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A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations
Applied Mathematics and Computation 216 (2010) 2635–2644
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A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations J.I. Ramos *, C.M. Garcı´a-López Escuela de Ingenierı´as, Universidad de Málaga, Doctor Pedro Ortiz Ramos, s/n, 29071 Málaga, Spain
a r t i c l e
i n f o
Keywords: Volterra’s integral equation Third-order nonlinear ordinary differential equations Periodic solutions Variation of parameters Transformation of independent variables
a b s t r a c t A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Nonlinear, autonomous, third-order ordinary differential equations have received some attention in the past because they may exhibit chaos and complex nonlinear behavior [1–3]. For these equations, the term ‘‘jerk”, i.e., the third-order derivative of the displacement, has been coined [4]. Nonlinear, autonomous, third-order ordinary differential equations may also exhibit periodic or limit cycle behavior [5]. This behavior has been analyzed in the past by means of harmonic balance methods [6,7], linearized harmonic balance procedures [8], asymptotic perturbation techniques which combine the harmonic balance procedure and the method of multiple time scales [9], the Linstedt–Poincaré procedure [10], the method of averaging of Krylov–Bogoliubov-Mitropolskii [10–12], parameter–perturbation Linstedt–Poincaré techniques [13,14] which employ an artificial or book-keeping parameter and expand both the solution and some constants that appear (or are introduced) in the differential equation in terms of this parameter, artificial parameter-Linstedt-Poincaré techniques [15,16] based on the introduction of a linear term proportional to the unknown frequency of oscillation and a new independent variable and the use of either the third-order equation or a system of a first-order and a second-order ordinary differential equations, etc. Parameter–perturbation methods are extensions of the homotopy perturbation technique which introduces an artificial parameter and expands both the solution and the unknown frequency of oscillation in terms of this parameter and has been applied to a variety of problems [17].
* Corresponding author. E-mail address: [email protected] (J.I. Ramos). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.108
J.I. Ramos, C.M. Garcı´a-López / Applied Mathematics and Computation 216 (2010) 2635–2644
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Ma et al. [13] analyzed the periodic solutions of the three jerk equations, i.e.,
x þ1x_ ¼ pxx_ €x;
ð1Þ
x þ1x_ ¼ px_ €x2 ;
_2
ð2Þ 2
_ x þ x 1Þ; x þ1x_ ¼ pxð
ð3Þ
_ subject to xð0Þ ¼ 0; xð0Þ ¼ B and € xð0Þ ¼ 0, that Gottlieb [6] had previously studied with a first-order harmonic balance procedure with p = 1, by means of a parameter perturbation technique which introduces the parameter p in Eqs. (1) and (2) and the parameter p and the terms 1x_ and px_ in Eq. (3) as shown above and expands both the solution and the coefficients 1 that multiply the (linear) velocity term in power series of a book-keeping parameter. On the other hand, Hu [14] rewrote Eq. (3) with p = 1 as
_ x_ 2 þ x2 Þ; x þ0x_ ¼ lxð
ð4Þ
which corresponds to Gotllieb’s equation for l ¼ 1 and then expanded both the coefficient 0 and the solution in power series expansions of the artificial parameter l. Both Ma et al. [13] and Hu [14] carried their analyses in terms of the original independent variable, t, and, for Eqs. (3) and (4) with p = 1 and l ¼ 1, respectively, obtained different two-term approximations to the frequency of oscillation, thus indicating that this (and higher-order) approximation depends on the constant being expanded in terms of the artificial parameter. By way of contrast, Ramos [15] considered Eq. (3) with p = 1, introduced the term x2 x_ in both sides of this equation and a new independent variable h ¼ xt, and an artificial parameter and expanded both the solution and the unknown frequency of oscillation in terms of this artificial parameter and, by requiring, that different approximations be free from secular terms, obtained good approximations to both the solution and the unknown frequency of oscillation. This procedure requires that the initial conditions be expanded in power series of the artificial parameter. Such an expansion can be avoided entirely by reducing the third-order nonlinear Eqs. (1)–(3) with p = 1 to systems consisting of a first-order linear and a second-order nonlinear ordinary differential equations and expanding both the solution and the frequency of oscillation in power series of the artificial parameter [16]. In this paper, we first present a Volterra integral formulation for nonlinear, autonomous, third-order nonlinear ordinary differential equations which are odd with respect to reflections in the dependent and independent variables and have periodic solutions. This formulation is based on the introduction of a linear term and a new independent variable and the method of variation of parameters, and it is shown that this formulation is characterized by two functions; one of which depends on the initial conditions, the new independent variable and the unknown frequency of oscillation, whereas the other is given by an integral which depends on the nonlinearities and the frequency of oscillation. By looking for series expansions for both the solution and the frequency of oscillation, we obtain a sequence of (integral) approximations which are required to be free from secular terms. This requirement provides successive terms in the series expansions for both the solution and the frequency of oscillation. The method presented here does not make use of artificial parameters and can be interpreted as a generalization of Adomian’s decomposition technique [18–20]. The paper has been arranged as follows. In the next section, a Volterra integral formulation for nonlinear, autonomous, third-order nonlinear ordinary differential equations which satisfy certain Lipschitz-continuity conditions and have periodic solutions is presented, whereas, in Section 3, the formulation is applied to the three examples previously analyzed by Gottlieb [6], i.e., Eqs. (1)–(3) with p = 1, by means of a first-order harmonic balance procedure, and its results are compared with those of parameter perturbation methods [13,14] and linearized and non-linearized harmonic balance procedures [6,8]. A final concluding section summarizes the main findings of the paper. 2. Formulation We shall consider the following autonomous, nonlinear, third-order ordinary differential equation
