World Applied Sciences Journal (): 64-69, 0 ISSN 88-495 IDOSI Publications, 0 DOI: 0.589/idosi.wasj.0..0.09 The Solution of Weakly Nonlinear Oscillatory Problems with No Damping Using MAPLE N. Hashim, N.S. Amin, A.A. Shariff, F.A. Manaf and R.M. Tahir Mathematics Section, Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia Department of Mathematics, Faculty of Science, University Technology of Malaysia, Johor, Malaysia Abstract: Weakly nonlinear oscillatory problems in engineering dynamics are frequently presented by nonlinear ordinary differential equations. Very few of such equations have exact solutions, thus the need for approximation techniques. In this study, the classical perturbation method, namely the Lindste-Poincare method (L-P method) is discussed. In most work, the L-P method is used to obtain uniformly valid approximate solutions to second order only, as perturbation procedure usually involves cumbersome algebraic manipulations and calculations which are time-consuming and prone to errors. Therefore, in this paper we manage to get the solution to fifth order by using our own built in Maple package based on Maple 4 and we include the graphical presentations of the results. Key words: Weakly nonlinear oscillator Nonlinear ODE perturbation method Lindste-Poincare method INTRODUCTION The Lindsted-Poincare method was introduced by Swedish Mathematician and astronomer Lindste in 88 and French Mathematician and theoretical Physicist Poincare in 886 [-4]. It involves a single time variable or scale and manages to obtain uniformly valid approximate solutions to nonlinear problems. In this paper, firstly we consider a weakly nonlinear oscillator with no damping described as, In 987, Rand and Armbruster [5] used the computer algebra system MACYSMA in a number of perturbation techniques for systems of coupled oscillators and bifurcation theory. In 996, Heck [6] and in 999 Chin and Nayfeh [7] had implemented L-P method using Maple V Release 4, while in 00, Lopez [8] had implemented L-P method using Maple 8. In all the papers mentioned using the method, only second order approximations were obtained at most. In the next section we will consider the Duffing equation to illustrate the L-P method. d y dy + 0 y = F y,, 0< () The Duffing Equation: Duffing equation is the most common example of weakly nonlinear oscillatory system where is a small positive parameter, often called a with no damping where 0 = and dy/ = 0. [9] also have perturbation quantity and 0 (we take, 0 = ) is an chosen the same example and solve it using homotopy angular frequency, which is often referred to as the perturbation method coupled with Laplace transform and natural frequency. In this case, the system is free of Padé approximants. Let s consider the initial value damping whereby, there is no resisting force, such as air problem for the equation, resistance. The periodic solution to the equation () is assumed to be of the form, d y + y = y, () m m+ y( t, ) = ym () t + O( ), () m= 0 with condition, 0 < t < and 0 <. The initial known as asymptotic expansion. conditions for the above problem, Corresponding Author: N. Hashim, Mathematics Section, Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia. Tel: +60-79675894. 64
and is a constant. We take initial conditions (4a,b) for y = sin + cos (4) ( ) ( ) World Appl. Sci. J., (): 64-69, 0 y(0) =, y (0) = 0 (4) The particular solution of () is; 8 the reason that these initial conditions are sufficiently general to cover all physical systems of interest [0]. Nayfeh [] and Mickens [] had solved the Duffing Then, the general solution of () is; equation with initial conditions (4a,b) using L-P method in second order expansion. According to the standard L-P y( ) = Acos + Bsin method, a new variable; (5) + sin + cos 8 = t, (5) Using the initial conditions (b,c), we obtain the is introduced, where is the frequency of the system that constants A= -/ and B = 0. Thus the solution of (5) is, depends on, that is = ( ). Equation () then becomes y( ) = sin y + y = y, > 0, (6) 8 (6) y(0) =, y (0) = 0 (7a, b) ( cos cos ) + We seek approximate solutions for y and in the form of Note that the solution of y contains a mixed-secular power series in as follows, term, which makes the expansion nonuniform. For a uniform expansion, we cannot permit such secular terms n y(, ) = y 0( ) + y ( ) +... + y n ( ) +... (8) to appear in y, y, y,. The secular term can be eliminated by choosing, n ( ) = + +... + n +... (9) = 0 = (7) where 0has been chosen to be unity, (0) = 0=. 