The Similar Structure Method for Solving Boundary Value Problems of a Three Region Composite Bessel Equation

Transcription

1 The Smlar Struture Method for Solvng Boundary Value Problems of a Three Regon Composte Bessel Equaton Mngmng Kong,Xaou Dong Center for Rado Admnstraton & Tehnology Development, Xhua Unversty, Chengdu 69, Chna College of Sene, Southwest Petroleum Unversty, Chengdu, 65, Chna Abstrat Ths paper studes the boundary value problem of three-regon omposte Bessel equaton Frstly, on the bass of smlar struture of the soluton of the boundary value problem of dfferental equatons, the smlar struture method for solvng the lass of omposte boundary value problems s put forward and ts steps are desrbed Seondly, the flow hart s ontruted and the orrespondng program s ompled Fnally, the new method s appled to solve a gven boundary value problem of threeregon omposte Bessel equaton The urve of the soluton of the boundary value problem s omputed The new method s smple and effetve to solve ths lass of boundary value problems Keywords - Boundary value problem; Three-regon omposte Bessel equaton; Smlar kernel funton; Funton of gude soluton I INTRODUCTION The boundary value problem of dfferental equaton s wdely appled n pratal problems, espeally n applaton of ol and gas well testng tehnology The theory and nterpretaton method of well testng analyss has been developed rapdly Therefore, the orrespondng solvng method of the boundary value problem of dfferental equaton s also smultaneously requred to be developed and mproved Wth the sustanable development and mprovement of the theory of smlar struture of soluton of dfferental equatons, ts applaton n the feld of engneerng s also more wdespread So t plays an role n the soluton to reservor flow model At the begnnng of ths entury, the thought of smlar struture of the soluton of the boundary value problem of dfferental equaton began to form n Ref 5 Some gratfyng results have been aheved On the bass of the smlar struture of soluton of boundary value problem of dfferental equaton, the researhers (Ref, Ref4, Ref6, Ref4) smplfed solutons of boundary value problems of seond-order lnear homogeneous dfferental equatons and seond-order partal dfferental equatons It an help to understand nherent the laws of analyt soluton Furthermore, the unfaton between the smlarty of soluton and the orrespondng numbers and shapes has harmonously mproved the trad of mathematal theory Wth the analyss epresson of fed soluton problems on a lass of omposte Bessel equaton and omposte modfed Bessel equaton, Ref and Ref4 ganed the soluton s formal smlarty Ths smlar struture of soluton eplaned that solutons of the lass of equatons an be gven by the produt of several fratons and the graphs have smlarty too By analyzng the soluton of the reservor pressure and dmensonless reservor pressure dstrbuton n Laplae spae, whh was amed at the well test analyt model of omposte reservor, Ref7 dsovered the smlar struture of the soluton s form of the omposte reservor n the three knds of outer boundary ondtons (nfnte, onstant pressure and losed) Furthermore, ths paper made the theoretal graph and analyzed the nfluene of the wellbore storage and skn effet on dmensonless reservor pressure and dmensonless bottom-hole pressure by usng numeral method of nverson Ref5 studed the smlar struture of soluton n the Laplae spae for a lass of omposte parabol partal dfferental equatons wth the nfnte outer boundary ondton and the knd of nner boundary ondton n onneton wth tme The work s essental to understand nherent laws of relevant engneerng sene and desgn pratal analyss software Amng at the boundary value problem of seond-order homogeneous lnear ordnary dfferental equatons(systems) and the med problem for seond order homogeneous lnear partal dfferental equatons(systems), Ref8 revewed some results of prelmnary eploratons wth regard to the smlar struture theory of ther solutons or ther Laplae spae solutons and analyzed the formaton dea of the smlar soluton and ts sgnfane of researh and took apart the relaton between the smlar struture of ts soluton and governng equaton and the boundary ondtons Ths paper also found a way to onstrut the soluton of defnte problems by makng use of the formula of smlar struture together wth the smlar kernel funton and ntrodues the estng ahevements whh have been appled to permeaton flud mehans The perolaton model of omposte reservor was establshed n Ref9, whh onsders the effetve well-bore radus and well-bore storage By Laplae transform, the DOI 5/ IJSSSTa798 8 ISSN: onlne, 47-8 prnt