_ €xÞ; x ¼ f ðx; x;
xð0Þ ¼ A;
_ xð0Þ ¼ B;
€xð0Þ ¼ C;
ð5Þ
where A, B and C are constants, the dot denotes differentiation with respect to t, and f is an odd function of x and t. Eq. (5) includes Eqs. (1)–(3) which as stated above coincide with those analyzed by Gottlieb [6] by means of a first-order harmonic balance procedure for p = 1. If f is a Lipschitz-continuous function, the Lindelöf–Picard theorem ensures the existence and uniqueness of a (local) solution to Eq. (5) in the neighborhood of t ¼ 0 [21,22] and this local solution may be extended for larger values of t by means of analytical continuation. _ € In this paper, we shall assume that f ðx; x; xÞ is Lipschitz-continuous so that there exists a locally unique solution to Eq. (5) which can be determined iteratively and we shall be interested in obtaining approximately the periodic solutions of Eq. (5). To this end, we first add to the left- and right-hand sides of Eq. (5) the linear term x2 x_ which is Lipschitz-continuous, so that Eq. (5) becomes
_ €xÞ þ x2 x_ gðx; x; _ €x; xÞ; x þx2 x_ ¼ f ðx; x;
xð0Þ ¼ A;
_ xð0Þ ¼ B;
€xð0Þ ¼ C;
ð6Þ
J.I. Ramos, C.M. Garcı´a-López / Applied Mathematics and Computation 216 (2010) 2635–2644
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where x denotes the unknown frequency of oscillation. Upon introducing the new independent variable h ¼ xt and using the method of variation of parameters, the solution to Eq. (6) can be written as
xðhÞ ¼ Fðh; x; A; B; CÞ þ Gðh; x; x0 ; x00 ; xÞ;
ð7Þ
where
Fðh; x; A; B; CÞ ¼ A þ
C
x2
ð1 cosðhÞÞ þ
B
x
sinðhÞ;
ð8Þ
and
Gðh; x; x0 ; x00 ; xÞ ¼
Z
h
hðxðsÞ; x0 ðsÞ; x00 ðsÞ; xÞð1 cosðh sÞÞds;
ð9Þ
0
where hðx; x0 ; x00 ; xÞ ¼ x13 gðx; x0 ; x00 ; xÞ; gðx; x0 ; x00 ; xÞ corresponds to the g function defined in Eq. (6) after making the change of independent variable mentioned above, and the prime denotes differentiation with respect to h. Eqs. (7)–(9) clearly indicate that the initial conditions are contained in F, whereas G contains the nonlinearities of Eq. (5) through h and g in Eq. (9). Both F and G contain the unknown frequency of oscillation x. The Volterra integral equation represented by Eqs. (7)–(9) can be solved in an iterative fashion in an analogous manner to the well-known Lindelöf–Picard method for first-order ordinary differential equations and its (local) convergence is ensured provided that g or h is a Lipschitz-continuous function. However, the frequency of oscillation in Eqs. (7)–(9) is not known and must be determined so that xðhÞ is a periodic function of period 2p. In order to determine both xðhÞ and x from Eqs. (7)–(9), one could treat these equations as a boundary-value problem by requiring that xðhÞ ¼ xðh þ 2pÞ; x0 ðhÞ ¼ x0 ðh þ 2pÞ and x00 ðhÞ ¼ x00 ðh þ 2pÞ; however, due to presence of the nonlinear function h in Eq. (9), the solution of such a boundary-value problem would require, in general, numerical techniques. In this paper, we determine the solution and frequency of oscillation of Eqs. (7)–(9) approximately by means of the following method. Assume that
xðhÞ ¼ x0 ðhÞ þ x1 ðhÞ þ x2 ðhÞ þ ;
ð10Þ
x ¼ x0 þ x1 þ x2 þ ;
ð11Þ
and substitute Eqs. (10) and (11) into Eqs. (7)–(9) to determine xi ðhÞ; 0 6 i, as
x0 ðhÞ ¼ Fðh; x0 ; A; B; CÞ; @F x1 ðhÞ ¼ ðh; x0 ; A; B; CÞx1 þ Gðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þ; @x @F 1 @2F ðh; x0 ; A; B; CÞx2 þ ðh; x0 ; A; B; CÞx21 x2 ðhÞ ¼ @x 2! @ x2 @G @G þ ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx1 ðhÞ þ 0 ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx01 ðhÞ @x @x @G @G þ 00 ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx001 ðhÞ þ ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx1 ; @x @x
ð12Þ ð13Þ
ð14Þ
etc., and require that xi ðhÞ for i ¼ 1; 2; . . ., be free from secular terms. This sequential approximation provides the values of xi ðhÞ and xi for i ¼ 0; 1; 2; . . ., which can be substituted into Eqs. (10) and (11) to determine the periodic solution and the frequency of oscillation of Eq. (5), respectively. Eqs. (7)–(14) can be interpreted as a generalization of Adomian’s decomposition technique [18–20] where both the solution and the unknown frequency of oscillation are expanded in the series of Eqs. (10) and (11), respectively. If x were assumed to be fixed, Eqs. (10) and (12)–(14) would reduce to the decomposition method [18–20] when this technique is written in terms of h rather than t. The formulation presented here provides trigonometric rather than the more conventional polynomial approximations to xi ðhÞ; 0 6 i, provided by Adomian’s decomposition method [18–20]. For example, for A ¼ C ¼ 0, Eq. (12) yields x0 ðhÞ ¼ xB0 sinðhÞ. An alternative derivation to the Volterra integral formulation presented above consists in first introducing a book-keeping parameter, , in Eq. (7), so that equation becomes
xðhÞ ¼ Fðh; x; A; B; CÞ þ Gðh; x; x0 ; x00 ; xÞ; which coincides with Eq. (7) when
ð15Þ
¼ 1 and then expanding xðhÞ and x as
xðhÞ ¼ x0 ðhÞ þ x1 ðhÞ þ 2 x2 ðhÞ þ
ð16Þ
x ¼ x0 þ x1 þ 2 x2 þ
ð17Þ
Substitution of Eqs. (16) and (17) into Eq. (15) yields a power series expansion in and, by setting the coefficients of this series that multiply the monomials n ; n ¼ 0; 1; . . ., to zero, one can easily obtain Eqs. (12)–(14); approximations to both the solution and the frequency of the oscillation can then be obtained by first requiring that xi ðhÞ; i ¼ 1; 2; . . ., be free from secular terms and then setting ¼ 1 in Eqs. (16) and (17).