8 8 Substituting equation (8) and (9) into equation (6) and (7a,b) and equating of like power of yields the following Thus, without the secular term, the equation (6) system of differential equations for successive becomes, approximations: y ( ) = ( cos cos ) (8) O() : y0 + y0 = 0, (0) According to Nayfeh [], to determine the condition y0 ( 0) = 0, y0( 0) = (0b,c) (7), we do not need to determine the particular solution first as done above. Instead, we only need to inspect the O( ): y + y = y0 y0, (a) inhomogeneous terms in () governing y and choose the coefficient of cos to be zero. y( 0) = y ( 0) = 0 (b,c) Substitute y 0( ) = cos and equation (8) into equation (8), we obtain the first order perturbation O( ) : y + y= y ( + ) y0 y0 y solution of () is, (a) y (, ) = cos + ( cos cos ) + (9) y 0 = y 0 = 0. (b,c) Equation (0a) with initial conditions (0b,c) has a where = t and w( ) = + ( /8) +. periodic solution y 0( ) = cos. Substituting y 0( ) = cos into equation (a) and using cos = ¼(cos + cos) Next, we solve the initial value problem (a,b,c) we obtain the simplified form as, to obtain the solution of second order expansion. The general solution to the resulting differential equation y + y = cos cos () for y (t) is, 65
5 ( ) ( ) y = cos 4cos + cos5 04 Therefore, to the second order expansion, the solution to equation () is, (, ) cos ( cos cos ) y = + 5 cos 4cos + 04 cos + t t y( t, ) = cos t + 8 56 t t + 8 cos 4 t 56 + 04 t t + 8 cos 4 t 56 + O ( ) World Appl. Sci. J., (): 64-69, 0 (0) The solution of the Duffing equation in fifth order expansion is obtained as follows, () 4 where = t and w( ) = + ( /8) - ( /56) +. However according to Nayfeh [], equation () can be written as, () which is valid when t = O( ). This is the solution of the Duffing equation in second order expansion. Next, we are going to illustrate the solution of the Duffing equation by L-P method using Maple software up to fifth order expansion. The Duffing Equation with Maple: Anyone who uses perturbation methods is struck almost immediately by the amount of algebraic manipulation. As an alternative, we can use a symbolic language like Maple. This mathematical software has a tremendous potential for constructing asymptotic expansions []. In this section, we will illustrate the initial value problem of Duffing equation () with initial conditions (4a,b) by using L-P method with Maple. All commands that we use for solving initial value problems are under the DEtools package in Maple based on Lopez's work [8]. In this work we manage to obtain all the solutions in the fifth order expansions and good agreement with Runge-Kutta Fifth Fourth order (rkf45) numerical solution. y where ( ), cos cos = + + cos 5 5 cos cos 04 8 + 5 + cos5 04 7 7 547 97 cos + cos 768 684 cos5 + cos7 048 768 + 7 7 9 9 67 5 cos cos 5488 048576 9 9 88 9 + + cos5 cos 7 5488 07 9 + cos9 048576 + O 5 ( ) 4 6 8 + + 8 56 048 = t 8 4 0 5 6549 + 777 644 0975 4 () We also present the numerical values of the approximate solutions for () together with the graphs for various values of and several expansions in different order. Table and Fig. show that the solutions do not vary much for values of which are less than 0.05. However, for = {-0., 0., 0.} show bigger deviation compared to the other values of. Note that, < 0 and > 0 represent pendulum type equations with a hard spring and a soft spring respectively []. Table (a) and Figure (a) shows that at the interval time 0 t 5 does not give any discrepancies for different order of expansion. While Table (b) and Figure (b) show that the higher order expansion gives remarkable accuracy. When t = 99, the relative error of a new result is the smallest compared to the Mickens and Lopez s result. From a quantitative point of view, we can conclude that the higher order approximation is important in giving the accurate result. Other Examples: In this section we will consider the other examples of weakly nonlinear oscillator with no damping. The following equation is a pendulum type equation, 66
World Appl. Sci. J., (): 64-69, 0 Table : The approximate solutions of Duffing equation in fifth order expansion for various values of tabulated in the interval 0 b t b ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- t 0.005 0.0 0.05 0.05 0. 0..066.05.009 0.94 0.8 0.577-0.858-0.88-0.955 -.068 -.7 -.60 -.986 -.99 -.999 -.994 -.99 -.56 4 -.60 -. -.06-0.809-0. 0.68 5 0.67 0.705 0.9.90.644.999 6.94.96.996.977.686 0.54 7.45.60.5 0.67-0.8 -.64 8-0.408-0.5-0.84 -.06 -.896 -.57 9 -.87 -.95 -.99 -.948 -.8 0.7 0 -.59 -.495 -.67-0.5 0.74.995 0.7 0. 0.786.44.998 0.45 Table (a): The approximate solutions of the Duffing equation at different orders of expansions tabulated in the interval 0 b t b 5 t rkf45 Lopez (00) O( ) Mickens (996) Error O( ) Lopez (00) Error 5 O( ) New results Error 0 0.00 0.00 0.00 0.8 0.87 0.006 0.809 0.00 0.8 0.00 -.7 -.66 0.004 -.708 0.00 -.7 0.00 -.99 -.9 0.004 -.98 0.00 -.99 0.00 4-0.0-0.7 0.06-0.96 0.005-0. 