An extension to the theory of trigonometric functions as exact periodic solutions to quadratic Liénard type equations

Transcription

1 An extension to the theory of trigonometric functions as exact eriodic solutions to quadratic Liénard tye equations D. K. K. Adjaï a, L. H. Koudahoun a, J. Akande a, Y. J. F. Komahou b and M. D. Monsia a1 a Deartment of Physics, University of Abomey-Calavi, Abomey-Calavi, 01.B.P. 56, Cotonou, BENIN monsiadelhin@yahoo.fr b Deartment of Industrial and Technical Sciences, ENSET-Lokossa, University of Abomey, Abomey, BENIN komaf@yahoo.fr This aer slightly extends the theory of exact trigonometric eriodic solutions to quadratic Liénard tye equations introduced earlier by the authors of the resent contribution. The extended theory is used to determine the general eriodic solutions to the Duffing equation and to some Painlevé-Gambier equations as illustrative examles. Finally the mathematical equivalence between the Duffing equation and the Painlevé-Gambier XIX equation has been highlighted by means of the roosed extended theory. Keywords: Duffing equation, Painlevé-Gambier equations, Liénard equations, eriodic solution, Jacobian ellitic functions. 1 Introduction The fundamental roblem of finding general eriodic solutions to nonlinear differential equations constitutes yet an active research field of mathematics since such solutions are only ossible for a few number of nonlinear differential equations. As many ractical roblems are modeled in terms of nonlinear equations, there aears then logic to be interested to mathematical theories which may rovide with a relative simlicity general eriodic solutions to these equations. In the matter the theory of exact trigonometric eriodic solutions to quadratic Liénard tye differential equations introduced recently by the authors of this aer seems to belong to this class of mathematical theories [1]. Indeed, this theory has the ability to linearize some classes of Liénard tye nonlinear differential equations in order to establish exact analytical solutions by means of the generalized Sundman transformation, and conversely to highlight some families of Liénard tye equations from linear differential equations. So an interesting finding of this theory was the detection of a general class of quadratic Liénard tye equations whose solutions are trigonometric eriodic functions. In this regard it is aroriate to state the fundamental question: Is it ossible to extend this theory by using a forcing function? The resent work assumes this ossibility. This fact may give the advantage to detect large classes of Liénard tye equations for which exact analytical solutions may be calculated. To do so, a brief review of the theory introduced by Monsia and his coworkers is given section ) and secondly, the extended theory under question is carried out section 3). Finally the general eriodic solutions to Duffing equation and some Painlevé-Gambier equations are determined section 4) and a conclusion for the research contribution is formulated. 1 Corresonding author. monsiadelhin@yahoo.fr 1

2 Theory by Akande et al. [1] Let us consider the general second order linear differential equation as y τ) + by τ) + a yτ) = 0.1) where rime denotes differentiation with resect to τ, a and b are arbitrary arameters. By alication of the generalized Sundman transformation F t, x) yτ) = F t, x), dτ = Gt, x)dt, Gt, x) 0.) x with F t, x) = gx) l dx, Gt, x) = exγϕx)) where l and γ are arbitrary arameters, and gx) 0 and ϕx) are arbitrary functions of x, to.1), one may obtain ẍ + l g x) gx) γϕ x))ẋ + bẋ exγϕx)) + a exγϕx)) gx) l dx gx) l = 0.3) as general class of mixed Liénard tye differential equations. The arametric choice b = 0 leads to ẍ + l g x) gx) γϕ x))ẋ + a exγϕx)) gx) l dx gx) l = 0.4) as general class of quadratic Liénard tye nonlinear equations. An interesting case of.4) is obtained by considering l = 0, viz ẍ γϕ x)ẋ + a x exγϕx)) = 0.5) The imortance of.5) is that it exhibits trigonometric functions as exact eriodic solutions but with amlitude-deendent frequency. In [1] it is shown for the first time that some existing Liénard tye equations may exhibit exact trigonometric eriodic solutions. One may also find that the quadratic Liénard tye equation ẍ + 1 ẋ 1 + x + x 1 + x = 0.6) used to model oscillation of a liquid column in a U-tube [] belongs to the class of equations defined by.5) by utting ϕx) = 1 ln1 + x), and γ = 1 [3], so that this equation may exhibit trigonometric functions as exact eriodic solution [3]. That being so the extended theory may be formulated. 3 Extended theory The objective of this section is to extend the receding theory to look for large classes of linearizable Liénard tye equations in order to ensure exact solutions which may be exressed exlicitly or by quadratures. In the revious theory the general second order linear equation is considered in the form of homogeneous equation with no forcing function. Instead in this section, let us consider the general second order linear equation.1) with a constant forcing term, y τ) + by τ) + a yτ) = c 3.1) where c is an arbitrary arameter. The alication of the generalized Sundman transformation.) to 3.1) yields ẍ + l g x) gx) γϕ x))ẋ + bẋ exγϕx)) + a exγϕx)) gx) l dx gx) l c exγϕx)) gx) l = 0 3.) The comarison of.3) with 3.) shows that the constant forcing function c contributes to the general class of mixed Liénard tye equation.3) by an additional term c exγϕx)) gx) l. This establishes the

