Computable Analysis of the Solution of the Nonlinear Kawahara Equation

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1 Diache Lu e al IJCSE April Vol Iue Compuale Aalyi of he Soluio of he Noliear Kawahara Equaio Diache Lu Jiai Guo Noliear Scieific eearch Ceer Faculy of Sciece Jiagu Uiveri Zhejiag Jiagu 3 Chia dclu@uj.edu.c apple_geg@6.com Arac I hi paper we udy he compuailiy of he oluio operaor of he iiial prolem for he oliear Kawahara equaio which i aed o he ype- urig machie. We will prove ha i Soolev pace for he oluio operaor: K C ; : i δ ρ δ ] -compuale. he cocluio eriche he heory of compuailiy. Key word he oliear Kawahara equaio iiial prolem compuaili ype- heory of effeciviy E Soolev pace Ⅰ.INODUCION A pree he compuailiy of oluio of he oliear evoluio equaio have ecome a impora opic o he worer of phyic mechaic life ciece applied mahemaic egieerig ad heoreical compuer. eearchig oudede ad compuailiy of he oluio of he oliear equaio will offer effecive ool for he applicaio of equaio erich heoreical foudaio of compuer ciece ad promoe he developme of compuer ofware. I 98 K.Weihrauch ad oher ealihed a compuaioal model called ype- heory of effeciviy E for hor. K.Weihrauch ad N. Zhog have udied he compuailiy of he geeralized fucio he KdV equaio ad he Schrödiger equaio 3] ] Diache Lu ad oher have udied he compuailiy of he mkdv equaio ]. he oliear Kawahara equaio wa fir propoed y Kawahara i 97 hi equaio ha wide applicaio i phyic uch a i he heory of mageo acouic wave i plama i he heory of log wave i hallow liquid uder ice cover ad o o. I hi paper we will dicu he oliear Kawahara equaio a follow: y u 3 au u β u γ u = c u 3 uy = φ y y where βγ are diperio coefficie a i oliear perurace coefficie ad c > i oud velociy. he paper i orgaized a follow. I Secio we maily review ome aic defiiio lemma ad cocluio of E which are releva o he proof of ecio3. Secio3 i devoed o he proof of he mai heorem. Ⅱ. PELIMINAIES hi ecio we will give a rief iroducio of E. For deail he reader ca refer o ]. Lemma. I Schwarz pace S he fucio a a i ρ δ δ compuale; i δ ρ ρ compuale; δ δ δ compuale. φ φ ad φ φ are 9

2 Diache Lu e al IJCSE April Vol Iue he fucio E E i δ ρ δ compuale for compuale m. 3 he Fourier raform iξ F: S S ϕ π ϕ d ad he ivere Fourier raform F : S i ξ S ϕ π e ϕξ dξ are oh δ δ compuale. Lemma. m e he fucio : C ; S S ua = ud i ρ δ ] ρρδ Lemma.3 ype coverio a compuale. ω Le δi : Σ X i e a repreeaio of he e X i i. Le f X X X : L : = f he f i δ δ δ m ad defie compuale coiuou if ad Defie g : N M M a follow: g = f g = f g where M N he he fucio g i υ γγ compuale. Suppoe h : M M i fucio : N M M a follow: γγ compuale defie a = = h = h he he fucio i υ γγ compuale. N he cocluio aou he compuailiy aove alo ca apply o mulidimeioal pace. Defiiio. 6] For a fied > ad ay fucio fucio: ω y : ] defie modular ω = up ω ] 3 ω ω = 4 L L y ω = ma{ ω } ω N oly if L i δ δ δ] compuale coiuou. δ Where mea he orm of Soolev pace W Lemma.4 primiive recurio Le γ : Y M ad γ : Y M are wo repreeaio υn i admiile repreeaio of N. he we have he followig propoiio: Suppoe f : M M i γγ compuale f : N M M M i υ γ γ γ compuale. N which i alo he orm of pace a follow: X C we coruc a fucio = { ω ]; ω < } he for ayu X u u i hold ha L L y u 6 6

3 Diache Lu e al IJCSE April Vol Iue Defie he operaor i : F f = W φ y = W φ y i f ˆ. Where { ; ˆ = f φ f L } f = f ˆ. L he he Cauchy prolem - are equivale o 3 u u u au u c 3 u β γ = y 7 = uy = G φ Ne we will give ome lemma ad heorem aou eimaor. heorem.6 6] he iiial value prolem of 9- i give y he for ay fied > whe for ay parameer γ β c >here ei a coa C > which oly deped o ad parameer c he i γ β uy = φ y y 8 he liear par of 7-8 i 3 u u u 3 u β γ c = 9 y uy = φ y y he ue he Fourier raform we oai he oluio of 9- i 3 ξ i γξ βξ c i ξ yξ ξ G = e e dξ dξ NoeG = G y hold ha 4 Gy C Lemma.7 6] For ay he eimaor W ϕ C ϕ L L 6 for ay fied > < holdwhere C i poiive coa which oly deped o γ β c. heorem.8 6] Suppoe p = q =. p = = he for ay fied > here p q q Suppoe W ϕ = G ϕ u 3 ei C which deped o γ β c i hold ha where u i he oluio of 9-. Le W y iθ φ = iθ 3 i γξ βξ c ξ ξ ξ e Wheθ = we have i ξ y e ξ ˆ φξ ξ dξdξ 4 W φ C φ 7 Ad for ay g L q p LL y q p L L y we have W g d C g q p τ τ q p L L y L L y τ 8 6

