EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING THE EXTENDED MULTIPLE RICCATI EQUATIONS EXPANSION METHOD

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1 IJRRAS 9 () Deceme EXACT SOUTIONS FOR NONINEAR PARTIA DIFFERENTIA EQUATIONS BY USING THE EXTENDED MUTIPE RICCATI EQUATIONS EXPANSION METHOD Mmo M. El-Bo Aff A. Zgo & Aml M. Else Demen of Memcs Fcl of scence Alen Unves Eg Demen of Memcs Fcl of scence(gls) El- A Unves Eg Demen of Memcs Fcl of scence(gls) El- A Unves Eg Eml: m_m_elo@oo.com; ff@oo.com ; mlm@oo.com ABSTRACT In s e e eene mlle Rcc eons enson meo s een se o consc sees of ole solon- le solons ole ngl fncon solons n comleon solon solons fo nonlne l ffeenl eons we on mn new es of comleon solon solons.e vos comnon of gonomec eoc fncon n eolc fncon solons vos comnon of gonomec eoc fncon n onl fncon solons vos comnon of eolc fncon n onl fncon solons. Kewos: Te eene mlle Rcc eons enson meo Dole solon- le solons Dole ngl fncon solons Comleon solon solons Te (+)- mensonl genele sl- low we eon W Zng (WZ) eon Non-lne eons.. INTRODUCTION In ecen es e ec solons of non-lne PDEs ve een nvesge mn os( see fo emle [- ] ) wo e neese n non-lne scl enomen. Mn owefl meos ve een esene ose os sc s e nvese sceng nsfom [] e Bcln nsfom [] e genele Rcc eon [] e Jco ellc fncon enson [] e eene n- fncon meo [78] e F-enson meo [9] e e-fncon enson meo [] e s- O-D.E. meo [] e eene sn-cos n sne-cosne meos [] e nce Pnleve enson [] e (G l G) enson meo [78] e omoo eon meo [9] n so on. Wng e l [] ve esene e mlle Rcc eons onl enson meo sng wo o moe vles wc ssf ffeen Rcc eons w ffeen mees. Recenl Ze e l [] ve consce mn new es of comleon solon solons o e genele cole nonlne Ho-Ssm eons sng Rcc eons meo. Te ojecve of e esen e s o se e eene mlle Rcc eons enson meo o consc sees of some es of velng wve solons nmel e ole solon- le solons ole ngl fncon solons n comleon solon solons fo some es nonlne l ffeenl eons. SUMMARY OF THE EXTENDED MUTIPE RICCATI EQUATIONS EXPANSION METHOD In s secon we wol le o olne e mn ses of s meo [] s follows: Se. We conse e followng nonlne l ffeenl eon w some scl fel : F... (.) Se. We noce moe genele ns n em of fne onl foml enson n e followng foms: wee n n n j j j (.) j ( j =... n) e consns o e eemne le wle e new vles ssf e followng wo Rcc eons: (.) () n 7

2 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 7 wee n ( = ) e consns sc. Te mees ξ n η e gven wee e consns. Se. Deemne e osve nege n of e foml olnoml solon (.) lncng e ges nonlne ems n e ges-oe l evve ems n e gven ssem eons n en we gve e foml solon. Se. Sse (.) no (.) long w (.) n (.) n en se ll e coeffcens of j ( j =...) of e eslng ssem s nmeo o e eo. We ge n ove-eemne ssem of nonlne lgec eons w esec o n j ( j =... n). Se. Solve e ove-eemne ssem of nonlne lgec eons. We en w elc eessons fo n j ( j =... n). Se. I s well nown e genel solons of e Rcc eons (.) n (.) e lse s follows: () wen n en n n co n n co. co co (.) () wen n en n n co n n co co co (.) () wen n en n n co n

