Max-Min Approach to Nonlinear Oscillators

Transcription

1 From the SelectedWorks of Ji-Huan He 8 Max-Min pproach to Nonlinear Oscillators Ji-Huan He, Donghua University vailable at:

2 Freund Publishing House Ltd. International Journal of Nonlinear Sciences and Numerical Simulation, 9(),7-, 8 Max-Min pproach to Nonlinear Oscillators Ji-Huan He Modern Ttile Institute, Donghua University, Shanghai 5, China jhhe@dhu.edu.cn bstract This paper suggests a novel method called max-min method. Maximal and minimal solution thresholds of a nonlinear problem can be easily found, and an approximate solution of the nonlinear equation can be easily deduced using He Chengtian s interpolation, which has millennia history. pplication of the method to nonlinear oscillators is systematically illustrated, illustrating amples show the present technology is very convenient and effective. Keywords: He Chengtian inequality, ncient Chinese Mathematics, Nonlinear Oscillation, Duffing equation, period. Introduction In most engineering problems, it is easy to find maximum/minimum thresholds of a solution of a nonlinear equation. We first consider a simple ample in the form: y y =, y( ) =. () The act solution is y ( x) = /( x). We give two crude trial functions: and It is obvious that y ( x) = y() = () y ( x ) = y () y () x= x. (3) = x = y ( x) < y( x) < y ( x). () From this maximum-minimum relationship, Eq.(), we can use an old inequality, He Chengtian inequality, which has millennia history, to find an approximate solution.. He Chengtian Inequality In an ancient history book, it writes( the Chinese version of this tt can be found in Refs.[,]) He Chengtian uses 6/9 as the strong, and 9/7 as the weak. mong the strong and the weak, Chengtian finds the fractional day(399/75) of the Moon by using the strong factor 5 and the weak factor. The statement is rather cryptic, in modern mathematical term, the statement can be plained as follows. ccording to the observation data, He Chengtian finds that days > Moon > 9 9 days. 7 Using the weighting factors ( 5 and ), He Chengtian(369?~7D) obtains The fractional day = = , so Moon = days. 75 He Chengtian actually uses the following inequality: If a d < x <, (6) b c where a,b,c and d are real numbers, then

3 8 J.H. He. Max-Min pproach to Nonlinear Oscillators a b ma < mb and x is approximated by ma x = mb nd d <, (7) nc c nd, (8) nc where m and n are weighting factors. The proof of the inequality can be found in details in Ref.[3], fascinating applications of the technology can be found in Refs.[,5,6,7]. He Chengtian (369?~7D) is a famous ancient Chinese mathematics and astronomer, he is an tremely important figure in development of mathematics, yet our Western colleagues know little about his mathematical achievements. Great classics, when revisited in the light of new developments, may reveal hidden pearls, as it the case with He Chengtian s interpolation. We re-write () in the form x < yx ( ) < (9) ccording to He Chengtian s interpolation, we set or m( x) n y( x; m, n) =, () m n x k y( x; k) =, () k where m and n are weighting factors, k=n/m. The value of k can be approximately determined by various approximate methods[8,9,]. mong others, hereby we use the residual method. Substituting () into () results in the following residual: x k R ( x; k) =. () k k Locating at x =. 5 we obtain.5 k R (.5; k) = = (3) k k k =.866. The approximate result at x=.5 is y (.5;.866) =.73. The 9.85% accuracy is remarkable good in view of the crudeness of the two trial functions. We can, of course, obtain a much better result by suitable choice of trial functions. It is interesting to note that m and n can also be functions of x. If we set m=, and n=x, then (6) happens to be the act solution! x x y( x;, x) = =. () x x 3. pplication to Nonlinear Oscillations Consider Duffing equation which reads u u ε u 3 =, u( ) =, u ( ) =, (5) where ε needs not be small in the present study, i.e. ε <. We re-write Eq.(5) in the form u ( ε u ) u =. (6) We choose a trial-function in the form u = cosωt, (7) where ω is the frequency to be determined. Observe that the square of frequency, ω, is never less than that in the solution ϕ ( t) cost (8) = of the following oscillation u = ( ε u ) u = u. (9) min In addition, ω never ceeds the square of frequency of the solution ϕ ( t) = cos ε t () of the following oscillation u = ( ε u ) u = ( ε ) u. () max Hence, it follows that ε < ω <. ()

