The Conversion a Bessel s Equation to a Self-Adjoint Equation and Applications

Transcription

1 Worl pplie Science Journal 5 (): , 0 ISSN IDOSI Publication, 0 The Converion a Beel Equation to a Self-joint Equation an pplication Delkhoh Mehi Department of Computer, Ilamic za Univerit, Barakan Branch, Barakan, Iran btract: In man application of variou Self-ajoint ifferential equation, whoe olution are complex, are prouce [, 5]. In thi paper, a metho for the converion Beel equation to Self-ajoint equation to provie a metho an then, a the invere of the tranformation Self-ajoint equation of the Beel equation. B which one can obtain analtical olution to Self-ajoint equation. Becaue thi olution, an exact analtical olution can provie to u, we benefite from the olution of numerical Self-ajoint equation [7, 9,,, 4, 6]. Ke wor: Beel equation Beel function Self-ajoint equation Self-ajointization factor INTRODUCTION ( ) + 0 a < x < b () In man application of cience to olve man ifferential equation, we fin that thee equation are Where h(x)>0 on (a, b) an, h are continuou Self-ajoint equation an olve relativel complex function on [a,b]. becaue the're force to ue numerical metho, which are containe everal error [7, 9,,, 4, 6]. Self-jointization Factor: B multiplin both ie a There are everal metho for olvin equation, econ orer linear homoeneou equation in a function there one of which can be een in the literature [4, 5, 6,, µ(x), it can be chane into a elf-ajoint equation. 5], where the chane of variable i ver complicate to Namel, we conier the followin linear homoeneou ue. equation: In thi paper, for olvin analtical Self-ajoint P + Q + R 0 () equation, we et a metho which the firt, Self-ajoint equation into a Beel equation an then uin the Where P(x) i a non-zero function on [a,b]. olution of Beel equation, for an exact an analtical B multiplin both ie in µ(x), we have olution a Self-ajoint equation. Before oin to the main point, we tart to introuce three followin equation: ( xpx ) + ( xqx ) + ( xrx ) 0 If we check the elf-ajoint conition, we have: Beel Equation: The Beel equation i a follow [,,, 8 an 0]: x + x + ( x p ) 0 Where P i a non-neative real number. To implif, we preent the eneral olution of Beel Equation a follow: x B Self-joint Equation: econ orer linear homoeneou ifferential equation i calle elf-ajoint if n onl if it ha the followin form [, an 5]: p Thu ( ( xpx ) ) ( xqx ) Q P P + P Q P Q Exp Px P Where i a real number that will be pecifie exactl urin the proce. If we multipl both ie of () an () equation in each other, then we have the followin form of elf-ajoint equation: () Correoponin uthor: Delkhoh Mehi, Department of Computer, Ilamic za Univerit, Barakan Branch, Barakan, Iran. Tel:

2 Worl ppl. Sci. J., 5 (): , 0 ( xpx) + ( xrx ) 0 From now on we will focu on the elf-ajoint equation hown in (). f f f f f f f f ( ) + ( f p ) 0 f f (8) Reuce a Beel' Equation to a Self-ajoint Equation: Now, we how that a Beel equation i chaneable to a elf-ajoint equation []. Let, the Beel equation i a follow: Y Y X X ( X p ) Y 0 X + X + We know that, thi equation ha the eneral olution a follow: B replacin of variable X f(x) an Y, x where f(x) an (x) are continuou an ifferentiable function an alo the followin relation: (4) Now, correponin to equation () we have Px f f f f Qx + f f f f Rx + f f f p Y B P(X) (5) B virtue of equation (), we have the elfajointization factor a follow: x Then, multiplin both ie of equation (8) b µ(x) an we have X f Y X f Y X f f Into the equation (4), we have f f + + f p 0 f f Thu, we have f f. + f p 0 f f B virtue of equation (4) an (5), we have a olution for the equation (6) a follow: (6) (x) (x) B (f(x)) (7) P Calculatin the erivative in (6) an implifin it to a econ orer linear homoeneou ifferential equation take the followin form: f f f + ( ) + ( f p ) 0 f f f Where i a elf-ajoint equation, i.e. correponin to the equation (), we have f hx f f f ( ) + ( f p ) f f x x B( f) p (9) (0) () With conier the equation (6) an (7), the equation (9) ha the eneral olution a follow: () Solvin Self-joint Equat-ion b Uin of Beel' Equa-Tion: Let, we have an equation a (). If we compare both ie of thi equation an equation (9), we can fin f(x) an (x) an then we have the eneral olution a the eneral olution of the equation (). From now, our main purpoe i to calculate f(x) an (x) mentione a in equation (9). 688

3 Worl ppl. Sci. J., 5 (): , 0 We conier the followin pecial cae which i ver important. Let f(x) an (x) be a follow: f r r h r hx hx r r ( ) + ( r p ) r r ( + r p ) ( x ) ( + r p ) () Where r an S are contant which will pecifie exactl in the proce. Thu, b virtue of the equation (0), we have (4) n we are able to calculate f(x) from relation (). B replacin of f(x) an (x) in equation (), we calculate the contant,p,r an, i.e. From equation (5) an (6) an hpothei that the coefficient of (x) i equal to one (B virtue of the eneral olution form of equation ()). we are able to calculate the contant, p, r an. Now, ince the function f(x) an (x) are pecifie we can a that we have the eneral olution, a the eneral olution of equation (), for the main equation () (a mentione at the beinnin of ection ). pplication EX : Solve the equation, x ( + x) ( ) ( ) ( + x) + a( + x) c 0 where,, c an a are contant. (Problem: Shear beam with variable cro-ection an itribute mae) [4-6]. Solution: B virtue of equation () we have h(x) ( + x), From equation (4) we have (7) Without lo of eneralit, we can aume that the coefficient of (x) i equal to : Thu, from equation () we have (8) + r ( p ) f r( + x) r p p r p (5) From equation () we have ( ) r + x ( f p ) () x + ( + x ) r ( + x) ( + x) + ( + x) ( f p ) ( + x) c a + r (6) a + ( f p ) ( + x) c + 689

