Solvability of nonlinear Klein-Gordon equation by Laplace Decomposition Method
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1 Vol. 84 pp Jly 5 DOI:.5897/JMCSR4.57 icle Nbe: 63F95459 ISSN Copyigh 5 hos ein he copyigh of his icle hp:// ficn Jonl of Mheics nd Cope Science Resech Fll Lengh Resech Ppe Solvbiliy of nonline Klein-Godon eqion by Lplce Decoposiion Mehod Mohed E.. Rbie Mheics Depen Fcly of Edcion-fif Shq Univesiy Sdi bi Mheics Depen Fcly of Science Sdn Univesiy of Science nd Technology Sdn. Received 9 Sepebe 4; cceped 6 Jne 5 In his sdy doin Decoposiion Mehod DM Modified Decoposiion Mehod MD nd Lplce Decoposiion Mehod LDM wee sed in solving nonline Klein-Godon eqion. I cn be esily conclded h hese hee ehods yielded ecly he se esls. Key wods: doin Decoposiion Mehod Modified Decoposiion Mehod Lplce Decoposiion Mehod Klein-Godon eqion nd Noise es phenoen. INTRODUCTION wide viey of physiclly significn pobles sch s nonline Klein-Godon eqion odeled by line nd nonline pil diffeenil eqions hs been he focs of eensive sdies fo he ls decdes. hge nbe of esech nd invesigions hve been invesed in hese scienific pplicions. Sevel ppoches sch s he chceisics ehod specl ehods nd pebion echniqes hve been eensively sed o eine hese pobles Wzwz 9. Solving of nonline eqions sing doin decoposiion ehod DM hs been done in Wzwz 9; 6; El-Wkil e l. 6; doin 994;984;986; bssy e l. 7; 4; Chel 99; Lesnic 6; 7; Wzwz ; Mohed nd Tig 3; 4 nd odified decoposiion ehod MD in Mohed nd Tig 3; 4. The i of his ppe is in wo folds: fisly o solve he nonline Klein-Godon eqion vi LDM DM nd MD. Secondly o show hese hee ehods yielded ecly he se esl. s we know he nonline Klein-Godon eqion coes fo qn field heoy nd descibes nonline wve inecion. The nonline Klein-Godon eqion in is sndd fo is F h Sbjec o he iniil condiions f g Whee h is consn is soce e nd E-il: ohed.elbie@gil.co. hos gee h his icle ein penenly open ccess nde he es of he Ceive Coons ibion License 4. Inenionl License
2 38 f. J. Mh. Cop. Sci. Res. F is nonline fncion of. In his wok he noise es phenoenon ws sed Wzwz 9 which povides jo dvnge in h i deonses fs convegence of he solion. I is ipon o noe h he noise es phenoenon y ppe only fo inhoogeneos pil diffeenil eqions; in ddiion his phenoenon is pplicble o ll inhoogeneos PDEs of ny ode. The noise es if eised in he coponens nd will povide in genel he solion in closed fo wih only wo sccessive ieions. Solion of Nonline Klein Godon Eqion by DM The decoposiion ehod will be eployed. The F nonline e will be eqed o he infinie seies of doin polynoils doin 994. In n opeo fo Eqion given by Whee nd L pplying o boh sides of 3 nd sing he iniil condiions o obin f g L h 3 We obin he ecsive elion: f g L h L L k k k k k h leds o: f g L h L L L L This coplees he deeinion of he fis few coponens of he solion. Bsed on his deeinion he solion in seies fo is edily obined. In ny cses closed fo solion obined condcively. Eple Given he following nonline Klein-Godon eqion: 7 following he discssion pesened bove we find: 5 6 L L F 4 Using he decoposiion seies fo he line e by nd he infinie seies of doin polynoils fo he F nonline e by 3 4 Cnceling he noise es 3! nd fo he coponen nd veifying h he eining noncnceled es fo he coponen sisfies he Eqion 7 hen he ec solion is 8 Whee e doin polynoils which clcled by Solion of Nonline Klein Godon Eqion by MD In n opeo fo Eqion becoes 9
