Applicatio of Homotopy Perturbatio Method for the Large Agle period of Noliear Oscillator Olayiwola, M. O. Gbolagade A.W., Adesaya A.O. & Akipelu F.O. Departmet of Mathematical Scieces, Faculty of Sciece, Olabisi Oabajo Uiversity, Ago-Iwoye, Ogu State, Nigeria. Departmet of Pure ad Applied Mathematics, Ladoke Akitola Uiversity of Techology, Ogbomoso, Nigeria. Abstract The homotopy perturbatio method is used to determie the period of oliear oscillator. The method produces the result eve for large amplitude. The result is compared with others i the literature. Keyword: homotopy perturbatio method; oliear oscillator; period.. Itroductio The study of oliear oscillators is of great importace to may scietific researchers i various fields. Various methods such as variatioal iteratio methods [],[-5], parameter expadig method [] have bee proposed. The homotopy perturbatio method (HPM), proposed first by He J i 999 for solvig differetial ad itegral equatios, liear ad oliear has bee the subject of extesive aalytical ad umerical studies. The method is a couplig of the traditioal perturbatio method ad homotopy i topology. This method has a sigificat advatage i that it provides a approximate solutio to a wide rage of oliear problems i applied scieces. I this paper, we apply HPM to obtai the frequecy of oliear oscillator. The solutio obtaied is of high accuracy which is valid for the whole solutio domai.
.. Homotopy Perturbatio Method. We cosider a geeral oliear oscillator of the form: m ' + ω u + kf ( u,, ') =...() accordig to [7], we expad m ad ω as follows: ω = ω + pω + p ω +... + p ω...(). m = + pm + p m +... + p m...() Where p is a homotopy parameter, ωi ad m i are ukow costats to be further determied... Mathematical Pedulum Whe frictio is eglected, the differetial equatio goverig the free oscillatio of the mathematical pedulum is give by: ' + ω si u =...() u ( ) = A, () =. where u is the agle of deviatio from the vertical equilibrium positio. g ω = ; is the acceleratio due to gravity ad l is the legth of the pedulum. l The model look simple if the approximatio si u ~ u is used, the equatio () becomes ' + ω u =...(5) To obtai more accurate result we modify the above equatio by puttig u si u ~ u...() Substitutig for equatio () i (5) we have; ω ' + ω u u =...(7)
.. Applicatio of HPM Equatio (7) ca be writte i the form of equatio () such that. ω u '' + ω u + ku =, k =...(8) Applyig equatio () i equatio (8), we have; ' + ( ω + pc ) u + pku The basic assumptio is that: =...(9) u ( t) = u + pu + p u + p u +... + p u =...() Whe p = equatio (9) becomes ' + ( ω + c ) u + ku Comparig equatio (8) ad (); =...() ω = ω + c...() Substitutig equatio () i equatio (9) ad equatig the coefficiets of like powers of p. '' u + ω u =, u () = A, () =...() '' u + ω u + c u + ku =, u () = A, () = The solutio to equatio () is u...() = Acosω...(5) t Puttig equatio (5) i equatio () ad elimiatig the secular term; we have. c = ka...(). From equatio () ω = ω + ka...(7) Therefore, ω = ω ( A )...(8) 8 The period (T) is therefore: π T =...(9) ω A 8
This compared favourably with the solutio obtai i [] ad the value give i [8]. 5.. Coclusio I this work, the homotopy perturbatio method has bee successfully applied to fid the approximate solutios for the oliear system. It is worth metioig that the method is capable of reducig the volume of computatioal work while still maitaiig high accuracy of the umerical result. This amouts to the improvemet of performace of approach. This method is relatively ew ad may lead to some ovel ad iovative applicatios i solvig liear ad oliear problems. The method which proved to be a powerful mathematical tool to oliear oscillators ca be used as a searchig tool for the period or frequecy of various oliear oscillatory systems.
Refereces:. Da-Hua shou, Ji-Hua He (7). Applicatio of Parameter-expadig method to strogly o-liear oscillators. Iteratioal Joural of o-liear sciece ad Numerical Simulatio 8(), -.. Dragaescu GE (). Applicatio of Variatioal Iteratio Method to Liear ad Noliear Visco-elastic models with Fractioal Derivatives, J. Math Phys. 7(8): Avt No 89.. He J(999). Homotopy perturbatio techique. Compt. Math. Appl. Mech. Eg., 78, 57-.. He JH (999). Variatioal Iteratioal Method a kid of o-liear aalytical techique, It. J. No-liear Mech., (): 99-78. 5. He JH, Wu XH (8). Costructio of Solitary Solutio ad Compactio like solutio by Variatioal Iteratio method, Chaos, solutio ad Fractals, 9(): 8-.. Ji-Hua He (7). Variatioal Iteratio method Some recet results ad ew iterpretatio. Joural of Computatioal ad Applied Mathematics 7, -7. No.8, 5-58. 7. JI-Hua He (). New iterpretatio of homotopy perturbatio method. It. jour. Of moder phy. B. vol. 8. Louis N. Had ad Jaet D. F(998). Aalytical Mechaics. (Cambridge Uiversity Press, New York). 5