2 eat solutons of reservor pressure and bottom-hole pressure were obtaned under dfferent boundary ondtons n Laplae spae Aordng to the theory of smlar struture of soluton, Ref9 defned the kernel funtons whh are only relevant to two lnear ndependent solutons of governng equatons and outer boundary ondtons, what s more, dfferent outer boundary ondtons orrespond to dfferent smlar kernel funtons There s a smlar struture among the solutons under three outer boundares Based on the analyss of the souton for a boundary value problem of the seond-order lnear homogeneous dfferental equaton,ref studed the smlar struture of soluton and smlar kernel funtons, and put forward a new method of solvng ths lass boundary value problem the smlar onstrutve method The method s an nnovatve dea and a smple effetve method of solvng the boundary value problem of the dfferental equaton The boundary value problem of two ponts was onsdered for seond-order lnear homogeneous dfferental equaton n Ref The estene and unqueness theorems were proved The smlar struture and smlar kernel funton of the solutons were found Based on the analyss of the soluton of a boundary value problem of seond-order omposte lnear homogeneous dfferental equaton, Ref studed smlar kernel funtons and the smlar struture of the soluton and proposed a new method for solvng ths lass boundary value problem Ref solved a lass of boundary value problems of the omposte frst Weber system Soluton wth a form of ontnued fraton produt to boundary value problem of the omposte frst Weber system was obtaned Then a new method was proposed solvng the omposte boundary value problem Smlar Construtng Method On the bass of smlar struture of soluton of a seondorder lnear dfferental equaton boundary value problem, Ref proposed a new smple soluton smlar onstrutve method of soluton and summed up ts detaled steps A mathematal model of fratal reservor wth spheral flow under three knds of outer boundary ondtons (nfnte, onstant pressure and losed) was set up, n whh nfluenes of skn fator and well-bore storage are taken nto onsderaton And then SCMS s appled to solve t In ths paper, the boundary value problem of three-regon omposte Bessel equaton s studed In seton, the boundary value problem of three-regon omposte Bessel equaton s gven In seton, on the bass of smlar struture of soluton of boundary value problem of dfferental equaton, a new method for solvng the lass of boundary value problem s proposed In seton, the steps of the method are summarzed In seton 4, the new method s appled to solvng a gven boundary value problem of three-regon omposte Bessel equaton and the urve of the soluton of the boundary value problem s drawn II PROPOSED BOUNDARY VALUE PROBLEMS AND PRELIMINARY KNOWLEDGE In ths paper, the followng boundary value problem of three-regon omposte Bessel equaton s studed:,, a b b d y y y y y y y y y Ey EF y D a y y, y y y y, y y My Ny d b b b b () Where D, E, F, M, N, a,b,, d,,,,,, are onstants and, M N and D J and Y,, are two lnear ndependent solutons of seond-order lnear dfferental y y y,,, equatons Jn, Y n are respetvely the frst and the seond lass of Bessel funtons of order n [5] Defnng a bnary funton as below: J Y Y J (),, + - m n mn m n m n Ths paper leads nto funtons of gude soluton as (4) (5) (6),,,, DOI 5/ IJSSSTa798 8 ISSN: onlne, 47-8 prnt

3 Where denotes nner regona b, denotes mddle regonb, denotes outer regon d III THE MAIN THEOREM AND ITS PROOF Theorem If the boundary value problem () has unque soluton, then solutons of nner, mddle and outer regons are epressed respetvely as follows (the detaled proof proess s presented n Append A): y D a E F F a a b (), bb, () y D E F a b,a, b,a, b F a b y D E F a F a, bb,,, b ab, ab, b, b,,,,, d, and Where are alled smlar kernel funtons of outer, mddle and nner regons respetvely and they are epressed as follows:,,,, M d N d d M d N d,,,,,, b, b, b,,,, b, b b, ab, b ab,,,,, () (4) (5) a b (6) Corollary In the boundary value problem (), f the outer boundary ondton s y d (e M, N ), the orrespondng smlar kernel funton of outer regon s d, d,,, Corollary In the boundary value problem (), f the y d (e M, N ), outer boundary ondton s s the orrespondng smlar kernel funton of outer regon d, d,,, Corollary In the boundary value problem (), f the left boundary ondton s the seond boundary ondton (e y ), the soluton of nner regon of the boundary a value problem () s the smlar kernel funton of nner regon (e ) Corollary 4 The frst ontnued fraton, whh belongs to the struture of the soluton (e Eq ()) to the boundary value problem (), has the followng property (The detaled proess of provng n Append A): y Fy a D E F a IV STEPS OF THE NEW METHOD Aordng to soluton proedures of the above boundary value problem, t s easy to ndue the steps of the new method for solvng the boundary value problem of threeregon omposte Bessel equaton Detaled steps are as follows: Step Solvng governng equatons Two lnear ndependent solutons J, Y,, of governng equatons are obtaned by solvng governng equatons of nner, mddle and outer regons of the boundary value problem () respetvely Step Construtng funtons of gude soluton We onstrut the funtons of gude soluton of nner, mddle and outer regons,,,, by usng two lnear ndependent solutons J,Y,, of governng equatons of nner, mddle and outer regons of the boundary value problem () respetvely, as shown Eq() Other funtons of gude soluton an be obtaned by alulatng partal dervatves of,,,, to, respetvely, as shown Eq(4)-(6) Step Construtng smlar kernel funtons of nner, mddle and outer regons DOI 5/ IJSSSTa798 8 ISSN: onlne, 47-8 prnt