3 extension of the theory by Akande et al. [1] to a wider class of Liénard tye nonlinear differential equations. The question now is to know: What haens to equations.4) and.5) with resect to this additional term? Let us consider then b = 0 and l = 0. The arametric choice b = 0 leads to ẍ + l g x) gx) γϕ x))ẋ + a exγϕx)) gx) l dx gx) l c exγϕx)) gx) l = 0 3.3) The comarison of.4) with 3.3) shows also the ersistence of the receding additional term in the general class of quadratic Liénard tye equations reresented by 3.3). The alication of l = 0 to 3.3) gives the class of quadratic Liénard tye equations ẍ γϕ x)ẋ + a x exγϕx)) c exγϕx)) = 0 3.4) which differs from the class of quadratic Liénard tye equations.5) by the additional term c exγϕx)). By noting yτ) = A 0 sinaτ + α) + c a 3.5) where a 0, the general solution to 3.1) with b = 0, the general solution to 3.4) becomes where A 0 and α are arbitrary arameters, and τ = φt) satisfies xt) = A 0 sinaφt) + α) + c a 3.6) dt = ex γϕx))dφt) 3.7) The solution 3.6) remains of eriodic form. For a convenient choice of ϕx) and γ the solution 3.6) may exhibit harmonic eriodic behavior but with a slift factor c a. This extends therefore the theory by Akande et al. [1]. It is then convenient to show the ability of the current theory to rovide general eriodic solutions to some existing Liénard equations. 4 Alications of theory This section is devoted to show the usefulness of the resent theory by considering some well known nonlinear equations. 4.1 Duffing equation The Duffing equation is one of the most investigated equations from mathematical as well as hysical viewoint. The motivation results from the fact that this equation arises in mathematical modeling of many roblems of mechanics and quantum mechanics, for examle. The Duffing equation is of the form [4] where ω 0 and are arbitrary arameters. ẍ + ω 0x + x 3 = 0 4.1) From the mechanical viewoint, the arameter > 0 is called hardening arameter while < 0 is called softening arameter, and = 0, gives the well known linear harmonic oscillator equation. The equation 4.1) may be recovered from the general class of quadratic Liénard equations 3.3). Indeed, the functional choice ϕx) = lnfx)) leads to which for fx) = x and gx) = x, yields ẍ + l g x) gx) γ f x) fx) )ẋ + a fx) γ gx) l dx gx) l cfx)γ gx) l = 0 4.) γ)ẋ ẍ + l x + a l + 1 x4γ+1 cx 4γ l = 0 4.3) 3

