The Similar Construction Method of Non-Homogeneous Boundary Value Problems for Second-Order Homogeneous Linear Differential Equations

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1 Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Volum 3 Issu Novm 5 PP 9-5 ISSN X Pint & ISSN Onlin Th Simil Constution Mtho o Non-Homognous Boun Vlu Polms o Son-O Homognous Lin intil Equtions Shn Ji ptmnt o Inomtion Engining Wuhn Businss Univsit Wuhn Chin s_ton@sohu.om Xu ongu Institut o Appli Mthmtis o XiHu Univsit Chngu Chin hnl@63.om Li Shunhu Institut o Appli Mthmtis o XiHu Univsit Chngu Chin lishunhu@63.om Astt: Fo lss o th non-homognous BVP o th son-o homognous lin intil qution s on th tho o th solutions simil stutu th impts o h pts o th BVP th qution n oun vlu onitions on th stutu o its solutions nl. A nw mtho to solv suh polms is psnt whih is in s simil onstutiv mtho n th til stps n mpls givn. Kwos: Boun vlu polm o intil qution Boun vlu onition Simil knl untion Simil stutu Simil Constution Mtho. INTROUCTION In th poss o solving lssil spg qutions sstms o spg qution som kins o son-o intil qutions n th sstms o son-o intil qution thi solutions o BVP hv th simil stutu [-5. Howv th BVP th stu is homognous ight oun vlu onition n th non-homognous oun vlu onition. Tho this pp ouss on th BVP o son-o homognous lin intil qutions whih non-homognous th lt n ight oun vlu onitions. Ou pln o this pp is s ollows. In S. stuis th simil stutu o th solutions o son- o homognous lin intil qutions stui n th poos o th unmntl thom givn. In S.3 nw mtho to solv suh BVP-Simil Constution Mtho is psnt n th til onstut stps iv nling n utiliing th stutu o th solutions. In S.4 n mpl is intou to monstt th onstut poss o solutions.. THE SIMILAR STRUCTURE OF THE SOLUTIONS OF BOUNARY VALUE PROBLEMS BVP in this pp is s ollow: p q [ [ wh oth l nums p C [ q C [. Fo inonvnint stu th pvious BVP is split into th ollowing two BVP: p q [ [ ARC Pg 9

2 Shn Ji t l. n [ [ p q Wh th ight vlu onition o th BVP is homognous n th light vlu onition o th BVP3 is homognous. Lmm Supposition Pinipl Th sum solution o BVP n 3 is th solution o BVP vi vs. Poo: Suppos tht Y Y th solutions o BVP n 3 lt Y Y Y so tk q. 4 into th Boun Vlu Polm q. : Y p Y q Y 5 Y Y 6 Y Y 7 u to qs. 5 to 7 Th sum solution o BVP n 3 is th solution o BVP vi vs. Fom Lmm solving th BVP n tnsom into solving th BVP n 3 sptivl. Th simil stutu o th solutions o BVP n 3 will isuss sptivl. Lmm B two linl inpnnt solutions n to th qution o BVP in th untion : 8 ptil ivtiv untion o : Thom I th BVP hs uniqu solution thn th solution hs th unii om s ollows i.. simil stutu o th solution : Y wh th simil knl untion : [ Poo: Suppos tht n two lin inpnnt solutions o th init Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg

3 Th Simil Constution Mtho o Non-Homognous Boun Vlu Polms o Son-O Homognous Lin intil Equtions qution in th Boun Vlu Polm q. Gnl solution o th qution o BVP : Y C C wh C C it onstnts whih tmin th oun onitions o BVP. B th lt oun vlu onition o q. [ C [ C n th ight oun vlu onition [ C C [ 4 th qs. 3 n 4 omin to otin lin qutions s ollows: C C Bus th Boun Vlu Polm q. hs uniqu solution thn [ [ [ [ B th Lmm n pss s: 3 5 [ 6 u to Cm Rul C n C : C C [ [ 7 Sustitut th q. 7 into th q. :. Th q. n otin t uing n soting th q. 8. Cooll Th ist tion o q. is solution o th Boun Vlu Polm whih lt oun vlu onition is hng s ollowing: Thom I th Boun Vlu Polm q. 3 hs uniqu solution thn th solution hs unii om i.. th solution hs simil stutu: Y 8 9 Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg

4 Shn Ji t l. wh th simil knl untion : Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg Th poo o th Thom is th simil to th Thom. Cooll Th ist tion o q. is solution o th Boun Vlu Polm whih ight oun vlu onition is hng s ollowing: B th lmm n th thoms n it is not iiult to gt th ollowing thom. Thom 3 I th BVP hs uniqu solution thn th solution hs th stutu s ollows: Y wh n in in th qs. 9 n. 3. THE SIMILAR CONSTRUCTION METHO OF SOLVING BVP. Though osving n nling th simil stutu qs. n 3 n th simil knl untion qs. n th simil stutus qs. n 3 hv th stutul similit n ontin th simil knl untion i t i =.. Fom th qs. 8 n th simil knl untion i t is otin th gui untion n th ltight oun vlu onition. 3. Fom th qs. 8 n 9th gui untion n otin two linl inpnnt solutions o th qution o BVP n th gnting untion o solutions is got intiting th gui untion. 4. Th thom 3 shows tht th oiints o lt n ight oun vlu onitions ssml to onstution th solutions o th BVP whih no long ns omplit lultions. B nling th ov itms it is s to otin nw mtho-simil Constution Mtho o solving th non-homognous BVP. Without th omplit ivtion n poo w just two linl inpnnt solutions o th Boun Vlu Polm qution to otin th solution o th qution o BVP n th oiints o oun vlu onitions th solution o BVP is onstut s ollows stps: Stp. Th q. 8 is viw s th untion o gui two linl inpnnt solutions n o th Boun Vlu Polm qutions. Stp. Th gnting untion o solutions is otin us o th q. 9 n th untion o gui. Stp 3. Th simil knl untion n gnt th gnting untion o solutions i j i j = n th oiints o oun vlu onitions n thn n lult. Stp 4. Th oiints n o oun vlu onitions n n ssml to otin th solution Y o th Boun Vlu Polm q. using th q. 3. 3

5 Th Simil Constution Mtho o Non-Homognous Boun Vlu Polms o Son-O Homognous Lin intil Equtions 4. APPLICATION EXAMPLE In this stion w will solv th ollowing BVP[5 with Simil Constution Mtho. [ C lim o R o R Wh In this pp th onstnt pssu o th outsi pool oun is just pov o th Boun Vlu Polm qs.4 5 n 6 n th ininit n los outsi pool oun th sm s ov ov. Whn out oun onition is onstnt pssu i.. th ight oun vlu onition th solution stps n onstut s ollowing. R Stp. Th q. 8 is viw s th untion o gui two linl inpnnt solutions n o th q.4. sinh 7 Stp. Th gnting untion o on solution is otin us o th q. 5 n th untion o gui. osh 8 Stp 3. Th lt oun vlu onition q.5 is ppopitl om s: [ CZ 9 Z W omp it with th Boun Vlu Polm q. n lt C thn n otin th simil knl untion: n R R sinh osh R R tnh R 3 Stp 4. Th oiints o oun vlu onitions n th simil knl untion ssml to otin: An... C Z 3 3 Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg 3

6 Shn Ji t l. So w gt th sult: C Th q. 34 is th sm s th qution in th n[6. Howv th simil onstution mtho is us o solving whih voi th ompl lultions. u to th lut solving stps o this mtho i this mtho is ppli to th nlsis sotw it will pl multipli t. ACKNOWLEGEMENTS This wok ws suppot th Ntul Sin K Pojts o th Sihun Eution Buu o Chin un Gnt No.ZA64. REFERENCES [ Li S.C. Hung B.G. Wng N.T. t l. Anlsis on th solution o wll tst mol o oul pmilit svoi in Chins Jounl o Southwst Ptolum Institut.36. [ Li S.C. Hung B.G. Li X.P. t l. Rsh o omposition svoi pssu istiution in Chins Fult-Blok Oil & Gs Fil. 86. [3 Li S.C. Hung B.G. Wng N.T. Lpl sp solution o th ottom hol pssu in oul-poosit omposit onin svoi in Chins Jounl o Xin Ptolum Institut Ntul Sin Eition.73. [4 Li S.C. A solution o tl ul poosit svoi mol in wll tsting nlsis in Chins Pogss in Eplotion Gophsis. 55. [5 Li S.C. Hung B.G. Li X.P. Anlsis o pssu istiution o wll in ul-poosit omtions in Chins Wll Tsting.5. [6 Li S.C. A mol solution o tsting nlsis in tl ul poosit svois with onstnt pssu out oun in Chins Ptolum illing Thniqus.3 3. [7 Li S.C. Wll tst mol o tl ul poosit los svoi in Chins Xinjing Ptolum Golog.4 3. [8 Zhng T.P. Li S.C. Xu Y. A mol solution o tsting nlsis in tl omposit svoi with onstnt pssu out oun Jounl o Nothst Noml Univsit in Chins.35Supp. 3. [9 Hung B.G. Li S.C.Li X.P. Non-Coss-Flow Multil Rsvoi Pssu Pomn istiution Snthsis Rsh in Chins Ptolum Golog & Oilil vlopmnt in qing [ Zhng J.J. Li S.C. On Kin o Funtion tht Posssss th Chtisti o Ftl in Chins Jounl o Chin Wst Noml Univsit Ntul Sin Eition [ Tin J.. Li S.C. Th Foml Similit o Solutions in th Lpl Sp on th Clss o Qusilin Ptil intil Eqution Mthmtil Tho n Applitions.4 4. [ Li S.C. Ji M.H. Th Foml Similit o Solutions on th Clss o intil Eqution Jounl o Eltoni Sin n Thnolog o Chin 4. [3 Zhng P.S. Li S.C. Zhng Y.F. Th Foml Similit o Solutions on th Clss o Qusilin Oin intil Eqution with th Boun Conitions inluing pmti in Chins Jounl o Nothst Noml Univsit Ntul Sin Eition [4 Ji M.H. Li S.C. Th Foml Similit o Solutions in th Lpl Sp on th Clss o Flui Flow intil Eqution Jounl o Eltoni Sin n Thnolog o Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg 4 34