0.00 5.64.6 0.0.646 0.00.644 0.00 Table (b): The approximate solutions of Duffing equation at different orders of expansions tabulated in the interval 95 b t b 00 t rkf45lopez (00) O( ) Mickens (996) Error O( ) Lopez (00) Error 5 O( ) New results Error 95 0.6 0.7 0.6 0.7 0.09 0.54 0.007 96 -.606 -.45 0.6 -.66 0.055 -.6 0.005 97 -.7 -.89 0.7 -.668 0.05 -.76 0.005 98 0.66 0.8 0.05 0.59 0.09 0.74 0.008 99.87.69 0.8.906 0.0.876 0.00 00.75.64 0.68.0 0.074.69 0.006 d y + y = y (5) Both equations will be solved with initial conditions (4a,b). In the literature, Jordan and Smith [0] had solved the problem in the equation (4) and obtained first order expansion using L-P method. Therefore, with the aid of Maple we obtain fifth order expansion as illustrated in the following Fig. (a) and (b). Here, we have shown our present solution is in agreement with the numerical solution by the method of Fehlberg fourth-fifth order Runge-Kutta. Jordan and Smith s result in the interval time 0 t 5 in Fig. (a) and 95 t 00 in Fig. (b) Fig. : The approximate solutions of Duffing equation deviate as time increasing. for various values of, with initial condition Fig. 4(a) and 4(b) show, we present result of the Πy(0) =, y (0) = 0 at the interval 0 t. equation (5) and it is in agreement with the numerical solution obtained from the maple package. Both solutions d y + y = y (4) indicate that the higher order solution gives more accurate results of the numerical solution compared to the solution Then, we consider the equation, obtained in O( ). 67
World Appl. Sci. J., (): 64-69, 0 Fig. (a): The approximate solutions of Duffing equation at different orders of expansions, with initial Fig. (b): The approximate solutions of equation (4) at conditions y(0) =, y (0) = 0 and = 0. in the different orders of expansions, with initial interval 0 t 5 conditions y(0) =, y (0) = 0 and = 0. in the interval 95 t 00 Fig. (b): The approximate solutions of Duffing equation at different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. at the Fig. 4(a): The approximate solutions of equation (5) at interval, 95 t 00 different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. in the interval 0 t 5 Fig. (a): The approximate solutions of equation (4) at Fig. 4(b): The approximate solutions of equation (5) at different orders of expansions, with initial different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. in the conditions y(0) =, y (0) = 0 and = 0. in the interval 0 t 5 interval 95 t 00 68
World Appl. Sci. J., (): 64-69, 0 CONCLUSIONS 5. Rand, R.H. and D. Armbruster, 987. Perturbation In this article, we have presented the approximate Methods, Bifurcation Theory and Computer Algebra. NY: Springer. solutions of weakly nonlinear oscillator with no damping 6. nd Heck, A., 996. Introduction to Maple. edition. by considering different initial conditions using L-P New York: Springer-Verlag. method. We implemented L-P method with Maple and 7. Chin, C.M. and A.H. Nayfeh, 999. Perturbation manage to get the periodic solutions in higher order Methods with Maple. Virginia: Dynamic Press. expansions and well-fitted with the numerical solution by 8. Lopez, R., 00. Advanced Engineering Mathematics. rkf 45 method. We also have considered the different Addison Wesley Longman Inc. values of and found that the solutions do not vary 9. Merdan, M., A. Yildirim, A. Gökdogan and much for values of that's less than 0.05. In future work, S.T. Mohyud-din, 0. Coupling of Homotopy we attempt to find the higher order solution for damping Perturbation, Laplace Transform and Padé oscillatory systems. Approximants for Nonlinear Oscillatory Systems. World Applied Sciences Journal, 6(): 0-8. ACKNOWLEDGEMENTS 0. Jordan, D.W. and P. Smith, 977. Nonlinearity Ordinary Differential Equations. Oxford: Claredon The authors gratefully acknowledge the financial Press. support provided by short term grants, PJP UM grant.. Nayfeh, A.H., 98. Introduction to Perturbation Techniques. New York: John Wiley and Sons. REFERENCES. Mickens, R.E., 996. Oscillations in Planar Dynamic Systems. Series on Advances in Mathematics for. Kevorkian, J. and J.D. Cole, 98. Perturbation Applied Sciences. Singapore: World Scientific Methods in Applied Mathematics. New York: Publishing Co. Pte. Ltd. Springer-Verlag.. Nayfeh, A.H., 99. Introduction to Perturbation. Holmes, M.H., 995. Introduction to Perturbation Techniques. New York: John Wiley and Sons. Methods. New York: Springer-Verlag.. De Jager, E.M. and F. Jiang, 996. The Theory of a Singular Perturbations. Amsterdam: Elsevier. 4. Sanchez, N.E., 996. The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems. Journal of Symbolic Computation, : 45-5. 69