4 ISSN: International Journal of Nonlinear Sciences and Numerical Simulation, 9(), 95-98, 8 9 ccording to He Chengtian s interpolation, we have m n( ε ) ω = = kε, (3) m n where m and n are weighting factors, k = n /( m n). So the frequency can be approximated as ω = kε. () Its approximate solution reads = / ut () cos ( kε ) t 3 ( ε ) = ε εu (6) (5) In view of the approximate solution, Eq.(5), we re-write Eq.(5) in the form u k u k u If, by chance, Eq.(5) is the act solution, then the right hand side of Eq.(6) is vanishing completely. Since our approach is only an approximation to the act solution, we set where obtain T / 3 ( )cos k ε u ε u ω tdt =, (7) T = π / ω. Substituting (5) in (7), we k = 3/ (8) Finally the frequency is obtained 3 ω = ε. (9) To illustrate the remarkable accuracy of the obtained result, we compare the approximate period with the act one [7] T = π T = (3) 3ε / π / ε d x k sin,with x ε k =. (3) ( ε ) What is rather surprising about the remarkable range of validity of (8) is that the actual asymptotic period as high accuracy. ε is also of T 3 π / d x lim = =.99 (3) ε T π.5sin x Therefore, for any value of ε, it can be easily proved that the maximal relative error is less than 5.66%. Example Consider the equation (Exercise. in Ref.[7]) ( u ) u u =, u( ) =, u () =.(33) We re-write (33) in the form = u. (3) u u If we choose the trial-function in the form u = cosωt, where ω is the frequency, then the maximal and minimal values of /( u ) are, respectively, and /( ). So we immediately obtain < ω <. (35) ccording to He Chengtian s interpolation, we set m n ω = =, (36) m n( ) k where m and n are weighting factors, k=n/(mn). So the frequency can be approximated as Setting ω =. (37) k Similarly we re-write Eq.(33) in the form we obtain ( k ) u u = k u u u T / (38) ( k u u u )cosωtdt = (39)

5 J.H. He. Max-Min pproach to Nonlinear Oscillators Its approximate frequency reads k=3/ () ω =. () Its approximate period can be pressed as Its act period reads[8] T In case 3 π T = = π ω 3 du = ln( ) ln( u ) [ ) /( u ]. () du =. (3) ln ( ), we have du lim T =. () (ln lnu) By transformation u=s, the above equation reduces to d s lim T =. (5) ln(/ s) By transformation s= p( x ), we have lim T In case = p( x )d x =, we have π.(6) T π 3 3π lim = = =.85. (7) T π 8 The accuracy of 8.5% when remarkable good. is pears in the great classics, such as Jiuzhang Suanshu(Nine Chapters), these pearls, when contacted with modern technologies, can shine marvelously. cknowledgement This material is based on work supported by the Program for New Century Excellent Talents in University under grand No. NCET-5-7. References. Qian, B.C. History of Chinese Mathematics, Science Publisher, Beijing,99(in Chinese). Ji,Z. Mathematics in the Northern-Southern, Sui and Tang Dynasties, Shijiazhuang, Hebei Science and Technology Publishing House, 999 (in Chinese) 3. He, J.H. Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, (6): -99. He, JH; Tang, H. Rebuild of King Fang BC musical scales by He's inequality, ppl. Math. Comput., 68 (5): He, JH. Mysterious pi and a possible link to DN sequencing, Int. J. Nonlinear Sci., 5 (): He, JH. Solution of nonlinear equations by an ancient Chinese algorithm, ppl. Math. Comput., 5 (): He, JH. He Chengtian's inequality and its applications, ppl. Math. Comput., 5 (): Nayfeh,.H. Introduction to Perturbation Techniques, John Wiley & Sons, New York, He JH. Nonperturbative methods for strongly nonlinear problems, dissertation.de-verlag im Internet GmbH Conclusion To conclude, we find that the ancient Chinese mathematics represents one of the most important fields of research in science and technology. There ist innumerable hidden