7 Th Simil Constution Mtho o Non-Homognous Boun Vlu Polms o Son-O Homognous Lin intil Equtions Chin.3 5. [5 Ji M.H. Li S.C. Th Simil Stutu o Solution intil Eqution on Boun Vlu Polm in Chins Collg Mthmtis in Chins [6 Zhng P.S. Li S.C. Zhng Y.F. Th Solution s Stutu o Tp o Oin intil Eqution Sstm with Clos Right Boun Conitions in Chins Jounl o Xihu Univsit [7 Chn Z.C. Liu P.H. LI S.C. Th simil stutu o omposit Bssl Eqution on i solution polm in Chins Jounl o Chongqing Thno Businss Univsit Ntul Sin Eition [8 Li S.C. Yi L.Z. Zhng P.S. Th Simil Stutu o intil Equtions on Fi Solution Polm in Chins Jounl o Sihun Univsit Ntul Sin Eition [9 Li S.C. Th Simil Stutu o Solution o Son-o Lin Homognous intil Equtions with Constnt Coiints on th Boun Vlu Polm in Chins Jounl o Xihu Univsit Ntul Sin Eition [ Zhng P.S. Li S.C. Th Foml Similit o Solutions on th Clss o Qusilin Oin intil Eqution Sstm inluing pmti in Chins Atomi Eng Pulishing Compn Mthmtis n its Applitions 7. [ Su J.P. Li S.. LI C.J. Th Simil o Solutions in th Lpl Sp o Composit Poli Ptil intil Eqution Jounl o Zohung Univsit 9 6:6-. [ Li S.C. Plimin Eplotion n Pospts o th Simil Stutu o Solutions o intil Equtions in Chins Jounl o Xihu Univsit Ntul Sin Eition. 9. [3 Li S.C.Th Simil Stutu o Solution to th Boun Vlu Polm o Son-o Lin Homognous intil Equtions in Chins Jounl o Xihu UnivsitNtul Sin Eition [4 Chi Y. Li S.C. Yn J. Simil Stutu o Bssl Eqution with Pmt on Boun Vlu Polm in Chins Jounl o lin Jiotong Univsit.35. [5 Chn Z.R. Li S.C.. Th Simil Stutu Mtho Solving th Boun Vlu Polm o Bssl Equtions Jounl o Sihun Noml Univsit Ntul Sin [6 Xu Li Li S.C. Shng C.C. Simil Stutu o Flow Etiv Wll Rius Mol though Homognous Rsvoi Solutions Jounl o Chongqing Ntul UnivsitNtul Sin.84. AUTHOR S BIOGRAPHY Shn Ji Fml om njingkou Cit o Hui PovinChin. Now ssoit posso with mst g woking in ptmnt o Inomtion Engining o Wuhn Businss Univsit. Min sh ous: mthmtis ppli mthmtis n omput pplitions. Intntionl Jounl o Sintii n Innovtiv Mthmtil Rsh IJSIMR Pg 5