4 The arametric choice l = γ = 1, reduces 4.3) to ẍ cx + a x3 = 0 4.4) which is the Duffing equation 4.1) by noting c = ω0, and a =. So the general solution to the Duffing equation 4.1) may be written according to 3.5) xt) = ε ω 0 + A 0 sinaφt) + α) 4.5) where dφt) εdt = ω 0 + A 0 sinaφt) + α) and ε = ±1. The evaluation of the integral resulting from 4.6) J = dφt) ω 0 + A 0 sinaφt) + α) 4.6) 4.7) deends on the value of ω 0 and. Therefore three distinct cases are considered in this aer [5]. Case 1: > 0, and ω 0 < 0 In this case where > 0 and ω 0 < 0, it is moreover assumed that ω 0 > A 0 > 0. By noting [5] = A 0 ω0 + A 0 = A 0 A 0 ω 0 4.8) and the integral J becomes [5] δ = arcsin 1 sinaφ + α) 4.9) J = 1 F δ, ) 4.10) a where F δ, ) is the ellitic integral of the first kind so that using 4.6) sin δ = sn aε ) A 0 t + C), 4.11) where snz, k) designates a Jacobian ellitic function and C an arbitrary arameter. In this way 1 sinaφ + α) = sn aε ) A 0 t + C), to say sinaφ + α) = 1 sn aε ) A 0 t + C), 4.1) 4

5 from which the general solution to the Duffing equation 4.1) becomes which reduces to [ xt) = ε ω 0 + A 0 4A 0 sn aε ] 1 t + C), ) xt) = ε A 0 [ 1 sn aε A 0 t + C), ) Using the identity k sn z, k) + dn z, k) = 1, the relation 4.14) becomes immediately xt) = ε ) A 0 aε dn t + C), Knowing that a = ε, 4.15) may be written in the form xt) = ε ) A 0 dn t + C), ] ) 4.14) 4.15) 4.16) By making A = A 0, and Ω = A 0, Ω = A, the general solution to 4.1) definitively reads where dnz, k) is a Jacobian ellitic function, and = A +ω 0 ) A. Case : < 0, and ω 0 > 0 xt) = εadn Ωt + C), ) 4.17) For < 0 and ω0 > 0, it is also assumed that ω 0 > A 0 > 0. So, the general solution to the Duffing equation 4.1) takes, using 4.17), the form xt) = εadc Ωt + C), ) ) where A = A 0, and Ω = A. The function dc is a Jacobian ellitic function. Case 3: > 0, and ω 0 > 0 In this case, it is assumed that 0 < ω 0 < A 0, 0 < ω 0 < A 0. So, the integral J may be written as [5] J = F δ, 1 aa 0 ) 4.19) where δ = arcsin [ ] A 0 1 sinaφ + α)) A 0 ω0 4.0) has the receding value, and F δ, 1 ) denotes the ellitic integral of the first kind. In this regard the following relationshi may be written, taking into consideration 4.6) [ A 0 1 sinaφ + α)) A 0 ω0 = sn aε A 0 t + C), 1 ] 4.1) sinaφ + α) = 1 A 0 ω0 [ sn aε A 0 t + C), 1 ] A 0 4.) 5

6 which may be rewritten sinaφ + α) = 1 sn aε A 0 t + C), 1 ) 4.3) In this ersective the general solution to 4.1) reads xt) = ε ω 0 + A 0 4A 0 sn aε A 0 t + C), 1 ) which may take the exression xt) = ε A 0 cn a A 0 t + C), 1 ) 4.4) Knowing a = ε, A = A 0 solution to Duffing equation 4.1) reduces immediately to the form, and Ω = a A 0, Ω = A, or Ω = A + ω0, the general xt) = εacn Ωt + C), 1 ) 4.5) where cnz, k) denotes a Jacobian ellitic function It would be now interesting to consider in the sequel of this work some Painlevé-Gambier equations. 4. Painlevé-Gambier XII equation This subsection is intended to carry out the general solution to the Painlevé-Gambier XII equation [6] for r = δ = 0, to say ẍ ẋ x qx3 x r δ x = 0 4.6) ẍ ẋ x qx3 x = 0 4.7) The Painlevé-Gambier equation 4.7) may be found from 3.4) by choosing ϕx) = ln x, γ = 1, a = q, and c =. Therefore the general solution to 4.7) takes the exression xt) = A 0 sinε qφt) + α) q 4.8) where ε = ±1, and dt = dφt) xt) dφt) dt = A 0 sinε qφt) + α) q 4.9) By integration the quantity J = dφt) A 0 sinε qφt) + α) q 4.30) leads to consider two distinct cases following the value of q. Case 1: q A 0 < 1 In this case the integral J becomes [5] J = q ε q3 A 0 q tg 1 6 ε tg qφt)+α) ) q 1 q A )