3 Whee nd The odified decoposiion ehod sggess h F h Opeing wih L on boh sides of Eqion 9 sbjec o Eqion nd sing he bove sspion we obin: g f Le be on he igh side of Eqion nd eqe he coefficiens of like powe of on boh sides we ge: nd hen he ecence elion given by: Hving deeined he coefficiens he solion in seies fo follow iediely. Eple: Conside he iniil nonline Klein-Godon poble 7 Rbie 39 Following he pevios discssion we find: Whee 3 k k The solion in seies fo given by Solion of Nonline Klein Godon Eqion by LDM Hee he LDM will be ipleened o Klein-Godon eqion.to illse he ehod conside he genel fo of Klien-Godon Eqion sbjec o he iniil condiion nd pplying Lplce nsfo denoed hogho his ppe by on boh sides of Eqion yields: Which gives 3
4 4 f. J. Mh. Cop. Sci. Res. So 7 Secondly sing he decoposiion seies fo he line e nd he infinie seies of doin polynoils fo he nonline e which gives Secondly sing he decoposiion seies fo he line e nd he infinie seies of doin polynoils fo he nonline e which gives 8 Whee is he doin Polynoils given by Eqion 5. Then Eqion 4 becoes Th leds o he ecence elion below 9 This gives he ecence elion 5 Clcle: pplying invese Lplce nsfo on he fis eqion of Eqion which gives 6 Clcle: pplying he invese Lplce nsfo o he Eqion 6 hen he eqied ecence elion is iediely obined which coplee he solion Fo Eqion 8: Fo Eqion : pplicion : Conside he nonline Klein-Godon Eqion 7 sing Eqion 6 sbjec o he iniil condiion we ge
5 Rbie 4 Fo he second Eqion pplying invese Lplce o obin Cnceling he noise es 3 3! nd fo he coponen nd veifying h he eining noncnceled es sisfies he eqion he ec solion pplicion : Conside he nonline Eqion + = Sbjec o he iniil condiions = = Following he nlysis pesened bove nd sing he given iniil condiions we obin he ecsive elion in he fo L = s L L pplying invese Lplce nsfo on he ls eqion leds o = ! ! nd By cnceling he noise es nd 56 fo he coponen nd veifying h he eining non-cnceled es of sisfies Eqion we find h he ec solion is given by = 3 3 Conclsion In his ppe we inodced Klein-Godon eqion nd solved i by sing DM nd MD hen pplied LDM.Clely hese hee ehods e vey effecive i ccelees he solions. If we cope i wih he ohe ehods i will be he bes. In ddiion he LDM y give he ec solions fo nonline PDEs. Moeove he noise es y ppe if he ec solion is p of he 'heoh coponen. L = 6 3 s ! 6 s6 + s 9 Conflic of Inees L k+ = s L k L k k The ho hs no decled ny conflic of inees. REFERENCES Clcle: : pplying invese Lplce nsfo on he fis eqion of h leds o = Clcle: : = = bssy T El-Twil M Sleh HK 4. he solion of KdV nd KdV eqions sing doin pde ppoiion. In. J. Nonline Sci. N. Sil. 54: bssy T El-Twil M Sleh HK 7. The solion of Bges' nd good Bossinesq eqions sing DM-Pdé echniqe. Chos Solions Rcls 33:8-6. doin G 984. new ppoch o nonline pil diffeenil eqions. J. Mh. nl. ppl. : doin G 986. Nonline Sochsic Opeo Eqions cdeic Pess Sn Diego New. Yok MR j:6. doin G 994. Solving Fonie Pobles of Physics: The Decoposiion Mehod Klwe Boson. Chel Y 99. Convegence of doin's ehod. Mh. Cop. Model. 4: El-Wkil S bdo M Elhnbly 6 doin decoposiion ehod fo solving he diffsion convecion ecion eqions. ppl. Mh. Cop. 77: Lesnic D 6. The decoposiion ehod fo iniil vle pobles.
6 4 f. J. Mh. Cop. Sci. Res. ppl. Mh. Cop. 8:6-3. Lesnic D 7. nonline ecion-diffsion pocess sing he doin decoposiion ehod. In. Con. He Mss Tnsfe 34:9-35. Mohed ER Tig ME 3. Solion of line syses of pil diffeenil eqions by doin nd Modified decoposiion ehods. si-pcific Sci. Cl. J. :56-6. Mohed MER Tig ME 4. sdy of soe syses of nonline pil diffeenil eqions by sing doin nd odified decoposiion ehods. f. J. Mh. Cop. Sci. Res. 76:6-67. Wzwz M. copionl ppoch o solion solions of he Kdosev-Pevishvili eqion. ppl. Mh. Cop. 3:5-7 Wzwz M 6.The odified decoposiion ehod fo nlyic een of Diffeenil Eqions. ppl. Mh. Cop. 73: Wzwz M 9. Pil Diffeenil Eqions nd Soliy Wves Theoy Highe Edcion Pess. Dewi Djie Beijing P. R. Chin.
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