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It must also be noted that Eq. (6) can be written as
x000 þ x0 ¼ hðx; x0 ; x00 ; xÞ;
xð0Þ ¼ A;
x0 ð0Þ ¼
B
x
x00 ð0Þ ¼
;
C
x2
ð18Þ
;
or derived from Eqs. (7)–(9) by employing the fundamental theorem of calculus [24–26]. In either case, the use of Eqs. (10) and (11) in Eq. (18) and the requirement that xi ðhÞ; i ¼ 1; 2; . . ., be free from secular terms provide the same results as those of the Volterra integral formulation presented here (cf. Eqs. (7)–(14)). Furthermore, the introduction of a book-keeping parameter, , in Eq. (18) yields
x000 þ x0 ¼ hðx; x0 ; x00 ; xÞ;
xð0Þ ¼ A;
x0 ð0Þ ¼
B
x
;
x00 ð0Þ ¼
C
x2
;
ð19Þ
which coincides with Eq. (18) for ¼ 1, and the use of Eqs. (16) and (17) in Eq. (19) provides power series expansions in for xi ðhÞ; xi ð0Þ; x0i ð0Þ and x00i ð0Þ. By setting the coefficients that multiply the monomials in n ; n ¼ 0; 1; . . ., in these series to zero, a third-order linear ordinary differential equation and its corresponding initial conditions are obtained and their solutions are identical to those of Eqs. (12)–(14) [23]. Furthermore, by requiring that these solutions be free from secular terms and using Eqs. (16) and (17) with ¼ 1, it is an easy exercise to show that this formulation is identical to the Volterra integral one presented in this paper, by means of the fundamental theorem of calculus [24–26]. 3. Presentation of results In this section, the Volterra integral technique presented in Section 2 is applied to three autonomous, non-linear, thirdorder ordinary differential Eqs. (1)–(3) with p = 1 which have analytical periodic solutions and have been previously studied by Gottlieb [6] by means of a first-order, i.e., a one-term, harmonic balance procedure, for A ¼ C ¼ 0 and B – 0 which are the initial conditions considered in this section. For Eqs. (1)–(3) with p = 1, it has been found that the Volterra formulation yields exactly the same results as an artificial parameter-Linstedt–Poincaré method [15,16] and this is consistent with the comments made in the previous section. In particular, this formulation yields the same results as the second-order reduction method with the variable independent h for Eqs. (1) and (2) with p = 1 [15] and the same algebraic results as the Linstedt–Poincaré method with the variable independent h for Eq. (3) with p = 1 [16]. For this reason, we shall not present the approximations to both the solution and the frequency of the oscillation here and refer the reader to Refs. [15,16]. Furthermore, for Eqs. (1)–(3) with p = 1, it has been found that the first-order approximation to the frequency of oscillation coincides with that obtained by Gottlieb [6] by means of a first-order harmonic balance procedure. In addition, for Eq. (1) with p = 1, the frequency of oscillation was found to be identical to that of Table 2 of Ref. [15] except for the errors corresponding to Ma et al. [13] second-order approximation for B ¼ 0:1, 0.2 and 0.5 which should read 0.00000264, 0.00000285 and 0.00059765, respectively, instead of 0.00000027, 0.00000029 and 0.0000598, respectively, as reported by Ma et al. [13] in their Table 1. The new errors reported here were obtained by employing Eqs. (16) and (19) of Ma et al. [13]. For Eq. (1) with p = 1, the series for x (cf. Eq. (11)) was found to converge for both B 6 1 and B P 1 by means of Mathematica 7 [27], and the method presented here provides a two-term approximation to the frequency of oscillation which is slightly more accurate than that of a one-term approximation [15]. This two-term approximation is slightly more (respectively, slightly less) accurate than that of a parameter perturbation procedure [13] for B 6 0:1 (respectively, B > 1). Figs. 1 and 2 show the three-dimensional phase diagram and the period of Eq. (1), respectively, as functions of B and indicate that the period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. _ ðx; z € Although not shown here, the projection of the three-dimensional phase diagram onto the ðx; y xÞ; xÞ and (y, z) planes yield an ellipse, an elongated (and rotated counter-clockwise) closed S curve and an ellipse, respectively, for B ¼ 1. For Eq. (1) with p = 1 and B ¼ 2 (not shown here), the projections of the phase diagram onto the (x, y) and (x, z) planes are analogous to those for B ¼ 1, whereas that onto the (y, z) plane exhibits an egg shape. For Eq. (2) with B ¼ 1:5 and p = 1, a typographical error in Table 4 of Ref. [15] for T 2 was found; T 2 should read 4.807901 instead of 4.808475 as reported in that reference. Moreover, the errors corresponding to Ma et al. [13] second-order approximation for B ¼ 0:1, 0.2 and 0.5 should read 0.00000586, 0.00001886 and 0.00525736, respectively, instead of 0.00000006, 0.00000213 and 0.000526, respectively, as reported by Ma et al. [13] in their Table 2. The new errors reported here were obtained by employing Eqs. (34) and (36) of Ma et al. [13]. For Eq. (2) with p = 1, it was found that the series for x (cf. Eq. (11)) converges for B 6 1 by means of Mathematica 7 [27]. Note, however, that the approximations provided by a first-order harmonic balance [6], parameter–perturbation-Linstetd– Poincaré [13] and artificial parameter-Linstetd–Poincaré methods [15] to the frequency of oscillation of Eq. (2) with p = 1 are not valid for B P 2, even though Eq. (2) has an analytical solution for B P 2. The failure of these techniques for Eq. (2) with B P 2 is associated with the fact that x and x_ exhibit very steep gradients _ plane is nearly a [15] for B greater than about 1.2 and the projection of the three-dimensional phase diagram onto the ðx; xÞ circumference for B 1 and exhibits very steep gradients of x_ at the locations where jxj is largest and very flat regions where _ is largest for B larger than about 1.2. As a consequence, the first-order approximation employed in harmonic balance projxj cedures and that resulting from both the Volterra integral formulation presented here and parameter perturbation methods, _ is very inaccurate for B greater than about 1.2. This has been verified both which correspond to a circumference in the ðx; xÞ,
J.I. Ramos, C.M. Garcı´a-López / Applied Mathematics and Computation 216 (2010) 2635–2644
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4 2 z 0 −2 −4 −2
−2 −1
−1 0
0 1
1 2
2
x
y
Fig. 1. Three-dimensional phase diagram for Eq. (1) with p = 1. (Short, medium and long curves correspond to B ¼ 0:1, 0.5 and 2.0, respectively. B ¼ 0:1, 0.5 and 2.0 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
6.5
6
5.5
T 5
4.5
4
0
0.5
1
1.5
2
2.5
B Fig. 2. Period T as a function of B for Eq. (1) with p = 1. Only the periods marked with circles have been determined.
numerically by means of a fourth-order accurate Runge–Kutta method and a Galerkin approximation for Eq. (2). When using the Runge–Kutta technique, it was observed that the time step must be reduced as B is increased in order to obtain very accurate trajectories; for example, the time step for B ¼ 2 was 0.001, whereas that for B ¼ 10 was 0.00001. In the Galerkin P approximation, Eq. (2) was first written in terms of h and xðhÞ was then approximated by xðhÞ Nn¼0 C ð2nþ1Þ cosðð2n þ 1ÞhÞ in the resulting equation; the residual of that equation was then required to be orthogonal in ½0; 2p to cosðð2n þ 1ÞhÞ and this provided a set of nonlinear equations for C ð2nþ1Þ which was solved iteratively by means of a Newton–Raphson procedure until jC kþ1 C kð2nþ1Þ j ð2nþ1Þ jC kð2nþ1Þ j
k 8 6 105 if jC kð2nþ1Þ j P 103 or jC kþ1 if jC kð2nþ1Þ j < 103 , for 0 6 n 6 N. For B 6 1, very accurate freð2nþ1Þ C ð2nþ1Þ j 6 10
quencies of oscillation where obtained with the Galerkin procedure and N ¼ 1; however, in order to obtain accurate results for larger values of B, the number of harmonics, N, had to be increased as B was increased. For example, for B ¼ 2; N ¼ 12, whereas, for B ¼ 5; N ¼ 400.
