Explicit Approximate Solution for Finding the. Natural Frequency of the Motion of Pendulum. by Using the HAM

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1 Applied Matheatical Sciences Vol. 3 9 no Explicit Approxiate Solution for Finding the Natural Frequency of the Motion of Pendulu by Using the HAM Ahad Doosthoseini * Mechanical Engineering Departent Engineering Faculty of Bu- Ali Sina University Haedan Iran Alireza Doosthoseini Faculty of Mechanical Engineering College of engineering University of Tehran Tehran Iran Ensiyeh Doosthoseini Faculty of Matheatics Departent of Applied Matheatics University of Tehran Tehran Iran Taha Abachi Faculty of Coputer Engineering Sharif University of Technology Tehran Iran Abstract In this work we use a powerful analytical ethod called hootopy analysis ethod (HAM) to derive the frequency of a nonlinear oscillating syste. Unlike the perturbation ethod the HAM does not require the addition of a sall physically paraeter to the differential equation. It is applicable to strongly and weakly nonlinear probles. Moreover the HAM involves an auxiliary paraeter h which renders the convergence paraeter of series solutions Controllable and increases the convergence and increases the convergence significantly. This article depicts that the HAM is * Corresponding author: Ahad Doosthoseini E-ail address: a_doosthoseini@yahoo.co

2 14 A. Doosthoseini et al an efficient and powerful ethod for solving oscillating systes. Keywords: Frequency; Nonlinear oscillation; Series solutions; Hootopy analysis ethod (HAM) Hootopy perturbation ethod (HPM) 1 Introduction Modeling of natural phenoena in ost sciences yields nonlinear differential equations the exact solutions of which are usually rare. Therefore analytical ethods are strongly needed. For instance one analytical ethod called perturbation involves creating a sall physically paraeter in the proble however finding this paraeter is ipossible in ost cases [1 ]. Generally speaking one siple solution for controlling convergence and increasing it does not exist in all analytical ethods. In 199 Liao [3] presented hootopy analysis ethod (HAM) based on fundaental concept of hootopy in topology [4-6]. In this ethod we do not need to apply the sall paraeter and unlike all other analytic techniques the HAM provides us with a siple way to adjust and control the convergence region of approxiate series solutions. HAM has been successfully applied to solve any types of nonlinear probles [7 8]. Basic idea of HAM In this work we apply the HAM to obtain the frequency of a nonlinear oscillating syste known as the duffing equation as follows θ() t + ω sin( θ()) t = (1) g where t denotes the tie the dot denotes derivative with respect to t ω = is l frequency and θ () t is the oscillator displaceent. As initial conditions we take θ( ) = a θ ( ) = () The exact frequency is [1] ω = π T 4 π T = dφ ω 1 k sin θ θ k = ω Now we expand sin(θ) in (1) aroundθ = and we obtain (3) (4) (5)

3 Explicit approxiate solution 15 3 θ () t θ() t + ω ( θ() t ) = 6 (6) to account for the nonlinear dependence of the frequency we explicitly exhibit the frequency ω of the syste in the differential equation. To this end we introduce the transforation. τ = ωt θ () t = u() τ (7) Hence equation (1) becoes 3 u ( τ ) ω u () τ + ω (() u τ ) = 6 (8) u( τ ) = a u ( τ ) = when τ =. (9) Where the prie indicates the derivative respect to τ. We note that the actual frequency of the syste now appears explicitly in the equation. Until now both u and ω are unknowns. u ( τ ) can be expressed in the for u( τ ) = b cos( nτ ). n = 1 n (1) Where bn is a coefficient. This provides us with the so-called solution expression g of u ( τ ). ω =. According to (1) and (9) it is obvious for us to choose such an l initial guess u ( τ ) = acos( τ ) (11) according to Equation (1). We choose the auxiliary linear operator ϕτ (; p) L[ ϕτ ( ; p)] = + ϕτ ( ; p) (1) τ which has the property Lc ( 1sinτ + ccos τ ) = (13) for any integration constants c 1 and c. Now with respect to equation (8) we define the nonlinear auxiliary function N as follows ϕτ (; p) Ω ( ) 3 N[ ϕτ ( ; p) Ω ( p)] =Ω ( p) +Ω ( ) ϕτ ( ; p) ϕ ( τ; p) τ 6 (14) where p [ 1is ] an ebedding paraeter ϕ( τ ; p ) is a kind of apping of the

4 16 A. Doosthoseini et al unknown function u ( τ ) and it is a function of τ and p Ω ( p) is kind of apping of unknown frequency ω and it is a function of p. Now we consider h deterine an auxiliary paraeter and H ( τ ) an auxiliary function respectively. h increases the results convergence. Then we construct the so-called zero-order deforation equation ( 1 p) L[ ϕ( τ; p) u ( τ)] = hpn[ ϕ( τ; p) Ω( p)] (15) subject to the initial conditions ϕ ( ; p) = a ϕ( ; p) = (16) τ it is obvious that the solution of equations (15) and (16) for p = and p = 1 as follows ϕ(; τ ) = u () τ Ω ( ) = ω (17) ϕ(;) τ 1 = u () τ Ω () 1 = ω. (18) When p increases fro zero to one ϕ( τ ; p ) fro the u( τ ) = acosτ to the u() τ of Eqs. (8) and (9) and Ω( p) fro ω to the unknown frequency ω. Expanding ϕ( τ ; p ) and Ω ( p) in Taylor series with respect to p and using (17) we have ϕτ (; p) = u () τ + u () τ p = 1 Ω ( p) = ω + ω p = 1 (19) () u 1 ϕ (; τ p) ( τ ) =! p 1 Ω( p) ω =! p p = p = (1-a) (1-b) we note that in the zero-order deforation equation ϕ( τ ; p ) and Ω ( p) are dependent upon the auxiliary h and auxiliary function H () τ. If the auxiliary linear operator the initial guess the auxiliary paraeter h and the auxiliary function H () τ are properly chosen so that the series (19) and () converge at p =1 using (18) then we have

