Similar Constructing Method for Solving the Boundary Value Problem of the Compound Kummer Equation

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1 Amercan Journal of Mathematcal Analyss, 05, Vol. 3, No., Avalable onlne at Scence and Educaton Publshng DOI:0.69/ajma-3--3 Smlar Constructng Method for Solvng the Boundary Value Problem of the Compound Kummer Equaton Duo Zhang, Shunchu L,*, Dongdong Gu College of Scence, Xhua Unversty, Chengdu, Chna Bejng Dongrunke Petroleum Technology Co., Ltd., Bejng, Chna *Correspondng author: Receved Aprl 0, 05; Revsed Aprl 3, 05; Accepted Aprl 8, 05 Abstract Ths artcle s devoted to analyze the compound Kummer equaton. Through the structure of left regon and rght regon smlarty kernel functon, have smlar structure of soluton for ths boundary value problems. The left feld soluton of ths knd of boundary value problems can be obtaned by combng the coeffcents of boundary value problems on the left and the left area smlar kernel functon. The rght feld soluton of ths knd of boundary value problems can be expressed by assemblng the left, the rght area smlar kernel functon, the left lead soluton functons and the coeffcents of the convergence condtons. Thus put forward the smple method whch was named smlar constructve method to solve the boundary value problems of the compound Kummer equaton. The put forward of ths method provdes the convenence for solvng ths knd of dfferental equaton boundary value problems. Keywords: boundary value problem, smlar structure, smlar kernel functon, smlar constructve method, compound Kummer equaton Cte Ths Artcle: Duo Zhang, Shunchu L, and Dongdong Gu, Smlar Constructng Method for Solvng the Boundary Value Problem of the Compound Kummer Equaton. Amercan Journal of Mathematcal Analyss, vol. 3, no. (05): do: 0.69/ajma Introducton The boundary value problems of dfferental equatons always are needed n the calculaton of many physcal and engneerng [,,3,4,5]. So t s not just the ntal value problems of dfferental equaton, boundary value problems are also the focus of our study. Compound boundary value problem of Kummer equaton s a mathematcal problem that closely assocated wth the practcal applcaton. But when n soluton of the compound boundary value problem of Kummer equaton, we often encounter the complcated solvng processe. So t s nconvenence for the practcal applcaton. Some boundary value problems of dfferental equatons can be wrtten n the form of soluton of contnued fractons whch was named smlar structure type. And ths concluson n 004 has been proved [6,7,8,9,0]. For example, the boundary value problem of Kummer equaton, the boundary value problem of Euler hypergeometrc equaton. Then the soluton of the boundary value problem for composte Kummer equatons whether t has the above propertes? Ths artcle wll study the followng compound confluence hypergeometrc equaton boundary value problem: xy'' + ( γ x) y' y ( 0 a < x < c) xy'' + ( γ x) y' y 0( c < x < b) [ Ey+ ( + EF) y'] x a D () y x c my x c y ' x c ny ' x c [Gy + Hy' ] x b 0 Where,, γ, γ, DERGH,,,,, abcmn,,,, are all known real constant, and γ, γ are not as ntegers, D 0, G + H 0, a < c< b.. Prelmnary Knowledge lemma [] The general soluton for xy '' + ( γ x) y ' y 0 s y AF( γ,, x) + Bx F( +,, x) γ γ Where γ s not an nteger, both A and B are real constants and ( ) k k F( γ,, x) x k 0 k!( γ ) k lemma Introducng two bnary functons then 00 ξ F(,, xf ) ( +,, ξ) x F( +,, xf ) (,, ξ) (, ) () (3)

