Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation

This paper shows that explicit and exact general periodic solutions for various types of Liénard equations can be computed by applying the generalized Sundman transformation. As an illustration of the efficiency of the proposed theory, the cubic Duffing equation and PainlevéGambier equations were considered. As a major result, it has been found, for the first time, that equation XII of the Painlevé-Gambier classification can exhibit, according to an appropriate parametric choice, trigonometric solutions, but with a shift factor.


Introduction
A large part of physics, engineering and applied science problems is solved by using linear models where the principle of superposition may be applied. However, this consideration is only valid as a first approximation, since a more realistic approach in the mathematical modeling of many processes and phenomena in physics and applied science leads to consider nonlinear problems often expressed in terms of nonlinear differential equations. The phenomena of oscillation,dielectric polarization, creep deformation and stress relaxation under loading in dynamic structures are examples of problems where the nonlinearity properties occur in different aspects [1]. In structural dynamics, for example, geometric nonlinearities are often due to large displacements or large deformations inducing not only stiffening but also nonlinear optical properties of the structure. The material nonlinearity is relative to the stressstrain constitutive law which implies the phenomena of dissipation of energy in heat, and the inertial nonlinearity results often from the quadratic terms in the first order derivative in the equations of motion or deformation [1,2]. Thus, the failure to take into account these nonlinearities can permanently affect the performance of these material structures during their service, which can become a danger for users and the environment. In this sense, good design of these structures or repair after damage requires to calculate the explicit and exact general solutions of the nonlinear differential equations describing their dynamics. In this perspective the explicit and exact general solutions of these nonlinear differential equations are more appropriate from the physical point of view as well as engineering application viewpoint than approximate and numerical solutions. In so doing, the periodic solutions constitute a particular class of solutions since many phenomena in the real world, such as mechanical oscillations of conservative systems, are of a periodic nature. Thus a special interest must be accorded to the explicit and exact general periodic solutions. However, it should be emphasized that there is no master mathematical method that works in all cases of calculation of explicit and exact general periodic solutions. It suffices to note the large and rich variety of mathematical methods existing in the literature for this purpose. Therefore the fundamental problem of finding explicit and exact general periodic solutions to nonlinear differential equations constitutes yet an active research field of mathematics. It is also known that such solutions are only possible for a few number of nonlinear differential equations. There appears then logic to be interested to mathematical theories which may provide with simplicity and elegance explicit and exact general periodic solutions to nonlinear differential equations. In the process of solving nonlinear second order differential equations, it is convenient to consider the linearization methods which transform the initial equation into a linear differential equation whose properties are well known. As such, one can note methods like point transformation, contact transformation and generalized Sundman transformation. The latter, according to Euler and Euler [3], is a more general transformation than the first two transformations of linearization. The generalized Sundman transformation is a nonlocal transformation that has been frequently used to determine the first integrals and solutions of a variety of Liénard-type differential equations [3][4][5]. Recently, Akande et al. [6] have formulated a generalized Sundman transformation applied to the harmonic oscillator equation to demonstrate for the first time, the existence of a class of quadratic Liénard type equations whose solutions are trigonometric functions, but with amplitude dependent frequency. Thus, it has been shown that equation XVIII of the Painlevé-Gambier classification [7] and its inverted version and other quadratic Liénard type equations admit trigonometric periodic solutions by applying the generalized Sundman transformation developed. It is also well known that the study of oscillation in many mechanical systems leads to consider the cubic Duffing equation consisting to introduce the cubic term in the displacement into the harmonic oscillator equation of the linear structural model in order to take into account the geometric nonlinearity properties. Recently it is found that the theoretical investigation of viscoelastic deformation in material structures taking into consideration simultaneously the geometric, material and inertial nonlinearities, leads to nonlinear problems described in terms of mixed or quadratic Liénard type equations [8]. One may note in this context that the quadratic Liénard type differential equations and the cubic Duffing equation have important applications in physics. The cubic Duffing equation plays an interesting role in many areas of physics like classical and quantum mechanics [9]. In this regard the forced cubic Duffing equation is often, for example, used to illustrate a number of nonlinear system characteristics such as jumps, subharmonic and superharmonic resonances in addition to the amplitude dependent frequency exhibited already by the free cubic Duffing oscillator. Let us now consider the cubic Duffing equation and equations XIX and XII of the Painlevé-Gambier classification. The general solution of the unforced cubic Duffing equation has often been calculated by means of specific methods based on the fact that this equation is an elliptic equation, that is to say, the Jacobian elliptic functions are solutions of this equation. On the other hand, Painlevé-Gambier equations XIX and XII have recently been studied by the application of generalized Sundman transformation [4]. The authors [4] were able to calculate the first integrals in addition to those already determined by Ince [7], and parametric solutions for these equations. However, no explicit and exact general periodic solution has been computed for these equations by the generalized Sundman transformation used by those authors [4]. The cubic Duffing equation belongs to the usual class of Liénard equations, whereas the Painlevé-Gambier XII and XIX equations are of the class of the quadratic Liénard type equations. According to the classification given by Ince [7], the equation XII is a canonical equation of type II, whereas the equation XIX is a canonical equation of type III. Although these equations have been already studied in the literature with more or less of success, it is important, to further identify their analytical properties, to investigate them by different types of analytical methods, with also a view on the simplicity, elegance and extent of application of the methods concerned. In other words, it is reasonable to ask whether other integration methods could not be applied for such equations, since these new methods may appear easy and more elegant to be applied in the resolution of the problem and may lead to new solutions in some cases. The problem now is to show that the explicit and exact general periodic solutions can be computed for these nonlinear differential equations by applying the generalized Sundman transformation proposed by Akande et al. [6]. The fundamental question to be solved, in the context of the generalized Sundman transformation proposed by Akande et al. [6], can then be formulated as follows: Can we calculate the explicit and exact general periodic solutions of the equations considered above? The present work supposes this fulfillment by applying the generalized Sundman transformation proposed by Akande et al. [6]. This fact will allow testing the efficiency of the proposed generalized Sundman transformation to ensure the calculation of explicit and exact general periodic solutions to a wide variety of nonlinear ordinary differential equations that are often encountered in mathematical modeling of physics and applied science problems. The present theory will have the advantage of generalizing the approach developed by Akande et al. [6] and other methods that exist in literature in the perspective of a large class of nonlinear differential equations of the Liénard type for which explicit and exact general periodic solutions can be computed by a well-documented linearization method, that is the generalized Sundman transformation. As practical implications, these results may be useful in numerical experiments involving these equations. To compute the explicit and exact general periodic solutions of these equations, a brief review of the theory introduced by Akande and his coworkers [6] is given (section 2) and secondly, the generalized theory under question is carried out (section 3). Finally the explicit and exact general periodic solutions to unforced cubic Duffing equation and for some Painlevé-Gambier equations are computed (section 4) and, a discussion and conclusion for the research contribution is carried out.
1 Review of the theory by Akande and coworkers [6] Let us consider the general second order linear differential equation as where prime denotes differentiation with respect to τ , a and b are arbitrary parameters. By application of the generalized Sundman transformation where l and γ are arbitrary parameters, and g(x) = 0 and ϕ(x) are arbitrary functions of x , to (1.1), one may obtainẍ as general class of mixed Liénard type differential equations.
The parametric choice b = 0, leads tö as general class of quadratic Liénard type nonlinear equations. An interesting case of (1.4) is obtained by considering The importance of (1.5) is that it exhibits trigonometric functions as exact periodic solutions but with amplitudedependent frequency. In [6] it is shown for the first time that some existing Liénard type equations may exhibit exact trigonometric periodic solutions. One may also find that the quadratic Liénard type equation x + 1 2ẋ used to model oscillation of a liquid column in a U-tube [10] belongs to the class of equations defined by (1.5) by putting ϕ(x) = 1 2 ln(1 + x), and γ = −1, so that this equation may exhibit trigonometric functions as exact periodic solution [11]. That being so the general theory under question may be formulated.

