Differential equations of amplitudes and frequencies of equally-amplitudinal oscillations of the second order

Transcription

1 Theoretical Mathematics & pplications, vol., no., 0, -4 ISSN: (print), (online) International Scientific Press, 0 Differential equations of amplitudes and frequencies of equally-amplitudinal oscillations of the second order Jelena Vujaković, Miloje Rajović and Dojcin Petković bstract Study of oscillations leads to a need to determine Sturm's two-amplitudinal but equally-frequent equations with positive coefficient (according to results from Prodi []). This is shown by Floké theorem [] and monograph Babakov []. The problem of finding of amplitudes and frequencies hasn't been solved even in Sturm theory in mrein et al. [4], because it is nonlinear problem even though originates from linear oscillations of the second order. In this paper are set up differential equations for amplitude and frequency function of solution of equally-amplitudinal oscillations, and are given theorems on solution existence as well as the important examples. Faculty of Sciences and Mathematics, L. Ribara 9, 80 K. Mitrovica, Serbia, enav@ptt.rs Faculty of Mechanical Engineering, Dositejeva 9, 6000 Kraljevo, Serbia, rajovic.m@mfkv.kg.ac.rs Faculty of Sciences and Mathematics, L. Ribara 9, 80 K. Mitrovica, Serbia, dojcin.petkovic@gmail.com rticle Info: Received : December, 0. Revised : January, 0 Published online : March 5, 0

2 4 Differential equations of amplitudes... Mathematics Subject Classification : 4E05, 4K Keywords: Differential equation, Oscillation, Frequency function, mplitude function, series iteration method Introduction and Preliminaries where Let linear homogenous differential equation of the second order y a x y b x y 0 (.) ax and bx are given continuous functions, satisfies conditions of Picard- Lindelöf theorem on existence and uniqueness of solution over half-open interval 0,. Introducing the substitution y zexp axdx (.) equation (.) reduces to canonical equation where It is known, that for positive z z 0 z (.) x a x a x bx. (.4) 4 x on 0, its integral diverges, so equation (.) has oscillatory solutions z and z which are continuous and differentiable. In the papers [5,6,7,8,9] by series iteration method we have shown that the solutions of equation (.) have following form x xdx xdx xdx z x x dx x dx x dx cos xdx xdx xx dx z sin x x x x x dx x dx x x dx (.5). (.6)

3 J. Vujaković, M. Rajović and D. Petković 5 For xk const. solutions are analytical function which are equal to Euclidean sine and cosine, while for arbitrary and positive x have a lot of similarities and mutual characteristics with sine and cosine. That is why we named functions (.5) and (.6) general or Sturm's sine and cosine with basis x. For them all analogies with standard trigonometry have been applied (zeros, inflections, Sturm's arrangements of zeros and extremes). From approximate formulas, which have proven in [6], we have cos x cos x x, (.7) x Here is obvious that sin x x x x x sin. (.8) o Frequency functions for both solutions are equal and approximately are G x x x. o mplitudes in general case are different. For cos x x is, and for, sin x x,. This verifies Prodi's theorem ([]): if x x, when x, then one solution remains bounded and second tends to zero. In other words, if x solutions are two-amplitudinal but equally-frequent. So, harmonic oscillations are equally-amplitudinal, but are they also only equally-amplitudinal or there are many types of them, we consider that these questions haven't been sufficiently elaborated and emphasized. Therefore, in this paper we shall try to determine simple equally-amplitudinal solutions with amplitude cos, sin y x x G x y x x G x (.9) x and unique common frequency G x.

4 6 Differential equations of amplitudes... From general differential equation of the second order y y y y y y 0 y y y expanding by first row and dividing by Wronskian we have, WW y y yy yy x G x yy yy yy yy y y y W W fter elementary calculation we obtain equation 0. y y G x y 0 (.0) x Gx x xgx x x Gx x xgx x Equation (.0) has sense if x 0 and G x 0, i.e., if 0, and if x, then we obtain no zeros on x has G x is monotonous. If from (.4) we calculate G x G x xgx. G x 4G x Comparing (.0) and (.) we have a x yy yy x G x, W x G x yy yy x x G x x. W x x Gx x b x G x Thus, frequency function x is the same for equations (.0) i (.). (.) (.) Let's notice that because of (.) the same cannot be applied for amplitudes exp a x dx x G x. If with y we mark amplitude because of equation (.0), and with z amplitude of canonical equation (.) then