4 Worl ppl. Sci. J., 5 (): , 0 a c + f p ( + x) ( ) ( ) a c + r ( + x) p ( + x) ( ) ( ) c + a r ( ) p c + (9) ( ) Where. p c + Equation (4) can be reuce to Euler equation when c +. EX.) Solve the equation ( e ) + ae 0 x cx Where,, c an a are contant [9-]. Solution: From equation () we have: x cx h(x) e, (x) e (5) p (0) From (4) we have r a ( ) () x exp( x) (6) From equation (8) an () we have a ( ) r ( ) a r c + c + () () B implifin an uin equation () an (6), a previou example, we have Thu, we have c, p c a r, c c Therefore, from equation (), (7), (8), (9), (0), () an () we have x ( + x) a f ( + x) c + c + Thu, b virtue of equation (9) an () the olution of equation I a follow: ( + x) + a( + x) 0 (x) (x)b (f(x)) p c (4) Thu a c x exp( x), f exp c Where. p c (x) (x) B (f(x)) p Equation (5) can be reuce to a ifferential equation with contant coefficient when c. EX.: Solve the equation n m + 0 x kx Where m, n an k are contant [9-]. (7) 690

5 Worl ppl. Sci. J., 5 (): , 0 Solution: B virtue of equation () we have: 4. llame.m, za.n, Solution of Thir Orer Nonlinear Equation b Talor Serie Expanion, Worl pplie n h(x) x, m (x) kx Science J., 4(): 59-6, 0, ISSN Borhanifar,., M.M. Kabir an. Hoein Pour, the previou example we can obtain: 0. Numerical Metho for Solution of the Heat Equation with Nonlocal Nonlinear Conition, Worl n pplie Science J., (): , 0, k x x, f x ISSN m n 6. Sweilam, N.H. an.m. Na, 0. Numerical where + Solution of Fractional Wave Equation uin Crank- Nicholon Metho, Worl pplie Science J., Thu (Special Iue of pplie Math): 7-75, 0, n ISSN k x x Bp 7. Gülu, M.,.Y. Öztürk an M. Sezer, 0. Numerical Solution of Sinular Intera-ifferential Equation Where n n p. with Cauch Kernel, Worl pplie Science J., m n+ (): 40-47, 0, ISSN Mohu-Din, T.,. Yilirim, M. Berberler an Equation (7) can be reuce to a ifferential M. Hoeini, 00. Numerical Solution of Moifie equation with contant coefficient when n m +. Equal With Wave Equation, Worl pplie Science J., 8 7): , 00, ISSN CONCLUSION 9. Qiuhen, L., C. Hon an L. Guiqin, 996. Static an Dnamic anali of traiht bar with The overnin equation for tabilit anali of a variable cro-ection bar ubject to variabl itribute axial loa, namic anali of multi-tore builin, tall builin an other tem are written in the form of a unifie elf-ajoint equation of the econ orer. Thee are reuce to Beel' equation in thi paper. The ke tep in tranformin the unifie equation to Beel' equation i the election of h(x) an (x) in equation (). Man ifficult problem in the fiel of tatic an namic mechanic are olve b the unifie equation propoe in thi paper. REFERENCES. rfken, G., 985. Self-joint Differential Equation. 9. in Mathematical Metho for Phicit, r e. Orlano, FL: caemic Pre, pp: Ganaria, M., 0. Weak elf-ajoint ifferential equation, J. Ph. : Math. Theor., 44: Mohu-Din, S.T., 009. Solution of Nonlinear Differential Equation b Exp-function Metho, Worl pplie Science J., 7(Special Iue for pplie Math): 6-47, 009, ISSN variable cro-ection, Computer & Structure, 59(6): Qiuhen, L., C. Hon an L. Guiqin, 96. nali of Free Vibration of Tall Builin, J. Enineerin Mechanic, 0(9): Demir, D., N. Bilik,. Konuralp an. Demir, 0. The Numerical Solution for the Dampe Generalize Reularize Lon-wave Equation b Variational Metho, Worl pplie Science J., (Special Iue of pplie Math): 08-7, 0, ISSN Hilerbran, F.B., 976. vance Calculu for pplication (-e), New Jere.. Javapour, S.H., 99. n Introuction to Orinar an Partial Differential Equation, Iran, lavi. 4. O Neil, P.V., 987. vance Enineerin Mathematic (-e), California, Waworth. 5. Olver, F. an L. Maximon, 00. Beel function, NIST Hanbook of Mathematical Function, Cambrie Univerit Pre, ISBN , MR748, Teukolk, S. an W.T. Vetterlin, 007. Section 6.5. Beel Function of Inteer Orer, Numerical Recipe: The rt of Scientific Computin (r e.), New York: Cambrie Univerit Pre, ISBN