4 Frstly, the smlar kernel funton of outer regon of the boundary value problem () an be strutured by usng funtons of gude soluton of outer regon and oeffents M, N of the homogeneous outer boundary ondton, as shown Eq(4) After that, the value of s alulated Seondly, the smlar kernel funton of mddle regon an be strutured by usng funtons of gude soluton of mddle regon, oeffents, of two onvergene ondtons of mddle and outer regon and, as shown Eq(5) After that, the value of b s alulated Fnally, the smlar kernel funton of nner regon of the boundary value problem () an be strutured by usng funtons of gude soluton of nner regon, oeffents, of two onvergene ondtons of nner and mddle regons and After that, the value of, as shown Eq(6) a s alulated Step4 Obtanng soluton of the boundary value problem Aordng to Eq(), Eq() and Eq(), solutons of the nner, mddle and outer regons are obtaned by assemblng oeffents D, E, F of the non-homogeneous nner boundary ondton, smlar kernel funtons,,, values of a,, b, funtons of gude soluton of nner and mddle regons, oeffents,,, of four onvergene ondtons of three regons, respetvely The flow hart (Fg ) of algorthm of steps of the above method s: Fg The flow hart of algorthm of the method DOI 5/ IJSSSTa ISSN: onlne, 47-8 prnt

5 The flow hart of algorthm learly presents the relatonshp between solutons of three regons and smlar kernel funtons, funtons of gude soluton, boundary ondtons and onneton ondtons It elaborately llustrates the soluton proedure of solvng the lass of boundary value problems of three-regon omposte Bessel equaton V THE EXAMPLE The followng boundary value problem s solved: y y y, 4 y y y y y y y 7y y y 4, y 4 y 4 4 y y 8, y 8 y 8 8 y y, 4 8 4, 8 (4) Comparng wth the boundary value problem () and (4), we know that,,, a, b 4, 8, d,,,,, D, E, F, M, N The boundary value problem (4) has unque soluton (the result s gven n Append B) Aordng to the new method, we solve the boundary value problem (4) Step Solvng governng equatons Two lnear ndependent solutons J, Y,, [5] of governng equatons are obtaned by solvng governng equatons of nner, mddle and outer regons of the boundary value problem (4), respetvely Step Construtng funtons of gude soluton Aordng to Eqs()-(6), funtons of gude soluton of nner, mddle and outer regons are strutured by usng two lnear ndependent solutons J, Y,, of governng equatons of nner, mddle and outer regons of the boundary value problem (4) respetvely as follows:, J Y Y J,, J Y Y J,, J Y Y J,, J Y Y J,, J Y Y J,,, J Y Y J J Y Y J,, J Y Y J J Y Y J,, J Y Y J J Y Y J JYYJJYYJ,, J Y Y J,, J Y Y J J Y Y J,, J Y Y J J Y Y J 4,, J Y Y J J Y Y J JYYJJYYJ Step Construtng smlar kernel funtons of nner, mddle and outer regons Aordng to the Eq(4), the smlar kernel funton of outer regon of the boundary value problem (4) s strutured as follows: Thus,, 8 8, 8,,,,, 8, 8,,, 8,, 8, 8, Aordng to the Eq(5), the smlar kernel funton of mddle regon of the boundary value problem (4) s strutured as follows: Thus,8 8, ,8 8 4,8,,,, 4,8 8 4,8,, 4,, 4,8 8 4,8 Aordng to the Eq(6), the smlar kernel funton of nner regon of the boundary value problem (4) s strutured as follows:,4 4,4 4,4 4,4,,,, DOI 5/ IJSSSTa ISSN: onlne, 47-8 prnt