3 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons co n co co (.7) wee n. Smll we cn fn oe genel solons wen n wc e ome ee fo convenence. (v) wen n en ( ). (.8) c c wee c n c e consns.. APPICATIONS In oe o llse e effecveness of e oose meo wo emles n memcl e cosen s follows.. W Zng (WZ) eon: W Zng (WZ) eon cn e wen s v w v v vv w w w vw v v. (.) wee w s e elevon of e we s e sfce veloc of we long -econ v s e sfce veloc of we long -econ. sng e eene mlle Rcc eons enson meo. B lncng e nonlne ems n e ges oe lne l evve ems of (.) we ge n n n n. Ts e nonlne W Zng (WZ) eon (.) ve e followng onl foml ec solons: v w. (.) wee n n ( =... ) e consns o e eemne le. we sse (.) long w (.) n (.) no (.) n se e coeffcens of e ems ( =... =...) o e eo el se of ove-eemne lgec eons w esec o n ( =... ). In e fom 7

4 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 7 9) 8) 7) ) ) ) ) ) ) ) 9) 8) 7) ) ) ) ) ) ) ) ) )

5 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 7 ) ) ) ) ) ) ) 9) 8) 7)

6 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 7 ) ) (.) we solve e ove eemne lgec eons. Conseenl we ge e followng esls: Cse (.) wee s consn. Cse (.) wee s consn Cse (.) wee s consn - - -

7 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 7 Cse (.7) wee s consn Cse (.8) wee s consn Cse (.9) wee s consn Cse 7 (.) wee s consn

8 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 77 Cse 8 (.) wee s consn. Accong o (.) (.) n e genel solons (.) - (.8) lse n e se we on e followng fmles of some new es of solons coesonng o cse fo e nonlne W Zng (WZ) eon (.) Fml. Wen en e ec solon of eon (.) ve e followng foms (.) n n - n n w v (.) co co - co co w v -

9 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 78 Fml. Wen. e ngl fncon solons of eon (.) ve e followng foms: n (.) co co - co co w v Fml. wen n. en we ve e followng esls: (.) - c c w c v c wee n ( = ) e consns sc. Te mees ξ n η e gven wee e consns. We sol on o e solons one n s e e no onl e (.)-(.). We onl ls some new ones coesonng o cse o sow o meo s effcenc n conscng e ec solons gonomec fncon solons of e nonlne W Zng (WZ) eon (.). (.) n n - n w v

10 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 79.. Te (+)- mensonl genele sl- low we eon: In s secon we conse e followng e (+)- mensonl genele sllow we eon n e fom. (.7) In s secon we s e Te (+)- mensonl genele sl- low we eon (.7) sng e eene mlle Rcc eons enson meo. B lncng e nonlne ems n e ges oe lne l evve ems of (.7) we ge n =. Ts e Te (+)- mensonl genele sl- low we eon eons (.7) ve e followng onl foml ec solons: (.8) wee n n e consns o e eemne le. we sse (.8) long w (.) n (.) no (.7) n se e coeffcens of e ems ( =... =...) o e eo el se of ove-eemne lgec eons w esec o n we solve e ove eemne lgec eons. Conseenl we ge e followng esls: (.9) Wee s consn. Accong o (.)(.) n e genel solons (.) - (.8) lse n e se we on e followng fmles of some new es of e ole solon- le solons ole ngl fncon solons n comleon solon solons coesonng o e(+)- mensonl genele sllow we eon. (.7): Fml. Wen. e ole solon- le solons of eon (.7) ve e followng foms: n n co n n co co co (.). -

11 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons 8 Fml. Wen. e ole ngl fncon solons of eon (.7) ev e followng foms: n n co n n co 7 co co 8 (.) Fml. Wen. e comleon solon solons of eon (.7) ve e followng foms n n 9 co n n co co co (..) Wee Fml. Wen we ve e followng esls: c c (.) Wee We sol on o e solons one n s e e no onl e (.)-(.). We onl ls some new ones coesonng o(+)- mensonl genele sl- low we eon o sow o meo s

