Analytical solutions of the simplified Mathieu s equation

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1 Analytical solutions of the simlified Mathieu s equation Nicolae MARCOV,1 Corresonding author AEROSPACE Consulting, B-dul Iuliu Maniu 0, Bucharest 06116, Romania 1 University of Bucharest, Faculty of Mathematics and Comuter Science, Str. Academiei nr. 14, sector 1, , Bucharest, Romania nmarcov@fmi.unibuc.ro DOI: / Received: 13 January 016/ Acceted: 05 February 016 Coyright 016. Published by INCAS. This is an oen access article under the CC BY-NC-ND license (htt://creativecommons.org/licenses/by-nc-nd/4.0/) Abstract: Consider a second order differential linear eriodic equation. The eriodic coefficient is an aroximation of the Mathieu s coefficient. This equation is recast as a first-order homogeneous system. For this system we obtain analytical solutions in an exlicit form. The first solution is a eriodic function. The second solution is a sum of two functions, the first is a continuous eriodic function, but the second is an oscillating function with monotone linear increasing amlitude. We give a formula to directly comute the sloe of this increase, without knowing the second numeric solution. The eriodic term of the second solution may be comuted directly. The coefficients of fundamental matrix of the system are analytical functions. Key Words: linear differential equation with Mathieu coefficient, arametric resonance, eriodic term of the solution. 1. PROBLEM FORMULATION Consider the following second order non-linear differential equation with resect to real dimensionless time t, d z Q( z 0 In what follows we assume the coefficient Q to be a real ositive continuous eriodic function with t real argument. The following function is a reasonable aroximation of the Mathieu s coefficient [1], [], [3]. 8q (1 cos Q 1, 1/9 < q < 1/9. () 1 q q cos t The set of solutions is two-dimensional real sace [4], [5], [6], [7]. The function x is a eriodic solution. cost q cos3t d x dx x, Q x 0, x(0) 1, (0) 0 (3) 1 q (1) INCAS BULLETIN, Volume 8, Issue 1/ 016, ISSN

2 Nicolae MARCOV 16 Let y be a second solution which satisfies, dy Q y 0, y(0) 0, dy (0) 1. For the characteristic coefficient (q), [8], the y function is a eriodic term of the solution y. (4) y = y t x. (5) The roblem is to give the analytical formula for this coefficient.. EXPLICIT CHARACTERISTIC COEFFICIENT We recast the equation (1) as a first-order system. The following system is obtained. dz dw w, Q w (6) Let u be the derivative of the eriodic solution x. Excet the case that q is zero, the derivative v of the oscillating solution y is not a eriodic function. dx du dy dv u, Q x, v, Q y. (7) The functions x, u, y, v have the following roerty x v u y = 1 (8) Consequently, it results the exression of the fundamental matrix [3] for system (6). x u y v 1, v u y. x The y function is also a solution of fist-order linear equation d y d y x u y 1, y(0) 0, (0) 1. (10) The eriodic term y is the solution of the following first-order linear differential equation. Indeed, substituting the function y in the equation above, we find this linear equation (11). dy y = y t x, v, v = v t u + x, x (v + t u + x) u (y + t u) = 1 where d y d y x x u y 1, y(0) 0, (0) 1. (11) cost dx sin t 1 3q 4q cos t, u 1 3q 1q t 1 q 1 q cos (1) x (9) INCAS BULLETIN, Volume 8, Issue 1/ 016

3 17 Analytical solutions of the simlified Mathieu s equation In order to comute the eriodic term we can also consider the fourth-order system (13). dv dv du dv t u, Q y Q( t x) u dx u, x(0) 1, du Q x, u(0) 0, dy v, y (0) 0, dv Q y u v (0) 1 (13) Let C be the variable constant of integration for the equation (11), when dc 1 y = C x C(0) = 0. (14) x We shall consider constant 1 q 1 3q We introduce the equivalent exressions for the exact x solution, cost tan t cost x( (1 3q) 1 ( 1)cos t (15) 1 q 1 tan t If we make the change of the variable of integration, then the function h(s) is the solution of the following equation, dh 1 s ( h( s), s tant, 4 r( s), r( s) C (16) ds 1 s s The rational function r has the equivalent exression 1 1 (1 ) s 1 r ( s) 1 1 (17) s s ( s ) s Hence we have an aroriate formula for r function. β s (1 )(1 3 ) 1 1 r ( s) 1 γ, β, γ. (18) s ( s ) Consequently, it results the equation of the function h. dh ds d ds s s s 4 3 s γ β, h(0) =0 (19) 1 s In order to obtain the value of the coefficient it is necessary to imose the following integral condition. 0 σ 3 3 β ds β σ 0 s 1 s (0) INCAS BULLETIN, Volume 8, Issue 1/ 016