7 such that ε qφt) + α = tg 1 1 q A 0 tg ε q3a 0 qt + C) q + qa 0 4.3) So the general solution to 4.7) becomes xt) = A 0 sin tg 1 1 q A 0 tg ε q3a 0 qt + C) + qa 0 q q 4.33) The solution xt) is real for q < 0. In this way, the solution 4.33) may clearly exhibit a harmonic eriodic behavior with a shift factor q. Case : q A 0 > 1 This case corresonds to [5] J = q ln ε q q3 A 0 which gives according to 4.9) ε qφt) + α = tg 1 q A 0 1 ε tg qφt)+α) ) q q A 0 tg ε qφt)+α) ) q + q A ex ε q 1 ex ε q q q3 A 0 q q3 A 0 t + C))) ) ) t + C)) + qa ) Therefore the general solution xt) may be exressed in the form )) q A ex ε xt) = A 0 sin tg 1 q q q3 A 0 t + C) ) + qa 0 1 ex ε q q q3 A q 0 t + C)) 4.36) For q < 0, the solution 4.36) is real and then may exhibit also a harmonic eriodic behavior with a shift factor q. Consider now as final illustrative examle a generalized Painlevé-Gambier XIX equation. 4.3 Generalized Painlevé-Gambier XIX equation The Painlevé-Gambier XIX equation is of the form [6] A general form of 4.37) may be written as ẍ 1 ẋ x 4x x = ) ẍ 1 ẋ x + a x cx = ) so that for a = 4, and c =, one may recover the Painlevé-Gambier XIX equation. The equation 4.38) may be obtained from 3.4) by setting γ = 1 4, and ϕx)=ln x. Thus the solution to 4.38) immediately takes the exression following 3.5) xt) = A 0 sinaφt) + α) + c a 4.39) 7

8 where dt = dφt) sinaφt) + α) + c a 4.40) This equation is of the same form as 4.6). So, three distinct cases may be distinguished Case 1: a > 0, c > 0, and c a > A 0 > 0 In this case [5] and δ = arcsin a = A 0 a A 0 + c ) 1 sinaφ + α) 4.41) 4.4) so the integral becomes J = dφt) sinaφt) + α) + c a 4.43) J = a A 0 F δ, ) 4.44) where F δ, ) designates the ellitic integral of the first kind. So according to 4.40) one may write a A 0 t + C) = F δ, ) In this regard, the general solution 4.39) becomes sin δ = sn a ) A 0 t + C), xt) = A ) 0 a dn t + C), 4.45) 4.46) which may take the exression where A = A 0, Ω = a A 0 xt) = A dn [Ωt + C), ] 4.47) Ω = aa, and may be rewritten as = a A c) a A Case : a < 0, c > 0 and 0 < c a < A 0 In such a situation, the integral [5] where δ = arcsin J = 1 F δ, 1 a A 0 ) 4.48) ) a A 01 sinaφ+α)) a a A 0+c, and = A 0 a A 0+c. So the use of 4.40) leads to a t + C) = F δ, 1 ) 4.49) 8

9 which may give from which ) a A 0 1 sinaφ + α)) a = sn a A 0 + c t + C), 1 ) sinaφ + α) = 1 a A 0 + c a sn a A 0 t + C), 1 Therefore the general solution 4.39) may take the exression [1 sn xt) = A 0 The general solution 4.5) may be also exressed as xt) = A 0 cn a a t + C), 1 t + C), 1 ) )] 4.50) 4.51) 4.5) 4.53) which becomes where A =. aa xt) = A cn t + C), 1 ) For a = i a, a = ±i a, the general solution xt) may take the form which becomes definitively xt) = A cn xt) = i ) a A t + C), 1 A 4.54) [ ] 4.55) cn a A t + C), 1 1 A xt) = ] 4.56) cn [Ωt + C), 1 1 a A where Ω =. By making a = 4, and c =, the exact doubly eriodic solution to Painlevé-Gambier XIX equation may be written as where A = 1, and = 1 A A + 1 A xt) = [ A cn + 1t + C), A xt) = cn [Ωt + C), Case 3: a > 0, c < 0 and 0 < c a < A 0 This case corresonds also to = a A 0 a A, δ = arcsin 0+c J = 1 a A +1 A +1 A +1 A +1 ] 4.57) ] 4.58) ) a A 01 sinaφ+α)) a A 0+c, and the integral A 0 F δ, 1 ) 4.59) 9

10 So the general solution 4.39) may be written as xt) = A 0 cn a t + C), 1 ) 4.60) xt) = A cn Ωt + C), 1 ) 4.61) where and A = A 0 4.6) Ω = aa 4.63) The arameter may also be exressed as = a A c) a A. That being so it is then ossible to show the equivalence between the Duffing equation and the generalized Painlevé-Gambier XIX equation. 5 Equivalence between equations This section is devoted to highlight the mathematical equivalence between the Duffing equation and the generalized Painlevé-Gambier XIX equation. The comarison of 4.6) with 4.40) as well as the comarison of resulting general solutions suggest this mathematical equivalence, to say the maing of the Duffing equation onto the generalized Painlevé-Gambier XIX equation and vice versa, the maing of the generalized Painlevé-Gambier XIX equation into the Duffing equation. In other words, the general solution to the Duffing equation may be obtained in terms of the solution to the generalized Painlevé-Gambier XIX equation and vice versa, the solution to the generalized Painlevé-Gambier equation may be calculated in terms of the general solution to Duffing equation. Formally consider the variable transformation x = W 5.1) due to the above general solutions to Duffing equation and general solutions to the generalized Painlevé- Gambier equation 4.38). The substitution of 5.1) into 4.1) yields Ẅ 1 Ẇ W + 4W + ω0w = 0 5.) which is the generalized Painlevé-Gambier XIX equation 4.38) by taking a = 4, and c = ω 0. So with that the equivalence between Duffing equation and the generalized Painlevé-Gambier XIX equation has been shown and a conclusion may be formulated for the resent work. Conclusion The vital roblem of finding exact eriodic solutions to nonlinear differential equations is still an active research field of mathematics. A slight extension of an earlier Liénard tye nonlinear differential equations theory is introduced in this aer for the determination of exact eriodic solutions as well as exact trigonometric eriodic solutions. By doing so the general solution to the Duffing equation as well as for some Painlevé-Gambier equations are in a clear and simle fashion determined. In this ersective as an interesting result, it has been shown that the Duffing equation is mathematically equivalent to the generalized Painlevé-Gambier XIX equation such that the solution of the last equation may be obtained in terms of the solution of the first and vice versa, the solution of the first may be determined in terms of the solution of the last equation. 10

11 References [1] J. Akande, D.K.K. Adjaï, L.H. Koudahoun, Y.J.F. Komahou, M.D. Monsia, Theory of exact trigonometric eriodic solutions to quadratic Liénard tye equations, vixra: V3 017). [] W.R.Utz, eriodic solutions of a nonlinear second order differential equation, SIAM J. Al. Math, 191), [3] J. Akande, D.K.K. Adjaï, L.H. Koudahoun, Biswanath Rath, Pravanjan Mallick, Rati Ranjan Sahoo, Y.J.F. Komahou, Marc D. Monsia, Exact quantum mechanics of quadratic Liénard tye oscillator equations with bound states energy sectrum, vixra: V1 017). [4] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations John Wiley and Sons New York), 1979). [5] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Seventh edition, 007. [6] E.L. Ince, Ordinary differential equations, Dover, New York