12 IJRRAS 9 () Deceme El-Bo & l. Solons fo Nonlne Pl Dffeenl Eons effcenc n conscng e ole sol- le wve solons ole gonomec fncon solons n comleon solon solons of e (+)- mensonl genele sl- low we eon (.7) SUMMARY AND CONCUSION In smm e eene mlle Rcc eons enson meo w smolc comon s eveloe o el w nonlne W Zng (WZ) eon (.)n Te (+)- mensonl genele sl- low we eon (.7). Ten wen lng e oose meo o e nonlne W Zng (WZ) eon (.)n Te (+)- mensonl genele sl- low we eon (.7) c ve of ec solons wc ncle () ole sol- le wve solons () ole gonomec fncon solons (c) comleon solon solons e one. O meo cn on e ee ffeen es of vellng wve solons ()() n (c) menone ove REFERENCES []. M.J. Alow n P.A. Clson Solons non-lne Evolon Eons n Invese Sceng Tnsfom Cmge Unv. Pess Cmge (99). []. M.R.M Bcln TnsfomonSnge-Velg Beln (978). []. C.Roges n W.F. Swc Bcln Tnsfomons Acemc Pess New Yo(98). []. Z.Wng n H.Q.Zng A new genele Rcc eon onl enson meo o clss of nonlne evolon eon w nonlne ems of n oe Al.M.Com 8 (7) 9-7. []. Z.Yn n H.Zng New elc sol wve solons n eoc wve solon fo Wm-Boe-K eon n sllow we Ps. ees A 8 () -. []. S. Z.F S.D. n Q.Zo Jco ellc fncon enson meo n eoc wve solons of nonlne wve eons Ps. ees A 89() 9-7. [7]. M.A.Ao Te eene n-meo n s lcons fo solvng nonlne scl moels Al. M. Com 9 (7) [8]. E.G.Fn Eene n- fncon meo n s lcons o nonlne eons Ps.ees A 77() -8. [9]. M.Wng n X. Eene F-enson n eoc wve solons fo e genele Zov eons Ps. ees A () 8-. []. M.Wng n X. Alcons of F-enson o eoc wve solons fo new Hmlonn mle eoncos Solons n Fcls () 7-8. []. J.H.He n X.H.W E-fncon meo fo nonlne wve eons Cos Solons n Fcls () []. E.Ysfogl New sol solons fo e MBBM eons sng E-fncon meo Ps. ees A 7 (8) -. []. X.Z. n M..Wng A s-ode meo fo fnng ec solons of genele KV-mKV eon w ge-oe nonlne ems Ps.ees A (7) - 8. []. M..Wng X.Z. n J..Zng S-ODE meo n sol wve solons fo geoe nonlne Sconge eon Ps. ees A (7) 9-. []. D.S.Wng Y.J.Ren n H.Q. Zng Fe eene sn-cos n sn-cos meos n new non- vellng wve solons of e (+)-mensonl sesve long wve eons Al. M.E-Noes ()7-. []. S..Zng B.W n S.Y.o Pnleve nlss n secl solons of genele Boe- K eons Ps. e. A () -8. [7]. M.Wng X. n J.Zng Te (G /G) enson meo n vellng wve solons of nonlne evolon eons n memcl scs Ps.ees A 7(8) 7-. [8]. S.Zng J.Tong n W.Wng A genele (G /G) enson meo fo e mkv eon w vle coeffcens Ps.ees A 7 (8) -7. [9]. J.H. He Te omoo eon meo fo nonlne oscllos w sconnes Al. M. Com. () []. J.H. He Comson of omoo eon meo n omoo nlss meoal. M. Com. () 7-9. []. Q.Wng Y.Cen n H.Zng A new Rcc eon onl enson meo n s lcon o (+)- mensonl Bges eon Cos Solons n Fcls () 9-8. []. A.K.Kll S.Y.Sovc n I.Ysen Inegl of Kesen-Ksl s sc cole KV-mKV eons: Sngl nlss n J.M. Ps. () []. D.S.Wng n H. Smolc comon n non-vellng wve solons of (+)- mensonl nonlne evolon eons Cos Solons n Fcls (7) 8-9. []. E.M.E.Ze T. A.Nofl n K.A.Geeel A sees of comleon solon solons fo nonlne Ho-Ssm eons sng e genele mlle Rcc eons onl enson meo Al. Anl. 87 (8)

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