6 Thus,4 4,4,,,,,4 4,4 Step4 Obtanng the soluton of the boundary value problem (4) Aordng to Eq(), Eq() and Eq(), solutons of nner, mddle and outer regons of the boundary value problem (4) an be obtaned respetvely as follows: y 7 4, 4, 4 4 8, 4,4, 8,8 y 7 4,,4,,4 y 8 Aordng to the flow hart of the algorthm of the method, orrespondng program s ompled by usng the MATLAB language Then the urve of the soluton (Fg) of the boundary problem (4) s drawn by runnng the program on the omputer as below: 7 4,,4,,4 8, 4,8, 4,8 Fg The urve of soluton of the boundary value problem (4) VI CONCLUSIONS () When dealng wth the boundary value problem of three-regon omposte Bessel equaton, only two lnear ndependent solutons J, Y,, of governng equatons of nner, mddle and outer regons respetvely are obtaned Then the boundary value problem an be solved by the obtaned method () It s lear that the method s a onvenent, effetve and reatve way to solve the boundary value problem of three-regon omposte Bessel equaton () Smlar strutures of soluton of nner, mddle and outer regons learly show the relatonshp between solutons and smlar kernel funtons, funtons of gude soluton that s generated by usng two lnear ndependent solutons of governng equatons, boundary ondtons and onneton ondtons (4) Aordng the new method, a orrespondng program s ompled It s appled to drawng a graph of the soluton of the boundary value problem (4) And the graph learly llustrates the soluton of the boundary value problem ACKNOWLEDGMENT Ths paper s supported by Natonal Natural Sene Foundaton (6787)Sentf Researh Fund of Shuan Provnal Eduaton Department of Chna under Grant (No6ZB6) and (No6ZA5) Fund of Shuan Provnal Sene and tehnology Department (6GZ 99)Laboratory of Intellgent Network Informaton Proessng ( Noszjj5-6/6) REFERENCES [] CHEN Z-hun, LIU Peng-hu, LI Shun-hu The Smlar Struture of Composte Bessel Equaton on Fed Soluton Problem (n Chnese) [J] Journal of Chongqng Tehno Busness Unversty (Natural Sene Edton), 6, ():-4(to 4) [] Cu-Cu Sheng, Jn-Zhou Zhao, Yong-Mng L, Shun-Chu L and Hu Ja Smlar Construton Method of Soluton for Solvng the Mathematal Model of Fratal Reservor wth Spheral Flow [J] Journal of Appled Mathemats, vol, Artle ID 98, 8 pages, [] Dong Xaou, L Shunhu, Gu Dongdong, Pu Jun, L Huhun Smlar Construtng Method for Solvng the Boundary Value Problem of the Composte Frst Weber equaton [J] Ameran Journal of Appled Mathemats and Statsts,, (4):76-8 [4] JIA Mnhu, LI Shunhu The Smlar Struture of Soluton Dfferental Equaton on Boundary Value Problem (n Chnese) [J] College Mathemats, 5, (5):7-9 [5] L Shunhu, Ja Mnhu The Formal Smlarty of Solutons on the Class of Dfferental Equaton [J] Journal of UEST of Chna, 4, (Supp):95-98 [6] LI Shun-hu The Formal Smlarty of Solutons to the Class of - order Partal Dfferental Equaton n the Laplae Spae (n Englsh) [J] Journal of Xhua Unversty (Natural Sene Edton), 7, 6(4):8-86 [7] LI Shun-hu, ZHENG Peng-she, ZHANG Yu-fe The Smlar Struture of Pressure Dstrbuton n the Composte Reservor (n Chnese) [J] Journal of Mathemats n Prate and Theory, 8, 8():-8 [8] LI Shun-hu Prelmnary Eploraton and Prospets of the Smlar Struture of Solutons of Dfferental Equatons (n Chnese) [J] Journal of Xhua Unversty (Natural Sene Edton),, 9():-6(to8) [9] L Quanyong, L Shunhu, L We, Tang Ybn Study on perolaton model of omposte meda reservor based on smlar struture of soluton (n Chnese) [J] Fault-Blok Ol & Gas Feld,, 8(5): 6-65 DOI 5/ IJSSSTa ISSN: onlne, 47-8 prnt