General theory
The objective of this section is to extend the preceding theory to look for large classes of linearizable Liénard type equations in order to ensure exact solutions which may be expressed explicitly or by quadratures. In the previous theory the general second order linear equation is considered in the form of homogeneous equation that is with no forcing function. Instead in this section, let us consider the general second order linear equation (1.1) with a constant forcing term, that is where c is an arbitrary parameter. The application of the generalized Sundman transformation (1.2) to (2.1) yields The comparison of (1.3) with (2.2) shows that the constant forcing function c contributes to the general class of mixed Liénard type equation (1.3) by an additional term − c exp(2γϕ(x)) . This establishes the extension of the theory by Akande et al. [6] to a wider class of Liénard type nonlinear differential equations. The question now is to ask: What happens to equations (1.4) and (1.5) with respect to this additional term? Let us consider then b = 0 and l = 0. The parametric choice b = 0 leads tö The comparison of (1.4) with (2.3) shows also the persistence of the preceding additional term in the generalized class of quadratic Liénard type equations represented by (2.3). The application of l = 0 to (2.3) gives the class of quadratic Liénard type equations which differs from the class of quadratic Liénard type equations (1.5) by the additional term −c exp(2γϕ(x)). By noting where a = 0, the general solution to (2.1) with b = 0, the general solution to (2.4) becomes where A0 and α are arbitrary parameters, and τ = φ(t) satisfies The solution (2.6) remains of periodic form. For a convenient choice of ϕ(x) and γ the solution (2.6) may exhibit harmonic periodic behavior but with a shift factor c a 2 . This extends therefore the theory by Akande et al. [6]. It is then convenient to show the ability of the current theory to provide explicit and exact general periodic solutions to some existing Liénard equations.

Applications of general theory
This section is devoted to show the usefulness of the present theory by considering some well known nonlinear equations.

Cubic Duffing equation
The cubic Duffing equation is one of the most investigated equations from mathematical point of view as well as physical viewpoint. The motivation results from the fact that this equation arises in mathematical modeling of many problems of mechanics and quantum mechanics, for example. The cubic Duffing equation is of the form [12] x + ω 2 0 x + βx 3 = 0 (3.1) where ω0 and β are arbitrary parameters.
From the mechanical viewpoint, the parameter β > 0 is called hardening parameter while β < 0 is called softening parameter, and β = 0, gives the well known linear harmonic oscillator equation. The equation (3.1) may be recovered from the general class of quadratic Liénard equations (2.3). Indeed, the functional choice ϕ(x) = ln(f (x)) leads toẍ which for f (x) = x 2 and g(x) = x, yields The parametric choice l = 2γ = 1, reduces (3.3) tö which is the Duffing equation depends on the value of ω 2 0 and β. Therefore three distinct cases are considered in this paper [13].
Case 1: β > 0, and ω 2 0 < 0 In this case where β > 0 and ω 2 0 < 0, it is moreover assumed that − where F (δ, p) is the elliptic integral of the first kind so that using (3.6) where sn(z, k) designates a Jacobian elliptic function and C an arbitrary parameter. In this way from which the general solution to the Duffing equation (3.1) becomes which reduces to (3.14) Using the identity k 2 sn 2 (z, k) + dn 2 (z, k) = 1, the relation (3.14) becomes immediately Knowing that a = ε √ 2β, (3.15) may be written in the form where dn(z, k) is a Jacobian elliptic function, and p 2 = 2(βA 2 +ω 2 0 ) βA 2 . The solution (3.17) differs from that obtained by some previous authors [14] with a specific method. This is in concordance with the fact that for an elliptic equation like the Duffing equation, different elliptic functions may be found as solutions [15]. Case 2: β < 0, and ω 2 0 > 0 For β < 0 and ω 2 0 > 0, it is also assumed that − ω 2 0 β > 2A0 > 0. So, the general solution to the Duffing equation (3.1) takes, using (3.17), the form 2|β|. The function dc(z, k) is a Jacobian elliptic function. This result differs from that computed by some previous authors [14].
Case 3: β > 0, and ω 2 0 > 0 In this case, it is assumed that 0 < | − So, the integral J may be written as [13] p has the preceding value, and F (δ, 1 p ) denotes the elliptic integral of the first kind. In this regard the following relationship may be written, taking into consideration (3.6) which may be rewritten In this perspective the general solution to (3.1) reads which may take the expression , or Ω = βA 2 + ω 2 0 , the general solution to Duffing equation (3.1) reduces immediately to the form where cn(z, k) denotes a Jacobian elliptic function. It can be noted that this result is in agreement with that found in [14] by a specific method. It would be now interesting to consider in the sequel of this work some Painlevé-Gambier equations.

Painlevé-Gambier XII equation
This subsection is intended to carry out the general solution to the Painlevé-Gambier XII equation [7] x −ẋ The reduced Painlevé-Gambier equation (3.27) may be found from (2.4) by choosing ϕ(x) = ln x 2 , γ = 1 2 , a 2 = −q, and c = β. Therefore the general solution to (3.27) takes the expression By integration the quantity leads to consider two distinct cases following the value of qA 0 β .
In this case the integral J becomes [13] So the general solution to (3.27) becomes The solution x(t) is real for q < 0. In this way, the solution (3.33) may clearly exhibit a harmonic periodic behavior with a shift factor − β q .
This case corresponds to [13] which gives according to (3.29) Therefore the general solution x(t) may be expressed in the form For q < 0, the solution (3.36) is real and then may exhibit also a harmonic periodic behavior with a shift factor − β q . Consider now as final illustrative example a generalized Painlevé-Gambier XIX equation.

Generalized Painlevé-Gambier XIX equation
The Painlevé-Gambier XIX equation is of the form [7] x − 1 2ẋ A general form of (3.37) may be written asẍ so that for a 2 = −4, and c = 2, one may recover the Painlevé-Gambier XIX equation. The equation (3.38) may be obtained from (2.4) by setting γ = 1 4 , and ϕ(x)=ln x 2 . Thus the solution to (3.38) immediately takes the expression following (2.6) This equation is of the same form as (3.6). So three distinct cases may be distinguished.
Case 1: a 2 > 0, c > 0, and c a 2 > A 0 > 0 In this case [13] p = 2a 2 A0 a 2 A0 + c becomes where F (δ, p) designates the Jacobian elliptic integral of the first kind. So according to (3.40) one may write In this regard, the general solution (3.39) becomes which may take the expression that is Ω = aA 2 , and p may be rewritten as p 2 = 2(a 2 A 2 −c) Case 2: a 2 < 0, c > 0 and 0 < | c a 2 | < A 0 In such a situation, the integral [13] where δ = arcsin which may give Therefore the general solution (3.39) may take the expression The general solution (3.52) may be also expressed as which becomes where A 2 = 2A 0 p 2 . For a 2 = i 2 |a 2 |, that is a = ±i |a 2 |, the general solution x(t) may take the form that is which becomes definitively where Ω = √ |a 2 |pA 2 , and i is the purely imaginary number.
By making a 2 = −4, and c = 2, the exact doubly periodic solution to Painlevé-Gambier XIX equation may be written as that is cn 2 Ω(t + C), This case corresponds also to p = 2a 2 A 0 a 2 A 0 +c , δ = arcsin a 2 A 0 (1−sin(aφ+α)) a 2 A 0 +c , and the integral So the general solution (3.39) may be written as that is and Ω = apA 2 (3.63) The parameter p may also be expressed as p 2 = 2(a 2 A 2 −c) That being so it is then possible to show the equivalence between the Duffing equation and the generalized Painlevé-Gambier XIX equation.

Equivalence between equations
which is the generalized Painlevé-Gambier XIX equation (3.38) by taking a 2 = 4β, and c = −2ω 2 0 . So with that the equivalence between Duffing equation and the generalized Painlevé-Gambier XIX equation has been shown and a discussion may be formulated for the present work.

Discussion
In the theory of nonlinear differential equations, the problem of calculating explicit and exact general periodic solutions often arises acutely, given the non-existence of a master solving method. In this work, it was proposed to check the efficiency of a generalized Sundman transformation supposed to solve certain types of nonlinear second order differential equations. Thus, the well known cubic Duffing equation in various branches of physics such as classical and quantum mechanics has been chosen to illustrate this purpose given, to our knowledge, that a generalized Sundman transformation has never been used to calculate its explicit and exact general periodic solutions. Usually the explicit general solution of the cubic Duffing equation is computed through specific methods which are based on the fact that all the twelve Jacobian elliptic functions are solutions of this equation [15]. By application of the proposed linearization transformation, explicit and exact general periodic solutions of the cubic Duffing equation were computed, as expected, as Jacobian elliptic functions with simplicity and elegance in the solution steps, which characterize the generalized Sundman transformation. In this perspective, explicit and exact general periodic solutions of Painlevé-Gambier XIX equation have been computed for the first time using a generalized Sundman transformation, as Jacobian elliptic functions, so that it has been possible to show the mathematical equivalence between the Painlevé-Gambier XIX equation and the cubic Duffing equation. On the other hand, it was possible to calculate explicit and exact general trigonometric solutions but with a shift factor to Painlevé-Gambier XII equation for the first time in accordance with a suitable parametric choice. With these convincing results, it would be possible now to conclude this research contribution.

Conclusion
The vital problem of finding explicit and exact general periodic solutions to nonlinear differential equations is still an active research field of mathematics. Explicit and exact general periodic solutions are computed for various types of Liénard equation by means of a generalized Sundman linearization approach. In this way, the explicit and exact general periodic solutions to the cubic Duffing equation as well as for some Painlevé-Gambier equations are with simplicity and elegance determined. In so doing, it has been demonstrated for the first time that for suitable parametric choice equation XII of the Painlevé-Gambier classification exhibits trigonometric periodic solutions but with a shift factor, which is a major finding in the investigation of the analytical properties of the Painlevé-Gambier equations. It is also for the first time that explicit and exact general periodic solutions have been calculated for Painlevé-Gambier XIX equation by a generalized Sundman transformation as Jacobian elliptic functions. In this perspective as an interesting result, it has been shown that the cubic Duffing equation is mathematically equivalent to the generalized Painlevé-Gambier XIX equation such that the solution of the last equation may be obtained from the solution of the former and vice versa. Thus it has been shown that the generalized Sundman transformation can be a powerful mathematical tool in the determination of explicit and exact general solutions to the problems of physics and applied science which are described in terms of nonlinear differential equations.