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For Eq. (2) with p = 1, it was found that the two-term approximation to the frequency of oscillation obtained with the Volterra integral formulation presented here is more accurate than that corresponding to a parameter perturbation approach [13] and also more accurate than a one-term approximation (cf. Eq. (11)). Figs. 3 and 4 illustrate the three-dimensional phase diagram and the period of Eq. (2), respectively, with p = 1 as functions of B and indicate that the period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. Although not shown here, the projection of the three-dimensional phase diagram for Eq. (2) with p = 1 and B ¼ 2 onto the (x, y) plane is characterized by flat regions near the locations where jxj and jyj are largest; the projection onto the ðx; zÞ plane has a more pronounced elongated (and rotated counter-clockwise) S shape than that for Eq. (1) with B ¼ 1, whereas the projection onto the ðy; zÞ plane has very steep gradients where jyj is largest. The projections of the phase diagram onto the (x, y) and (y, z) planes for Eq. (2) with p = 1 and B P 1:2 clearly indicate the reasons for the failure of the one-term approximation to the frequency of oscillation predicted by harmonic balance and parameter perturbation methods and the Volterra integral formulation presented in this paper for this equation. As indicated in Section 2, the Volterra formulation as well as the other techniques mentioned previously predict a one-term approximation to the periodic solution consisting of only one frequency which results in an ellipsoidal phase diagram in the (x, y) plane, whereas the projection of the phase diagrams corresponding to Eq. (2) with p = 1 and B P 1:2 are not ellipses at all and their accurate prediction requires the use of several harmonics. Moreover, as indicated previously, the time step in Runge–Kutta methods and the number of harmonics in the Galerkin procedure for the numerical solution of Eq. (2) had to be decreased and increased, respectively, as B was increased, because the (x, y)-phase diagrams became steeper at the locations where jxj is largest as B was increased. In Table 1, T i ¼ P2i p denotes the period corresponding to i-terms of the series approximation to the frequency of oscilj¼0
xj
lation (cf. Eq. (11)) for Eq. (3) with p = 1. T 1 and its corresponding error are not reproduced because they are identical to those presented in Table 2 of Ref. [16]. Table 1 indicates that the two-term approximation to the frequency of oscillation calculated with the method presented in this paper is more accurate than Hu [14] parameter perturbation approach and Wu et al.’s [8] second-order linearized harmonic balance procedure for 0:1 6 B 6 20; it is also slightly more (respectively, slightly less) accurate than Ma et al.’s [13] parameter perturbation method for 0:1 6 B < 1 (respectively, 1 < B 6 20). Table 1 also shows that the parameter perturbation method of Ma et al.’s [13] predicts different frequencies of oscillation than that of Hu [14]. It must be pointed out that the errors of T 2 corresponding to the parameter perturbation method of Ma et al. [13] presented in Table 1 do not coincide with those reported by these authors in their Table 3; neither do they coincide with those reported in Table 2 of Ref. [16] which are identical to those of Table 3 of Ref. [13]. The errors corresponding to Ma et al.’s parameter perturbation method [13] presented in Table 1 of this paper have been obtained from Eqs. (50)–(52) of Ref. [13] and do not coincide with those reported by these authors in their Table 3. The results presented in Table 1 differ from and are more accurate than those presented in Table 1 of Ref. [16] which were obtained by means of an artificial parameter-Linstedt–Poincaré method applied to the corresponding third-order nonlinear
5
z 0
−5 −2
−2 −1
−1 0
0 1 y
1 2
2
x
Fig. 3. Three-dimensional phase diagram for Eq. (2) with p = 1. (Short, medium and long curves correspond to B ¼ 0:1, 0.5 and 1.5, respectively. B ¼ 0:1, 0.5 and 1.5 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
J.I. Ramos, C.M. Garcı´a-López / Applied Mathematics and Computation 216 (2010) 2635–2644
2641
6.5
6
5.5
T
5
4.5
4
3.5
0
0.5
1
1.5
2
2.5
B Fig. 4. Period T as a function of B for Eq. (2) with p = 1. Only the periods marked with circles have been determined.