7 [] L Shunhu, Lao Zhjan Construtng the Soluton of Boundary Value Problem of the Dfferental Equaton wth ts an Arbtrary Nontrval Soluton (n Chnese) [J] Journal of Shuan Unversty (Natural Sene Edton),, 49(6):9- [] L Shun-hu, Wu Xao-qng Several Important Propertes for the Boundary Value Problems of Seond-order Lnear Homogeneous Dfferental Equaton (n Chnese)[J] Journal of Xhua Unversty (Natural Sene Edton),, (): -6 [] L Shunhu The Smlarty Struturng Method of Boundary Value Problems for Composte Dfferental Equatons [J] Journal of Xhua Unversty (Natural sene Edton),, ():-6 [] Lu Shsh, Lu Shda Speal Funton [M] Bejng: Chna Meteorologal Press, [4] LIU Peng-hu, CHEN Z-hun, LI Shun-hu Smlar Struture of Fed Soluton Problems for Composte Abnormal Bessel Equatons (n Chnese) [J] Journal of Xhua Unversty (Natural Sene Edton), 6, 5():-6 [5] SU Jan-peng, LI Shun-hu, LI Cheng-je The Smlar of Solutons n the Laplae Spae of Composte Parabol Partal Dfferental Equaton [J] Journal of Zaozhuang Unversty, 9, 6():6- APPENDIX A In ths seton, the detaled proof of theorem s dsussed J and Y v v are two lnear ndependent solutons of seond-order lnear homogeneous dfferental y y y,,, equatons It s unversally known that general solutons of governng equatons of nner, mddle and outer regons of the boundary value problem () are [5] (,,) y AJ BY (A) By substtutng Eq(A) nto nner, mddle and outer boundary ondtons and four onneton ondtons of the boundary value problem (), we obtan the followng equatons respetvely: EJ a EF J a J a A a EY a EF Y ay a B D a J b A Y b B J b A Y b B (A) (A) J bj b A Y b Y b B b b b b J b J b A Y b Y b B J A Y B J A Y B (A4) J J A Y Y B J J A Y Y B MJ d N J d J d A MY d d d N Y d Y d B (A5) (A6) (A7) Aordng to the estene and unqueness of soluton of the boundary value problem (), we know that the oeffent determnant of lnear equatons (Eqs(A)- (A7)) about undetermned oeffents s not equal to zero, and E M, a, b, b,,, dm, a, b, b,,, d M, a, b, b,,, d M, a, b, b,,, d N, a, b, b,,, d N, a, b, b,,, d N,a, b, b,, d, N, ab,, b,, d, EF M, a, b, b,,, d M, a, b, b,, d, M, ab,, b,, d, M, ab,, b,, d, N, ab,, b,,, d N,a, b,b,, d, N, ab,, b,, d, N, ab,, b,, d, (A8) Values of A, B, A, B, A, B an be obtaned by usng the Cramer rule as follows: A DY b Y b Y b b E, b a b,,,, (A9) E, a b b EF, a b EF, a b b B DJ b J b J b b E b, a b,,,, E, a b b EF, a b EF, a b b (A) D A M Y, b b, d M Y Y, b b,,,, d, NY, bb,, d, N Y Y, bb,, d, (A) DOI 5/ IJSSSTa ISSN: onlne, 47-8 prnt

8 D B MJ, b, b,, d M J J, bb,, d, NJ, bb,, d, N J J b, b, d,, (A) D A MY d N Y d Y d, b b, d (A),, D B MJ dn J d J d, b, b,, d (A4) By substtutng values of A, B, A, B, A, B (Eqs(A9)- (A4)) nto Eq(A) and usng the smlar kernel funton of outer regon Eq(4), the smlar kernel funton of mddle regon Eq(5) and the smlar kernel funton of nner regon Eq(6), solutons of nner, mddle and outer regons of the boundary value problem () are obtaned respetvely, e Eq(), Eq() and Eq() APPENDIX B In ths seton, the boundary value problem (4) has unque soluton, whh s proved J and Y are two lnear ndependent solutons yy y governng equaton of nner regon J and Y are two lnear ndependent solutons governng equaton of mddle regon y y y J and Y are two lnear ndependent solutons governng equaton of outer regon y y 4 y Aordng to Append A, the orrespondng oeffent matr of lnear equatons about undetermned oeffents A, B, A, B, A, B s: J C 7J Y 7Y J 4 Y 4 J 4 Y 4 J 4 Y 4 J 4 J 4 Y 4 Y 4 J 8 Y 8 J 8 Y 8 J 8 J 8 Y 8 Y 8 J 8 J 8 Y 8 Y J J Y Y The row smplest form of matr C s below: It s lear thatrc 6 numbers of undetermned oeffents So the boundary value problem of (4) has unque soluton DOI 5/ IJSSSTa ISSN: onlne, 47-8 prnt