5 Explicit approxiate solution 17 u() τ = u () τ + u () τ = 1 ω = ω + ω. = 1 () (3) Now for siplicity we define the vectors of u (τ ) and ω as follows u { u ( τ) u ( τ)... u ( τ)} = 1 ω { ω ω... ω }. = 1 Differentiating the zero-order deforation equation (15) and (18) ties with respect to p then dividing the by! and finally setting p= we have the socalled th-order deforation equation in the following for Lu [ ( τ ) x u 1] = hh( τ) R ( u 1 ω 1 ) (4) Subject to initial conditions u ( ) = u ( ) = (5) where 1 1 N[ ϕτ ( ; p) Ω( p)] 1 k R( u ω ) ( ωω i k i) u 1 1 = = k + ω 1 1 ( 1)! p k= i= where p= ω 1 k u ( uu ) u 6 1 i k i 1 k k= i= (6) 1 x = 1 > 1. It should be noted that we assue H () τ = 1. we should note that both u ( τ ) andω are unknown. But we have one equation (4) for u ( τ ).Then another algebraic equation ust be given so as to deterine ω 1 Regarding the nature y proble we can understand that R( u 1 ω 1 ) should be expressed as follows μ R ( u ω ) = β ( ω )cos[( k + 1 ) τ] (7) 1 1 k 1 k =

6 18 A. Doosthoseini et al where βk is a coefficient μ is a integer dependent on the. Regarding to (13) when β ( ω 1) the solution of th-order (4) disobeys the solution expression (1). Then we should set β ( ω 1) = which provides us another algebraic equation for getting approxiation is given as follows (8) ω 1 The Nth-order u() τ u () τ + u () τ N = 1 (9) N ω ω + ω 1 = 1 (3) It should be noted that the HPM is a particular fro of the analytical HAM that is for h = 1 HAM will equal HPM [3]. 3 Result Analysis The objective of this study is to calculate the value of ω and copare it with exact values in Nayfeh [1]. The auxiliary paraeter h controls the convergence of series solutions. Once we obtainω we will be able to plotω versus h ( ω h curve) to acquire the appropriate value of h to find the exact solution. In this paper we plot ω h curves for a =. 1 l = 1 after 15 iterations for ω. The optial h for the curve is zero. The ω. values obtained through HAM (after 15 iterations for estiating ω.) HPM (after 15 iterations for estiating ω.) and the exact solution are shown in table [1]. The solution resulting fro HAM coincides with the exact solution. Whereas for h = 1(HPM) the solution is uch weaker than those of HAM. Table 1 The values of ω. a=. 1 l = 1 values π ω = exact 98. HAM T exact ω 98. ω 7 HPM

7 Explicit approxiate solution h Fig.1. The 15th-order approxiation of ω. versus h 4 Conclusions In this paper we utilized the powerful ethod of hootopy analysis to obtain the frequency of an oscillating syste. We achieved a very good approxiation with the actual solution of the considered syste. In addition this technique is algorithic and it is easy to ipleentation by sybolic coputation software such as Maple and Matheatica. Different fro all other analytic techniques it provides us with a siple way to adjust and control the convergence region of approxiate series solutions. Unlike perturbation ethod the HAM does not need any sall paraeter. It shows that the HAM is a very efficient ethod. We sincerely hope this ethod can be applied in a wider range. References [1] N. Ali Hasan Introduction to Perturbation Techniques John Wiley & Sons New York [] N. Ali Hasan Probles in Perturbation John Wiley & Sons [3] S.J. Liao The proposed hootopy analysis technique for the solution of nonlinear probles Ph.D. ThesisShanghai Jiao TongUniversity199.

8 13 A. Doosthoseini et al [4] Liao S. J. (1995): An approxiate solution technique which does not depend upon sall paraeters: a special exaple. International Journal of Nonlinear Mechanics [5] Liao S. J. (1997): An approxiate solution technique which does not depend upon sall paraeters (Part ): an application in fluid echanics. International Journal of Nonlinear Mechanics Vol. 3 No [6] Liao S. J. (1999): An explicit totally analytic approxiation of Blasius viscous flow probles. International Journal of Non-Linear Mechanics Vol. 34 No [7] M.M.Rashidi G.Doairy A.DoostHosseini S.Dinarvand Explicit Approxiate Solution of the Coupled KdV Equations by Using the HAM. Int. Journal of Math. Analysis Vol. 8 no [8] Wen Jianin Cao Zhengcai Sub-haronic resonances of nonlinear Oscillations with paraetric excitation by eans of the hootopy analysis Method Phys. Lett. A 371 (7) 47. Received: August 8