Table 1 The period ðT i Þ and relative error ðEi ¼ 100ðT i T e Þ=T e Þ of the i-term approximation (cf. Eq. (11)) for Eq. (3) with p = 1, where T e denotes the exact period obtained by integrating numerically Eq. (3) with p = 1 by means of a fourth-order accurate Runge–Kutta method with a time step equal to 0.001. The two-term approximations to the period of oscillation obtained by Hu et al. [14] and Ma et al. [13] by means of a parameter perturbation method are also presented; these methods predict the same T 1 as the technique presented in this paper and a first-order harmonic balance procedure [6]. The errors corresponding to Ma et al. parameter perturbation method [13] have been obtained from Eqs. (50)–(52) of Ref. [13] and do not coincide with those reported by these authors in their Table 3. The periods obtained with a second-order linearized harmonic balance procedure [8] are also shown. B
Te
T 2 ðE2 Þ
T 2 ðE2 Þ [14]
T 2 ðE2 Þ [13]
T 2 ðE2 Þ [8]
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0
25.359725 17.495410 10.210761 6.283185 3.508793 1.468638 0.739762 0.370580
25.56407 (0.806) 17.61452 (0.681) 10.24328 (0.318) 6.28319 (0.000) 3.50486 (0.112) 1.46715 (0.101) 0.73907 (0.094) 0.37024 (0.092)
25.60228 (0.956) 17.63225 (0.782) 10.24768 (0.362) 6.28319 (0.000) 3.50239 (0.182) 1.46528 (0.229) 0.73803 (0.234) 0.36971 (0.235)
24.51794 (-3.319) 17.09799 (-2.272) 10.14660 (-0.628) 6.28319 (0.000) 3.50968 (0.025) 1.46796 (0.046) 0.73930 (0.063) 0.37033 (0.067)
24.6677 (2.729) 17.2251 (1.545) 10.1884 (0.2196) 6.28319 (0.000) 3.50554 (0.0927) 1.46650 (0.1454) 0.738632 (0.1527) 0.370008 (0.1544)
ordinary differential equation in the original independent variable, i.e., t, after introducing the term x2 x_ in the left and right sides of the original equation. As indicated in Section 3.1 of Ref. [16], a Linstedt–Poincaré method applied to the third-order ordinary differential equation in the original independent variable does not provide approximations to the frequency of oscillations; it provides approximations to the differences between the squares of successive approximations to the frequency of oscillations because that method uses an equation analogous to Eq. (11) for x2 rather than for x. The differences in the values of T 2 reported in Table 1 obtained with different methods are to be expected because, as stated in the Introduction, the parameter perturbation methods of Ma et al. [13] and Hu [14] are based on the expansion of some constants that appear or are introduced in the governing equations, whereas the method presented in this paper is based on the expansion of the unknown frequency of oscillation. Moreover, the parameter perturbation methods of Ma et al. [13] and Hu [14] use the original independent variable (t), whereas the technique presented here uses a new independent variable ðhÞ. Figs. 5 and 6 illustrate the three-dimensional phase diagram and the period of Eq. (3), respectively, with p = 1 as functions of B and indicate that period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. For B ¼ 1, the projection of the three-dimensional phase diagram onto the (x, y) plane has an elliptical shape, whereas that onto the (y, z) plane is characterized by flat regions of large (respectively, small) slope where jyj (respectively, jzj) is largest. On the other hand, the projection onto the (x, z) plane has a rotated closed S shape. It is worth noting that, for Eqs. (1) and (2) with p = 1, the absolute value of the errors of the two-term approximation to the frequency of oscillation determined by the formulation presented here increases as the initial velocity, i.e., B, is increased; a similar behavior is observed in parameter perturbation methods [13,14]. However, for Eq. (3) with p = 1, the absolute value of the errors of the two-term approximation to the frequency of oscillation provided by parameter perturbation
J.I. Ramos, C.M. Garcı´a-López / Applied Mathematics and Computation 216 (2010) 2635–2644
2642
300 200 100 z
0 −100 −200 −300 −40
−2 −20
−1 0
0 20
1 40
2
y
x
Fig. 5. Three-dimensional phase diagram for Eq. (3) with p = 1. (Short, medium and long curves correspond to B ¼ 1, 5 and 20, respectively. B ¼ 1, 5 and 20 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
30
25
20
T 15
10
5
0
0
5
10
15
20
B Fig. 6. Period T as a function of B for Eq. (3) with p = 1. Only the periods marked with circles have been determined.
methods first decreases as B is increased from B ¼ 0:1, then reaches a minimum value at B ¼ 1 for which the exact solution of Eq. (3) with p = 1 (in the absence of round-off errors) is obtained, and then increases as B is increased, although it tends to an almost constant value for B > 20. On the other hand, for Eq. (3) with p = 1, the absolute value of the errors corresponding to the two-term approximation presented here, first decreases as B is increased for B 6 1, then increases (cf. the periods for B ¼ 1 and B ¼ 2 in Table 1), and then decreases for B > 2, although it also tends to a constant value for B > 20. Eqs. (1)–(3) with p = 1 and Eq. (4) with l ¼ 1 clearly indicate that the parameter perturbation methods employed by both Ma et al. [13] and Hu [14], respectively, are based on the introduction of artificial parameters in the governing equations and the expansion of both the solution and some constants that appear in these equations in terms of this artificial parameter; in addition, these authors worked with the original independent variable, i.e., t, and differential equations. By way of contrast, the method proposed here is based on the introduction of a linear term, i.e., x2 x_ on both sides of Eq. (5), a new independent variable, i.e., h ¼ xt, and an integral formulation (cf. Eqs. (7)–(9)) and does not make use at all of an artificial parameter. Furthermore, as the results shown in Table 1 clearly indicate, the two-term approximation of the power series expansion to the
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frequency of oscillation given by the parameter perturbation methods of Ma et al. [13] and Hu [14] depends on the constant being expanded in terms of the artificial parameter (cf. Eqs. (3) and (4)). This is not the case for the method presented in this paper because it does not make use of artificial parameters. 4. Conclusions A Volterra integral formulation based on the introduction of a term equal to the velocity times the square of the unknown frequency of oscillation in both sides of the equation, a new independent variable linearly proportional to the unknown frequency of oscillation and the method of variation of parameters, is presented and used to determine, in series expansions, both the periodic solution and the frequency of oscillation of three autonomous, nonlinear, third-order nonlinear ordinary differential equations which have analytical solutions, and its results are compared with those of a first-order harmonic balance and two parameter perturbation techniques. It has been shown that the method presented in this paper provides a one-term approximation to the frequency of oscillation which is identical to those of a first-order harmonic balance procedure and the first approximation of parameter perturbation procedures, and, for the second equation considered here, the two-term approximation results in a more accurate frequency of oscillation than a parameter perturbation method based on the expansion