4 Diache Lu e al IJCSE April Vol Iue Ⅲ. MAIN ESUL heorem 3. he iiial value prolem 7-8 defie a oliear map : K C ; which i from he iiial valueϕ o he oluiou ad he oluio operaor i δ ρ δ ] compuale. If he iiial value prolem of 7 iuy he i he eighorhood of apply coracig mappig priciple he iiial value prolem of 7 ca e coruced uually we egi hi from =. Suppoe he oluio u ha ee coruced i he ierval ]. Ne we will prove how o eed he oluio from he eighorhood = iiial value v :. Ne we coider he equaio wih he Le X = ; { u X < } i a all of X ad he radiu i he here ei = γ β c > a ad = γ β c > which mae he a map A ad G are coracig i he eighorhood of. Form 7 we have 4 W C Whe = i 8 ad comie wih 6 we have G ac y y ddyd ac 3 v v v av v c 3 v β γ = y 9 Where C i a coa deped oγ β c. Liewie whe = i 8 we have vy = y he equivalece iegral equaio of 9- i v v = W a W τ v τ dτ Defie wo map A adg G C C : ; ; i defied G = a W τ τ dτ A : C ; C ; A i defied = W G 3 G a W τ τ dτ LL y = ac ac ac LL y So from 6 we have ddy d ddyd 7 G ac 6 6

5 Diache Lu e al IJCSE April Vol Iue From 4 6 ad comie wih i hold ha A 7 C acma{ } 7 7 Noe θ = ma{ } he for ay fied iiial value y le = ad C ee i hold ha 4Caθ < 8 From7we have A < 4 i.e. A X he for ay X A A = G G ac θ 9 O he ai of ad 9he fied poi i he coracig map A aifie v = ad i he oluio of he iiial value of he iegral equaio 7. Proof. By lemma. ad. he fucio: A i ρ δ ρ δ ] ρ δ compuale. Lemma 3.3 he map F A : ρ δ v ρ δ ] compuale. N i Where A = A A i he h ieraio. Proof. By Lemma.4 ad A i compuale we ca ge i. Ne we prove A i compuale i. Le u i he fied poi of A coruc a equece of ieraio a follow: u = A = A A. Selec a uequece of aural umer { j i } i aifie j u u. Becaue which S i dee herefore if we ca compue A i pacehe we i o here ei S aifie ca compue he oluio of iiial value i he followig lemma will how ha he rericio of A o he Schwarz pace S a dee ue of hi rericio will alo e deoed a A. i compuale. Lemma 3. he rericio of he operaor A o S CS ; CS ; S :. Le o we have. u A = A A = W W = W A i ρ δ ρ δ ]ρ δ ] compuale. C u j i A = 63

6 Diache Lu e al IJCSE April Vol Iue u A A A j i ji u u Lemma 3.4 he map F : v θ ] i ρδ ρδ compuale. Becaue = u v i he oluio of 7 i he ierval ] o he oluio i eeded form ] θ o ]. θ Suppoe a β γ c are compuale real umer ad i compuale oc θ are compuale. For ieger z Z we compue he oluio u z θ a ime z θ. Defie ϕ = ϕ = ϕ ϕ = F θ ϕ θ By Lemma 3.4 ϕ = u θ i compuale from ad ϕ ecaue fucio F. i primiive recurio of he From θ we ca compue z Z uch ha zθ z θ. Apply Lemma 3.4 we ca compue u z θ z θ u z θ where F z θ u z θ F =. u ad I Lemma 3.4 we ca prove ha for = he oluio of he iiial value prolem of 7-8 i compuale. So we prove he heorem 3.. ACKNOWLEDGMEN eearch wa uppored y he Naioal Naure Sciece Foudaio of Chia No: 673 ad he Ouadig Peroal Program i Si Field of Jiagu No: 988 ad he Jiagu Provice Naural Sciece Foudaio of Uiveriy No: KJD. EFEENCES ] Diache Lu ad Qigya Wag. Compuig he oluio of he m-koreweg-devrie equaio o urig machie Elecroic Noe i heoreical Compuer Sciece ] K.Weihrauch. Compuale Aalyi. Spriger Berli. 3] N.zhog K.Weihrauch. Compuailiy of geeralized fucio. J.Aoc. for Compuig Machier 3 4: ] K.Weihrauch N. Zhog. Compuig he Soluio of he Koreweg - devrie Equaio wih Arirary Preciio o urig Machie. heore. Compu. Sci 33-3: ] K.Weihrauch N. Zhog Compuig Schrödiger propagaor o ype- urig machie Joural of Compleiy 6 : ] AO huagpig CUI hagi. Eiece ad uiquee of oluio o oliear Kawahara equaio. Aal of Mahemaic 3A:-8. 7] Zhao Xiagqig. Local olvailiy of Cauchy prolem for he Kawahara-BO equaio. Applied mahemaic - a joural of Chiee uiveriie 9 43: