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ] [( ( ) ( ) ( ) ) ( ( ) ( ) ( ) )(( ( ) ( ) ( ) ) ( ( ) ( ) ( ) )) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) (( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ))]} (4) W consrcd compr programs for applcaon of CTCS scm and scm (4) for Eampl 3. T prsnd nmrcal rsls and graps n Fg. 5 and Fg. 6 llsra rspcv approma solon and rlav rrors a slcd ms pons w svral grd szs. Fg. 5. T Eac solon of Eampl 3 n grapcal form a.5 Fg. 6. T Solon of Eampl 3 n grapcal form sng scm (4) a.5 T abov grapcal llsraons sow a grap for approma solon n Fg. 6 loos sam as grap for ac solon n Fg. 5 d o occrrnc of smallr rrors. 6

9 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar Tabl 7 Rlav rrors for CTCS scm a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..) Tabl 8 Rlav rrors for scm (4) a slcd ms pons w svral grd szs ( ) (.5.5) (.5.5) (.75.75) (..) Tabl 9 Comparson of avrag rlav rrors bwn CTCS scm and scm (4) Scm CTCS scm Scm (4) Tabl 7 Tabl 8 and Tabl 9 dmonsras rlav rrors and avrag rlav rrors for CTCS scm and approma scm for Eampl 3 a slcd grd szs rspcvly. T rlav rrors of approma solon approac zro as grd sz rdcs. Manwl avrag rlav rrors n Tabl 9 ar smallr wn grd sz rdcs o zro. Ts vdncs ndca bo scms ar convrgng. Frrmor as grd szs bcom smallr nmrcal approma solons closr o ac solon. Ts bo scms ar sabl and conssn as grd sz approacs zro. Ts sows a scm (4) s mor accra an CTCS scm. 4. Conclson T nmrcal prmns for modfd scm av dmonsrad good prformanc for slcd nonlnar nomognos Kln-Gordon problms. A comparav sdy on avrag rlav rrors bwn CTCS scm and nonsandard fn dffrnc procdr a slcd grd szs was don. As grd szs bcom fnr lvl of accracy ncrass. Hnc w can concld a nonsandard fn dffrnc scm a s assocad w armonc man avragng n (5) s ffcv and sows sgnfcan mprovmn n solvng nonlnar Kln-Gordon qaons ovr sng mods. T scm s also obsrvd o b locally sabl and convrgn. Rfrncs [] Comay E. Dffcls w Kln-Gordon Eqaon. Apron no. 3 (4): 8. [] Cang Donald C. "A Classcal Approac o Modlng of Qanm Mass." Jornal of Modrn Pyscs 4 no. (3): 3. [3] Dgan Md Abar Mobb and Zor Asgar. "For-ordr compac solon of nonlnar Kln- Gordon qaon." Nmrcal Algorms 5 no. 4 (9):