of the coefficient that multiplies the linear velocity term in power series of an artificial parameter for all initial velocities analyzed here. The technique presented here also predicts a more accurate frequency of oscillation for the third nonlinear third-order ordinary differential equation considered in this paper than a parameter perturbation method that expands zero in terms of power series of an artificial parameter for all the initial conditions analyzed in this paper; it also predicts slightly more (respectively, slightly less) accurate frequencies of oscillation than a parameter perturbation method that introduces the velocity in both sides of the equation and expands the coefficient, i.e., one, that multiplies this term in power series of an artificial parameter, for initial velocities less (respectively, greater) than one. The method presented here has also been found to be more accurate than the second approximation of a linearized harmonic balance procedure for all the initial conditions considered in this paper. It has also be shown that the Volterra integral formulation and the series expansion techniques presented here provide series solutions to both the periodic solution and frequency of oscillation but do not make use of artificial parameters, whereas parameter perturbation methods employ artificial parameters, provide power series expansions in the artificial parameter for both the solution and the frequency of oscillation and predict a two-term approximation to the frequency of oscillation which depends on the constant which is expanded in terms of the artificial parameter. Acknowledgements The research reported in this paper was supported by Projects FIS2005–03191 and FIS2009–12894 from the Ministerio de Ciencia y Educación y Ministerio de Ciencia e Innovación, respectively, of Spain and fondos FEDER. The authors are grateful to the reviewers for their helpful and critical comments which have resulted in an improved manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
J.C. Sprott, Some simple chaotic jerk functions, American Journal of Physics 65 (1997) 537–543. S.J. Linz, Nonlinear dynamical models and jerky motion, American Journal of Physics 65 (1997) 523–526. S.J. Linz, Newtonian jerky dynamics: some general properties, American Journal of Physics 66 (1998) 1109–1114. S.H. Schot, Jerk: the time rate of change of the acceleration, American Journal of Physics 46 (1978) 1090–1094. H.P.W. Gottlieb, Simple nonlinear jerk functions with periodic solutions, American Journal of Physics 66 (1998) 903–906. H.P.W. Gottlieb, Harmonic balance approach to periodic solutions of non-linear jerk equations, Journal of Sound and Vibration 271 (2004) 671–683. H.P.W. Gottlieb, Harmonic balance approach to limit cycles for nonlinear jerk equations, Journal of Sound and Vibration 297 (2006) 243–250. B.S. Wu, C.W. Lim, W.P. Sun, Improved harmonic balance approach to periodic solutions of non-linear jerk equations, Physics Letters A 354 (2006) 95– 100. A. Maccari, The dissipative nonlocal oscillator, Nonlinear Dynamics 16 (1998) 307–320. R.J. Mulholland, Nonlinear oscillations of a third-order differential equation, International Journal of Non-Linear Mechanics 6 (1971) 279–294. H.R. Srirangarajan, P. Srinivasan, B.V. Dasarathy, Ultraspherical polynomials approach to the study of third-order non-linear systems, Journal of Sound and Vibration 40 (1975) 167–172. H.R. Srirangarajan, B.V. Dasarathy, Study of third-order non-linear systems – Variation of parameters approach, Journal of Sound and Vibration 40 (1975) 173–178. X. Ma, L. Wei, Z. Guo, He’s homotopy perturbation method to periodic solutions of nonlinear Jerk equations, Journal of Sound and Vibration 314 (2008) 217–227. H. Hu, Perturbation method for periodic solutions of nonlinear jerk equations, Physics Letters A 372 (2008) 4205–4209. J.I. Ramos, Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations, Applied Mathematics and Computation 15 (2010) 4304–4319. J.I. Ramos, Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations, Nonlinear Analysis Series B: Real World Applications, in press. Ji–H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 (2006) 1141–1199. G. Adomian, Stochastic Systems, Academic Press Inc., New York, 1983. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press Inc., New York, 1986. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994. P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. W. Kelley, A. Peterson, The Theory of Differential Equations: Classical and Qualitative, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004.
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[23] J.I. Ramos, C.M. García–López, Periodic solutions of some nonlinear jerk equations, in: J.F. Silva Gomes, Shaker A. Meguid (Ed.), In IRF’2009, Integrity, Reliability and Failure (Challenges and Opportunities), Ediçoes INEGI, Porto, Portugal, 2009, pp. 311–312. [24] T.M. Apostol, Calculus, One-Variable Calculus with an Introduction to Linear Algebra, second ed., vol. 1, John Wiley & Sons, New Yok, 1967. [25] W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill Book Company, New Yok, 1986. [26] D.V. Widder, Advanced Calculus, second ed., Dover Publications, Inc., New Yok, 1989. [27] Mathematica 7, Wolfram Research, Inc., Champaign, Illinois, USA, 2009.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations J.I. Ramos *, C.M. Garcı´a-López Escuela de Ingenierı´as, Universidad de Málaga, Doctor Pedro Ortiz Ramos, s/n, 29071 Málaga, Spain
a r t i c l e
i n f o
Keywords: Volterra’s integral equation Third-order nonlinear ordinary differential equations Periodic solutions Variation of parameters Transformation of independent variables
a b s t r a c t A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Nonlinear, autonomous, third-order ordinary differential equations have received some attention in the past because they may exhibit chaos and complex nonlinear behavior [1–3]. For these equations, the term ‘‘jerk”, i.e., the third-order derivative of the displacement, has been coined [4]. Nonlinear, autonomous, third-order ordinary differential equations may also exhibit periodic or limit cycle behavior [5]. This behavior has been analyzed in the past by means of harmonic balance methods [6,7], linearized harmonic balance procedures [8], asymptotic perturbation techniques which combine the harmonic balance procedure and the method of multiple time scales [9], the Linstedt–Poincaré procedure [10], the method of averaging of Krylov–Bogoliubov-Mitropolskii [10–12], parameter–perturbation Linstedt–Poincaré techniques [13,14] which employ an artificial or book-keeping parameter and expand both the solution and some constants that appear (or are introduced) in the differential equation in terms of this parameter, artificial parameter-Linstedt-Poincaré techniques [15,16] based on the introduction of a linear term proportional to the unknown frequency of oscillation and a new independent variable and the use of either the third-order equation or a system of a first-order and a second-order ordinary differential equations, etc. Parameter–perturbation methods are extensions of the homotopy perturbation technique which introduces an artificial parameter and expands both the solution and the unknown frequency of oscillation in terms of this parameter and has been applied to a variety of problems [17].