2 Amercan Journal of Mathematcal Analyss 40,0 ( x, ξ) 00 ξ γ γ γ ξ F( +, +, xf ) ( +,, ) (4) ( γ ) x F( +,, xf ) (,, ξ) + x F( +,3, xf ) (,, ξ) 0 00 ξ ( γ ) ξ F(,, xf ) ( +,, ξ) + (5) ξ + F(, γ, xf ) ( + γ,3 γ, ξ ) γ x F( +,, xf ) ( +, +, ξ) 0, ( x, ξ) ( γ ) ξ F( +, +, x) F( +,, ξ) + ξ + F( +, γ +, x) F( +,3, ξ) (6) ( γ ) x F( +,, x) F( +, +, ξ) + x F( +,3, x) γ F( +, +, ξ) Proof: Based on the dfferental propertes of the Kummer functon we can know that so d F ( γ,, x ) F ( +, γ+, x ) (7) dx γ 00,0 ( x, ξ) ξ F( +,, ξ) F(,, x) ( γ ) x F( +,, xf ) (,, ξ) x F(,, ξ) F( +,, x) ξ γ γ γ ξ F( +, +, xf ) ( +,, ) ( γ ) x F( +,, xf ) (,, ξ) γ + x F( +,3, xf ) (,, ξ) Thus: Smlarly verfable (5) and(6). 3. The Man Theorem and Its Proof Theorem If boundary value problem () has a unque soluton, n that way the left area ( a < x< c) s: y D ϕ( x) E + R+ ϕ( a) R + ϕ( a ) The rght area ( c< x< b) soluton s: y D E + R+ ϕ( a) R + ϕ( a ) 0,(,) cc ϕ ( x) mϕ () c, (,) ac n,0 (, cb) (8) (9) Where ϕ ( x) s defned as smlar kernel functon of the rght area : ϕ G0,0( xb, ) + H0,( xb, ) ( x) ( c< x< b) G,0 (, cb) + H, (, cb) (0) Where ϕ ( x) s defned as smlar kernel functon of the left area : ϕ mϕ( c) 0,( xc, )- n0,0( xc, ) ( x) ( a < x< c) mϕ( c), ( ac, )- n,0 ( ac, ) () Proof: Frst of all, Based on the lemma we know that the general soluton for are as follows xy'' + ( γ x) y' y ( 0 a < x < c) () xy'' + ( γ x) y' y 0( c < x < b) (3) y AF (, γ, x) + Bx F( +, γ, x) (4) y AF (, γ, x) + Bx F( +, γ, x) (5) Where A, B, A, Bare real constants. Accordng the left boundary condtons [ Ey+ ( + ER) y'] x a D, we can know that A [ EF(, γ, a) + ( + ER) F( +, γ+, a)] γ + B [ Ea F( +,, a) (6) + ( + ER)( γ ) a F( +,, a) + + ( + ER) a F( +, 3 γ, a)] D γ Accordng to the contnuty condtons y x c my x c, y ' x c ny ' x cwe can get AF (, γ, c) + Bc F( +,, c) ma F(, γ, c) mbc F( +, γ, c) 0 (7)