10 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar [4] Kong Lnga Jngng Zang Yng Cao Yal Dan and Hong Hang. "Sm-plc symplcc parond Rng Ka Forr psdo-spcral scm for Kln Gordon Scrödngr qaons." Compr Pyscs Commncaons 8 no. 8 (): [5] Bülbül Brna and Mm Szr. "A nw approac o nmrcal solon of nonlnar Kln-Gordon qaon." Mamacal Problms n Engnrng 3 no. 6 (3): 9. [6] Sao Wnng and Xonga W. "T nmrcal solon of nonlnar Kln Gordon and Sn Gordon qaons sng Cbysv a mslss mod." Compr Pyscs Commncaons 85 no. 5 (4): [7] Vong Sawng and Zbo Wang. "A compac dffrnc scm for a wo dmnsonal fraconal Kln Gordon qaon w Nmann bondary condons." Jornal of Compaonal Pyscs 74 (4): [8] Yn Fang Tan Tan Jnqang Song and Mn Z. "Spcral mods sng Lgndr wavls for nonlnar Kln Sn-Gordon qaons." Jornal of Compaonal and Appld Mamacs 75 (5): [9] Wang Qanfang and DaZan Cng. "Nmrcal solon of dampd nonlnar Kln Gordon qaons sng varaonal mod and fn lmn approac." Appld mamacs and compaon 6 no. (5): [] Kr S. A. and Al Sayfy. "A spln collocaon approac for nmrcal solon of a gnralzd nonlnar Kln Gordon qaon." Appld Mamacs and Compaon 6 no. 4 (): [] Go P. F. K. M. Lw and P. Z. "Nmrcal solon of nonlnar Kln Gordon qaon sng lmn-fr p-rz mod." Appld Mamacal Modllng 39 no. (5): [] Hssan Arsad Sral Haq and Maran ddn. "Nmrcal solon of Kln Gordon and sn-gordon qaons by mslss mod of lns." Engnrng Analyss w Bondary Elmns 37 no. (3): [3] Dgan Md and Al Sor. "Nmrcal solon of nonlnar Kln Gordon qaon sng radal bass fncons." Jornal of Compaonal and Appld Mamacs 3 no. (9): 4-4. [4] Kan ASV Rav and K. Arna. "Dffrnal ransform mod for solvng lnar and nonlnar Kln Gordon qaon." Compr Pyscs Commncaons 8 no. 5 (9): [5] Ksn Yıldıray Sma Srv and Galp Oranç. "Rdcd Dffrnal Transform Mod for Solvng Kln Gordon Eqaons." In Procdngs of World Congrss on Engnrng no. Jly (): 6. [6] Alomar A. K. Mod Salm Md Nooran and ROSLINDA MOHD Nazar. "Approma analycal solons of Kln-Gordon qaon by mans of omoopy analyss mod." Jornal of Qaly Masrmn and Analyss JQMA 4 no. (8): [7] Iqbal S. M. Idrs Abdl Mad Sddq and A. R. Ansar. "Som solons of lnar and nonlnar Kln Gordon qaons sng opmal omoopy asympoc mod." Appld Mamacs and Compaon 6 no. (): [8] Yldrm Am Syd Tasf Moyd-Dn and D. H. Zang. "Analycal solons o plsd Kln Gordon qaon sng modfd varaonal raon mod (MVIM) and Bobar polynomals panson scm (BPES)." Comprs & Mamacs w Applcaons 59 no. 8 (): [9] Sar Fam and Md Dgan. "Nmrcal solon of Kln Gordon qaon va H s varaonal raon mod." Nonlnar Dynamcs 5 no. - (8): [] Baa B. "A varaonal raon mod for solvng nonlnar Kln Gordon qaon." Asral. J. Basc Appl. Sc 3 (9): [] Grs K. F. "A smpl consrcon of nonsandard fn-dffrnc scms for small nonlnar sysms appld o SIR modls." Comprs & Mamacs w Applcaons 66 no. (3): [] Mcns Ronald E. Applcaons of Nonsandard Fn Dffrnc Scms. Sngapor: World Scnfc Pblsng Co. P. Ld.. [3] Mcns Ronald E. "Dscrzaons of nonlnar dffrnal qaons sng plc nonsandard mods." Jornal of Compaonal and Appld Mamacs no. (999): [4] Mcns Ronald E. Inflnc of Spaal Dscrzaons on Nonsandard Fn Dffrnc Scms for Nonlnar PDE s. Inrnaonal Jornal of Applcaon Scnc Comprs 6 (999b): [5] Mcns Ronald E. A Nonsandard Fn Dffrnc Scm for a PDE Modlng Combson w Nonlnar Advcon and Dffson. Mamacs and Comprs n Smlaon 69 (5): [6] Hasna Mazn O. and M-S. Alon. "Applcaon of armonc man sascs o nd-o-nd prformanc of ransmsson sysms w rlays." In Global Tlcommncaons Confrnc. GLOBECOM'. IEEE vol. pp IEEE. [7] Jng Cang H. Soo Ya Ba and Yong P. Km. "Appromad solon on proprs of scavngng gap drng prcpaon sng armonc man mod." Amosprc Rsarc 99 no. 3 (): [8] Wazwaz Abdl-Mad. "On nmrcal solon of Gorsa problm." Appld mamacs and compaon 59 no. (993):

11 Jornal of Advancd Rsarc n Appld Scncs and Engnrng Tcnology Volm 6 Iss (7) 9-9 Pnrb Aadma Bar [9] Wazwaz Abdl-Mad. Paral dffrnal qaons and solary wavs ory. Sprngr Scnc & Bsnss Mda. [3] Kasron Noran Mod Agos Salm Nasr S Salma Yasran and Karl Isandar Oman. "Nmrcal solon of a lnar Kln-Gordon qaon." In Elcrcal Elcroncs and Sysm Engnrng (ICEESE) 3 Inrnaonal Confrnc on pp IEEE 3. [3] Dba E. Y. and S. A. Kr. "A dcomposon mod for solvng nonlnar Kln Gordon qaon." Jornal of Compaonal Pyscs 4 no. (996): [3] Wazwaz Abdl-Mad. "T modfd dcomposon mod for analyc ramn of dffrnal qaons." Appld Mamacs and Compaon 73 no. (6): [33] Rasdna Jall Md Gasm and R. Jallan. "Nmrcal solon of nonlnar Kln Gordon qaon." Jornal of Compaonal and Appld Mamacs 33 no. 8 ():

Implementation of the Extended Conjugate Gradient Method for the Two- Dimensional Energized Wave Equation

Implementation of the Extended Conjugate Gradient Method for the Two- Dimensional Energized Wave Equation Lonardo Elcronc Jornal of raccs and Tchnolos ISSN 58-078 Iss 9 Jl-Dcmbr 006 p. -4 Implmnaon of h Endd Cona Gradn Mhod for h Two- Dmnsonal Enrd Wav Eqaon Vcor Onoma WAZIRI * Snda Ass REJU Mahmacs/Compr