* Corresponding author. E-mail address: [email protected] (J.I. Ramos). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.108
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Ma et al. [13] analyzed the periodic solutions of the three jerk equations, i.e.,
x þ1x_ ¼ pxx_ €x;
ð1Þ
x þ1x_ ¼ px_ €x2 ;
_2
ð2Þ 2
_ x þ x 1Þ; x þ1x_ ¼ pxð
ð3Þ
_ subject to xð0Þ ¼ 0; xð0Þ ¼ B and € xð0Þ ¼ 0, that Gottlieb [6] had previously studied with a first-order harmonic balance procedure with p = 1, by means of a parameter perturbation technique which introduces the parameter p in Eqs. (1) and (2) and the parameter p and the terms 1x_ and px_ in Eq. (3) as shown above and expands both the solution and the coefficients 1 that multiply the (linear) velocity term in power series of a book-keeping parameter. On the other hand, Hu [14] rewrote Eq. (3) with p = 1 as
_ x_ 2 þ x2 Þ; x þ0x_ ¼ lxð
ð4Þ
which corresponds to Gotllieb’s equation for l ¼ 1 and then expanded both the coefficient 0 and the solution in power series expansions of the artificial parameter l. Both Ma et al. [13] and Hu [14] carried their analyses in terms of the original independent variable, t, and, for Eqs. (3) and (4) with p = 1 and l ¼ 1, respectively, obtained different two-term approximations to the frequency of oscillation, thus indicating that this (and higher-order) approximation depends on the constant being expanded in terms of the artificial parameter. By way of contrast, Ramos [15] considered Eq. (3) with p = 1, introduced the term x2 x_ in both sides of this equation and a new independent variable h ¼ xt, and an artificial parameter and expanded both the solution and the unknown frequency of oscillation in terms of this artificial parameter and, by requiring, that different approximations be free from secular terms, obtained good approximations to both the solution and the unknown frequency of oscillation. This procedure requires that the initial conditions be expanded in power series of the artificial parameter. Such an expansion can be avoided entirely by reducing the third-order nonlinear Eqs. (1)–(3) with p = 1 to systems consisting of a first-order linear and a second-order nonlinear ordinary differential equations and expanding both the solution and the frequency of oscillation in power series of the artificial parameter [16]. In this paper, we first present a Volterra integral formulation for nonlinear, autonomous, third-order nonlinear ordinary differential equations which are odd with respect to reflections in the dependent and independent variables and have periodic solutions. This formulation is based on the introduction of a linear term and a new independent variable and the method of variation of parameters, and it is shown that this formulation is characterized by two functions; one of which depends on the initial conditions, the new independent variable and the unknown frequency of oscillation, whereas the other is given by an integral which depends on the nonlinearities and the frequency of oscillation. By looking for series expansions for both the solution and the frequency of oscillation, we obtain a sequence of (integral) approximations which are required to be free from secular terms. This requirement provides successive terms in the series expansions for both the solution and the frequency of oscillation. The method presented here does not make use of artificial parameters and can be interpreted as a generalization of Adomian’s decomposition technique [18–20]. The paper has been arranged as follows. In the next section, a Volterra integral formulation for nonlinear, autonomous, third-order nonlinear ordinary differential equations which satisfy certain Lipschitz-continuity conditions and have periodic solutions is presented, whereas, in Section 3, the formulation is applied to the three examples previously analyzed by Gottlieb [6], i.e., Eqs. (1)–(3) with p = 1, by means of a first-order harmonic balance procedure, and its results are compared with those of parameter perturbation methods [13,14] and linearized and non-linearized harmonic balance procedures [6,8]. A final concluding section summarizes the main findings of the paper. 2. Formulation We shall consider the following autonomous, nonlinear, third-order ordinary differential equation
_ €xÞ; x ¼ f ðx; x;
xð0Þ ¼ A;
_ xð0Þ ¼ B;
€xð0Þ ¼ C;
ð5Þ
where A, B and C are constants, the dot denotes differentiation with respect to t, and f is an odd function of x and t. Eq. (5) includes Eqs. (1)–(3) which as stated above coincide with those analyzed by Gottlieb [6] by means of a first-order harmonic balance procedure for p = 1. If f is a Lipschitz-continuous function, the Lindelöf–Picard theorem ensures the existence and uniqueness of a (local) solution to Eq. (5) in the neighborhood of t ¼ 0 [21,22] and this local solution may be extended for larger values of t by means of analytical continuation. _ € In this paper, we shall assume that f ðx; x; xÞ is Lipschitz-continuous so that there exists a locally unique solution to Eq. (5) which can be determined iteratively and we shall be interested in obtaining approximately the periodic solutions of Eq. (5). To this end, we first add to the left- and right-hand sides of Eq. (5) the linear term x2 x_ which is Lipschitz-continuous, so that Eq. (5) becomes
_ €xÞ þ x2 x_ gðx; x; _ €x; xÞ; x þx2 x_ ¼ f ðx; x;
xð0Þ ¼ A;
_ xð0Þ ¼ B;
€xð0Þ ¼ C;
ð6Þ
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where x denotes the unknown frequency of oscillation. Upon introducing the new independent variable h ¼ xt and using the method of variation of parameters, the solution to Eq. (6) can be written as
xðhÞ ¼ Fðh; x; A; B; CÞ þ Gðh; x; x0 ; x00 ; xÞ;
ð7Þ
where
Fðh; x; A; B; CÞ ¼ A þ
C
x2
ð1 cosðhÞÞ þ
B
x
sinðhÞ;
ð8Þ
and
Gðh; x; x0 ; x00 ; xÞ ¼
Z
h
hðxðsÞ; x0 ðsÞ; x00 ðsÞ; xÞð1 cosðh sÞÞds;
ð9Þ
0
where hðx; x0 ; x00 ; xÞ ¼ x13 gðx; x0 ; x00 ; xÞ; gðx; x0 ; x00 ; xÞ corresponds to the g function defined in Eq. (6) after making the change of independent variable mentioned above, and the prime denotes differentiation with respect to h. Eqs. (7)–(9) clearly indicate that the initial conditions are contained in F, whereas G contains the nonlinearities of Eq. (5) through h and g in Eq. (9). Both F and G contain the unknown frequency of oscillation x. The Volterra integral equation represented by Eqs. (7)–(9) can be solved in an iterative fashion in an analogous manner to the well-known Lindelöf–Picard method for first-order ordinary differential equations and its (local) convergence is ensured provided that g or h is a Lipschitz-continuous function. However, the frequency of oscillation in Eqs. (7)–(9) is not known and must be determined so that xðhÞ is a periodic function of period 2p. In order to determine both xðhÞ and x from Eqs. (7)–(9), one could treat these equations as a boundary-value problem by requiring that xðhÞ ¼ xðh þ 2pÞ; x0 ðhÞ ¼ x0 ðh þ 2pÞ and x00 ðhÞ ¼ x00 ðh þ 2pÞ; however, due to presence of the nonlinear function h in Eq. (9), the solution of such a boundary-value problem would require, in general, numerical techniques. In this paper, we determine the solution and frequency of oscillation of Eqs. (7)–(9) approximately by means of the following method. Assume that