5 J. Vujaković, M. Rajović and D. Petković 7 from obvious equality max y max x G x max z follows y x G x z. (.) max Equation (.0) shall be identical with its canonical if applies x G x a x x G x 0. This differential equation gives the first integral of form x G x c const G x (.4)., 0. (.5) Regarding the linearity of solution (.9), equation (.), it sufficient in equation (.5) to take c. So, problem of differential equations of frequency and amplitudes we first observe for narrower classes of functions. First, for the most general case ax 0 solution Gx is not simple. Second, more elementary, when condition (.4) applies, i.e., a x 0, connection of amplitude and frequency is given by equation (.5). Main Results In this Section in the form of theorem we present differential equations for amplitudes and frequencies. Theorem.. General unambiguous amplitude x, of differential equations (.0), that is (.4), which solutions are given by equation (.9), with successive approximations of Picard is obtained from differential equation 4 c xbxx. (.) x

6 8 Differential equations of amplitudes... Proof. Let a general equation (.) is given, where a x and bx are given continuous functions which satisfies Lipschitz condition. Comparing (.) and (.0) we obtain nonlinear system of differential equations of the second order x G x ax, x G x x x G x x bx G x. x x Gx x From here, we obtain the equation of the fourth order for Out of the first equation of the system we found x x or for (.) Gx. exp a x dx Gxc dxc. (.) Substituting (.) in second equation of the system (.) we obtain differential equation of amplitude For ax 0 x exp 4 a x dx xax xbxxc c. (.4), (.4) is nonlinear differential equation of the second order. Evaluating this we find two more constants c and c 4. Let ax 0. Since equation (.4) transforms into equation (.), for it is important not just solution existence but also possible quadratures. From here x, where x b x x 0. Substituting is given with (.). Equation (.) has solutions for x u it can be presented as system of equations

7 J. Vujaković, M. Rajović and D. Petković 9 Since,, x u f x u 4 c xu bxx f x, u,. x x and bx are continuous functions, f and f are also continuous, and Lipschitz condition gives f x u f x u c b x 4,,,, max, c bx k 4 min Here with procedure described in [6,7,8], it is easy to find x.. Theorem.. Nonlinear equation (.) on condition that G x 0 and x continuous function, has solution which approximately can be found by Picard's successive approximations method. Proof. For given according to ax and b x let's solve first equation of the system (.) x x c Gx exp axdx. (.5) When this value, along with x x we replace in second equation of and the system (.), after necessary calculation and cancellation, we obtain nonlinear differential equation of the third order for frequency function G x G x G x bxgx a x ax G x 4 G x. (.6) 4 From here, by solving we obtain not just frequency function, G x G a x b x, but also periods of possible equally-amplitudinal solutions of general linear homogeneous differential equation (.).

8 0 Differential equations of amplitudes... s in formula (.) we already have constants c and c, by solving (.6) we would obtain two more new constants c 5 and c 6 by what we are finalizing this problem. Let now a x 0, i.e. let condition (.4) applies and its consequence is (.5). Then (.6) becomes nonlinear equation of the third order G x G x bxgx x G x 4G x. (.7) Since is difficult to solve this equation with quadratures, the only thing left is to estimate the existence of possible solutions. Substituting we obtain system of two differential equations of the first order,, G x v w f x v w w G xv w xv v f x, v, w. v G x v from (.7) Let's notice that second equation of the system is nonlinear, but is not unbounded due to condition Gxv 0. This system satisfies Lipschitz condition, which can be seen from the evaluation w w f x v w f x v w v v x v vv v,,,, v v v v x v vv v w max w min. v v Let N is the least by module common multiple of Gx v, and x bx max, min. Since v v is continuous and bounded function, so due to v 0 function w is like that too. Then x v vv v k, so afore mentioned estimation finally becomes