3 4 Amercan Journal of Mathematcal Analyss A F( +, γ+, c) na F( +, γ +, c) γ γ + B [( γ) c F( +,, c) + + c F( +,3, c)] (8) γ nb [( γ) c F( +,, c) + + c F( +,3 γ, c)] 0 γ By the rght boundary condtons of the complex confluence hypergeometrc equaton boundary value problem () that [Gy + Hy' ] x b 0 we can get A [ GF(, γ, b) + H F( +, γ +, b)] γ + B [ Gb F( +,, b) ` + H( γ ) b F( +,, b) + + Hb F( +,3 γ, b)] 0 γ (9) Gettng lnear equatons about of undetermned coeffcents A, B, Aand B by combnng equaton (6), (7), (8) and (9). If we assume that complex confluence hypergeometrc equaton boundary value problem () has a unque soluton, then there wll be a determnant of coeffcent of lnear equatons not be 0,mean that 0.And after calculaton usng (3), (4), (5), (6) for smplfyng we can get mg [ 0,0(, cb) + H0,(, cb)] [ E0,( ac, ) + ( + ER),( ac, )] ng [,0 (, cb) + H, (, cb)] [ E0,0( ac, ) + ( + ER),0( ac, )] (0) Usng Kramer rule to solve the equatons set whch s composed of (4), (5), (6) and (7) equatons, we can get D A { m [ G0,0( cb, ) + H0,( cb, )] [( γ ) c F( +,, c) + + c F( +,3, c)] γ n [ G,0 (, cb) + H, (, cb)] c F( +,, c)} () D A [ Gb F( +,, b) + H( γ ) b F( +,, b) + + Hb B γ γ γ 0, F( +,3, b)] ( cc, ) GF(, γ, b) D 0, + H F( +, γ +, b) γ (,) cc (3) (4) Then y, y can be obtaned f we get A, B, A, B substtuton of(4),(5). And f we smplfy y, y usng the smlar kernel functon of the left and the rght area, we can get the soluton of the left and the rght area (8)and(9), respectvely. 4. Gve an Example Draw the mage of followng boundary value problem soluton: xy'' + (0.5 x) y' y ( 0 < x < ) xy'' + (.5 x) y' y 0( < x < 3) [ y+ y'] x y x y x y' x y' x [y + y' ] x 3 0 (5) Accordng to the prelmnary knowledge we can get the general soluton of the target equaton as follows. y AF (,0.5, x) + Bx F(.5,.5, x) (6) - y AF (,.5, x) + Bx F(0.5,0.5, x) (7) Through the comparson of the toundary value problem () and the boundary value problem(5) we found that,, γ 0.5, γ.5, D, E, F, G, H, a, b 3, c, m, n, usng MATHEMATICA language program to draw the soluton of the boundary value problem n the specfed mage defnton doman as follows: D B { mg [ 0,0(, cb) + H0,(, cb)] + + γ F(, γ, c),0, ng [ ( cb, ) + H ( cb, )] F(, γ, c)} () Fgure. Cuver of soluton to boundary value problem(5) (left area)

4 Amercan Journal of Mathematcal Analyss 4 Fgure. Cuver of soluton to boundary value problem(5) (rght area) By changng the parameters of D and G n the condton of observaton and analyss of the mage change. Fgure 3. The nfluence of parameter D on curve( rght area) Fgure 6. The nfluence of parameter G on curve( rght area) 5. Conclusons and Understandng ) Wth the defnton of the left, the rght area of smlar kernel functon (0) and (), the solutons of boundary value problem () can be wrtten as the contnued fracton structure. In the progress of solvng the boundary value problem for composte Kummer equaton, we can get the left area and the rght area soluton drectly by constructng smlarty kernel functon. Ths smplfes the solvng process, mproves the soluton effcency, contrbutes to the researchers comple correspondng analyss software. ) From the Fgure 3 and Fgure 4 we can see that the greater absolute value of the parameter D the greater functon curve fluctuatons n the left area and the rght area. 3) From the Fgure 4 we can see that the mage functon on the X axs symmetry when D were obtaned from the two opposte number. 4) From the Fgure 5 we can see that the greater absolute value of the parameter G the greater functon curve fluctuatons n the left area. 5) From the Fgure 5 we can see that n the rght area regardless of how to change the parameters of the G mage through a fxed pont and the pont coordnates for (.7, ). Acknowledgment Fgure 4. The nfluence of parameter D on curve( rght area) Ths work s supported the by the Scentfc Research Fund of the Schuan Provncal Educaton Department of Chna (Grant No. ZA64). References Fgure 5. The nfluence of parameter G on curve( left area) [] Gaoxong-Wang, Zhmng-Zhou, Smng-Zhu, Shousong-Wang. Ordnary Dfferental Equaton[M]. Bejng:Hgher Educaton Press, 978. [] L Shun-chu. The Smlarty Structurng Method of Boundary Value Problems of the Composte Dfferental Equatons (n Chnese) [J]. Journal of Xhua Unversty (Natural Scence Edton), 03, 3(4): 7-3. [3] L Shun-chu, Wu Xao-qng. Several Important Propertes for the Boundary Value Problems of Second-order Lnear Homogeneous Dfferental Equaton (n Chnese)[J]. Journal of Xhua Unversty (Natural Scence Edton), 03, 3(): 3-6. [4] LI Shun-Chu, Lao Zh-Jan. Constructng the Soluton of Boundary Value Problem of the Dfferental Equaton wth t s an

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