4 Nicolae MARCOV 18 The characteristic coefficient may be found using the formula 1 8q β( ) (1 3q) (1) when q, 1 3q 1 8q (1 q) (1 3 q) Let be the integral eriodic function 3. SECOND EXPLICIT SOLUTION t ( sign () 0 1 ( 1)cos t The restriction of on the interval [0, /) is a known function. Integrating the given equation (14) above, we obtain the exression of C constant variable. 4 γ cos t C( C ( β (, C ( 1 tant. (3) 1 ( 1)cos t From (15) we can use the equivalent exression of the x exact solution Hence γ cos t cost x 4 1 tant 1 1 t 1 ( 1)cos t cos (4) C y C x = [1+ ( 1) cos t + cos t ] sin t = ) cos t ] sin t Consider the following identity Therefore y can be written as 1 / y( : [ + ( 1 /) cos t ] sin t (5) From this it results the eriodic term of the y solution. Therefore It is easy to find the derivative v v y ( = y( + x( ( (6) y( = y( + x( [ ( + t ] (7) dy β β sin t cost (8) Consequently, we deduce successively INCAS BULLETIN, Volume 8, Issue 1/ 016

5 19 Analytical solutions of the simlified Mathieu s equation v d y v x 1 ( 1) cos t sign u t ( ) v = v + (/) cos t x + u (, v = v + t u (9) The characteristic matrix is the sum of a eriodic matrix and an oscillating matrix x u y x v u v y β t β x 0 x 0 u 0 x β ( β t 0 u 0 x 0 u (30) x y (31) u v (β/ )cost The matrix and are unimodular matrix. Finally, the solution of the system (6) has the following exression z( z(0) ( w( w(0) 4. RESULTS The coefficients of the two-order differential linear system are real continuous functions. The unique variable coefficient is a function Q (q, in which q is real small arameter. In this articular case it is known an analytical exlicit solution (x, u) T. In order to find the exression of the characteristic coefficient, [8], [9], it was imosed an exlicit necessary imroer integral condition. So it was found the exlicit analytical formula for the characteristic coefficient of one articular two-order differential system. Consequently, it was found the second exlicit solution (y,v) T. The vector (ytx,vtu) is a eriodic vector. The comonents of the fundamental matrix have exlicit exressions in which y and v are trigonometric olynomial functions and ( is a definite integral on the real interval (0,. For directly calculating the eriodic term y, the fourth-order system (13) is useful, but if and only if the arameter is equal with the characteristic coefficient, [10], [11]. REFERENCES [1] É. Mathieu, Mémoire sur le mouvement vibratoire d une membrane de forme ellitique, Journal de Mathématiques Pures et Aliquées, Tome XIII (e série), , Bachelier Imrimeur Librairie, Paris, [] E. Janke, F. Emde, F. Lösch, Tafeln Höherer Funktionen, B. G. Teubner Verlagsgesellschaft Stuttgart, [3] R. P. Voinea, I. V. Stroe, Introduction in the theory of dynamical systems (in Romanian language), Ed. Acad. Române, ISBN , Bucharest, 000. [4] Vo-Khac Khoan, Distribution Analyse de Fourier Oérateurs aux Derivées Partielles, (cours), Librairie Vuibert, Paris, 197. [5] H. Brezis, Analyse fonctionnelle. Théorie et alications 1983, Dunod, Paris. [6] T. Petrila, C. Gheorghiu, Finite element methods and alications, Ed. Acad. Române, Bucharest, [7] A. Carabineanu, Theoretical Mechanics, (in Romanian language), Matrix Rom, Bucharest, 006, ISBN (10) , ISBN (13) [8] P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhauser Verlag, INCAS BULLETIN, Volume 8, Issue 1/ 016

6 Nicolae MARCOV 130 [9] J.P. Tian and J. Wang, Some results in Floquet theory, with alication to eidemic modeling, Alicable Analysis, vol. 94, 015, , htt://dx.doi.org/ / [10] E. Scheiber, M. Luu, Matematici seciale: rezolvarea roblemelor asistată de calculator cu exemlificări in Derive, Mathcad, Male, Mathematica, (in Romanian language), Ed. Tehnică, Bucureşti, 1998, ISBN [11] I. Paraschiv-Munteanu, D. Stănică, Analiză numerică, (in Romanian language), Editura Universităţii din Bucureşti, 008, ISBN INCAS BULLETIN, Volume 8, Issue 1/ 016