9 J. Vujaković, M. Rajović and D. Petković fxv,, w fxv,, w k vv Nw w qvv w w. Thus, equation (.7) satisfies Lipschitz condition. Now we can apply a series iteration method and we can easely find solution. Classification of problems. Three types of problems x or a x and b x, has sense only for Solving of system of differential equations (.), for amplitude frequency function Gx, and for given some subclasses of these equations. That means, that we can take additional conditions but in a way that equations (.4) and (.6) apply. Example.. The most important place in classification of problems have equally amplitudinal oscillations, i.e. harmonic oscillations. Let the linear function G x x is given. From Gx, 0 and from (.6) we have bx a x ax G x G x Especially, for ax 0, (see (.)), follows b x y y 0 has solutions oscillations. From ax 0, that is, from solution x const.. 4. Now, canonic equation y cos x, y sin x which are harmonic x G x c const we obtain one. c for equation of amplitude (.). Example.. Let linear function G x x is given, ax 0 and b x x const.. The equation of the second order (.), for amplitude,

10 Differential equations of amplitudes..., or equivalent to becomes x x x x x 4 x. (.) This is differential equation which explicitly does not contains x. Substituting x p and variables: dp x p equation (.) becomes equation with separate d x x pdp x d x. From here we obtain first integral p k x, where k is integrating constant. Further we have p k x d x. Second separation of variables, as second x dx integral, gives xk x d x 4 k x x. Finally, by solving we obtain k k 4 x sin x k 4, where k and k are integration constants. Equally amplitudinal solutions are y x x and, cos. x can have zeros if and y x x, sin sin x k x does not exist when this sinus is less then zero. This is not in opposition to continuality guarantied by Theorem. (for x 0 solutions y and y, and zeros ). Namely, then we have opposite x mean vertical tangents in solutions. s a matter of fact, those are connections of two solutions y and y. It means, when y stops, further follows continuation with y which is of opposite sign. Therefore, continuality continues and sustains, even when x 0. We, therefore, conclude that three types of problems are possible:

11 J. Vujaković, M. Rajović and D. Petković o The simplest is to suppose that frequency G x is known, because then F x can be only found by derivatives G x, G x, G x. x can be easily found from (.5), but solutions in more complex cases are also possible, as shown in Example.. Knowing of form that equation (.) for x can be assumed, but in a way x is satisfied. From here x is determined, and from (.7) and Gx. o Knowing of x can also be assumed, but then x must be found from nonlinear equation (.), which is not trivial at all. It is more difficult to find Gx since for Gx we have nonlinear equation of the third order. o Without some, even minimal accuracy and specialization, or for x or for Gx, this nonlinear problem is quadrature unsolvable in general case. For Physics, however, the most important is connection given by ax 0, so, if x G x c const. CKNOWLEDGEMENTS. The authors are thankful to the Ministry of Science and Environmental Protection of Serbia. References [] G. Prodi, Un osservazione sugli intergali dell equazione y xy 0 caso nel x per x, Rend. ccad. Lincei, 8, (950), [] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, nn. Sci. École Norm. Sup.,, (88),

12 4 Differential equations of amplitudes... [] I. M. Babakov, Theory of oscillation, Nauka, Moscow, 960 (in Russian). [4] W.O. mrein,.m. Hinz and D.B. Pearson, Sturm-Liouville Theory, Past and Present, Birkhäuser Verlag, Basel, Switzerland, 005. [5] D. Dimitrovski and M. Mijatović, series iteration method for solving ordinary differential equations, Monography, Hadronic Press, Florida, US, 997. [6] M. Lekić, Sturm theorems throuhg iterations, Dissertation, University Kosovska Mitrovica, Faculty of Sciences and Mathematics, 007 (in Serbian). [7] D. Dimitrovski, M. Rajović and R. Stojiljković, On types, form and supremum of the solutions of the linear differential equation of the second order with entire coefficients, pplicable nalysis and Discrete Mathematics,, (007), [8] J. Vujaković, M. Rajović and D. Dimitrovski, Some new results on a linear equation of the second order, Computers and Mathematics with pplications, 6, (0), 87-84, doi: 0.06/j.camwa [9] D. Dimitrovski, D. Radosavljević, M. Rajović and R.Stojiljković, Sturm theorems for second order linear nonhomogeneous differential equations and localization of zeros of the solution, cta Math. Hungar., (-) (0), 5-6, doi: 0.007/s