xðhÞ ¼ x0 ðhÞ þ x1 ðhÞ þ x2 ðhÞ þ ;
ð10Þ
x ¼ x0 þ x1 þ x2 þ ;
ð11Þ
and substitute Eqs. (10) and (11) into Eqs. (7)–(9) to determine xi ðhÞ; 0 6 i, as
x0 ðhÞ ¼ Fðh; x0 ; A; B; CÞ; @F x1 ðhÞ ¼ ðh; x0 ; A; B; CÞx1 þ Gðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þ; @x @F 1 @2F ðh; x0 ; A; B; CÞx2 þ ðh; x0 ; A; B; CÞx21 x2 ðhÞ ¼ @x 2! @ x2 @G @G þ ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx1 ðhÞ þ 0 ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx01 ðhÞ @x @x @G @G þ 00 ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx001 ðhÞ þ ðh; x0 ðhÞ; x00 ðhÞ; x000 ðhÞ; x0 Þx1 ; @x @x
ð12Þ ð13Þ
ð14Þ
etc., and require that xi ðhÞ for i ¼ 1; 2; . . ., be free from secular terms. This sequential approximation provides the values of xi ðhÞ and xi for i ¼ 0; 1; 2; . . ., which can be substituted into Eqs. (10) and (11) to determine the periodic solution and the frequency of oscillation of Eq. (5), respectively. Eqs. (7)–(14) can be interpreted as a generalization of Adomian’s decomposition technique [18–20] where both the solution and the unknown frequency of oscillation are expanded in the series of Eqs. (10) and (11), respectively. If x were assumed to be fixed, Eqs. (10) and (12)–(14) would reduce to the decomposition method [18–20] when this technique is written in terms of h rather than t. The formulation presented here provides trigonometric rather than the more conventional polynomial approximations to xi ðhÞ; 0 6 i, provided by Adomian’s decomposition method [18–20]. For example, for A ¼ C ¼ 0, Eq. (12) yields x0 ðhÞ ¼ xB0 sinðhÞ. An alternative derivation to the Volterra integral formulation presented above consists in first introducing a book-keeping parameter, , in Eq. (7), so that equation becomes
xðhÞ ¼ Fðh; x; A; B; CÞ þ Gðh; x; x0 ; x00 ; xÞ; which coincides with Eq. (7) when
ð15Þ
¼ 1 and then expanding xðhÞ and x as
xðhÞ ¼ x0 ðhÞ þ x1 ðhÞ þ 2 x2 ðhÞ þ
ð16Þ
x ¼ x0 þ x1 þ 2 x2 þ
ð17Þ
Substitution of Eqs. (16) and (17) into Eq. (15) yields a power series expansion in and, by setting the coefficients of this series that multiply the monomials n ; n ¼ 0; 1; . . ., to zero, one can easily obtain Eqs. (12)–(14); approximations to both the solution and the frequency of the oscillation can then be obtained by first requiring that xi ðhÞ; i ¼ 1; 2; . . ., be free from secular terms and then setting ¼ 1 in Eqs. (16) and (17).
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It must also be noted that Eq. (6) can be written as
x000 þ x0 ¼ hðx; x0 ; x00 ; xÞ;
xð0Þ ¼ A;
x0 ð0Þ ¼
B
x
x00 ð0Þ ¼
;
C
x2
ð18Þ
;
or derived from Eqs. (7)–(9) by employing the fundamental theorem of calculus [24–26]. In either case, the use of Eqs. (10) and (11) in Eq. (18) and the requirement that xi ðhÞ; i ¼ 1; 2; . . ., be free from secular terms provide the same results as those of the Volterra integral formulation presented here (cf. Eqs. (7)–(14)). Furthermore, the introduction of a book-keeping parameter, , in Eq. (18) yields
x000 þ x0 ¼ hðx; x0 ; x00 ; xÞ;
xð0Þ ¼ A;
x0 ð0Þ ¼
B
x
;
x00 ð0Þ ¼
C
x2
;
ð19Þ
which coincides with Eq. (18) for ¼ 1, and the use of Eqs. (16) and (17) in Eq. (19) provides power series expansions in for xi ðhÞ; xi ð0Þ; x0i ð0Þ and x00i ð0Þ. By setting the coefficients that multiply the monomials in n ; n ¼ 0; 1; . . ., in these series to zero, a third-order linear ordinary differential equation and its corresponding initial conditions are obtained and their solutions are identical to those of Eqs. (12)–(14) [23]. Furthermore, by requiring that these solutions be free from secular terms and using Eqs. (16) and (17) with ¼ 1, it is an easy exercise to show that this formulation is identical to the Volterra integral one presented in this paper, by means of the fundamental theorem of calculus [24–26]. 3. Presentation of results In this section, the Volterra integral technique presented in Section 2 is applied to three autonomous, non-linear, thirdorder ordinary differential Eqs. (1)–(3) with p = 1 which have analytical periodic solutions and have been previously studied by Gottlieb [6] by means of a first-order, i.e., a one-term, harmonic balance procedure, for A ¼ C ¼ 0 and B – 0 which are the initial conditions considered in this section. For Eqs. (1)–(3) with p = 1, it has been found that the Volterra formulation yields exactly the same results as an artificial parameter-Linstedt–Poincaré method [15,16] and this is consistent with the comments made in the previous section. In particular, this formulation yields the same results as the second-order reduction method with the variable independent h for Eqs. (1) and (2) with p = 1 [15] and the same algebraic results as the Linstedt–Poincaré method with the variable independent h for Eq. (3) with p = 1 [16]. For this reason, we shall not present the approximations to both the solution and the frequency of the oscillation here and refer the reader to Refs. [15,16]. Furthermore, for Eqs. (1)–(3) with p = 1, it has been found that the first-order approximation to the frequency of oscillation coincides with that obtained by Gottlieb [6] by means of a first-order harmonic balance procedure. In addition, for Eq. (1) with p = 1, the frequency of oscillation was found to be identical to that of Table 2 of Ref. [15] except for the errors corresponding to Ma et al. [13] second-order approximation for B ¼ 0:1, 0.2 and 0.5 which should read 0.00000264, 0.00000285 and 0.00059765, respectively, instead of 0.00000027, 0.00000029 and 0.0000598, respectively, as reported by Ma et al. [13] in their Table 1. The new errors reported here were obtained by employing Eqs. (16) and (19) of Ma et al. [13]. For Eq. (1) with p = 1, the series for x (cf. Eq. (11)) was found to converge for both B 6 1 and B P 1 by means of Mathematica 7 [27], and the method presented here provides a two-term approximation to the frequency of oscillation which is slightly more accurate than that of a one-term approximation [15]. This two-term approximation is slightly more (respectively, slightly less) accurate than that of a parameter perturbation procedure [13] for B 6 0:1 (respectively, B > 1). Figs. 1 and 2 show the three-dimensional phase diagram and the period of Eq. (1), respectively, as functions of B and indicate that the period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. _ ðx; z € Although not shown here, the projection of the three-dimensional phase diagram onto the ðx; y xÞ; xÞ and (y, z) planes yield an ellipse, an elongated (and rotated counter-clockwise) closed S curve and an ellipse, respectively, for B ¼ 1. For Eq. (1) with p = 1 and B ¼ 2 (not shown here), the projections of the phase diagram onto the (x, y) and (x, z) planes are analogous to those for B ¼ 1, whereas that onto the (y, z) plane exhibits an egg shape. For Eq. (2) with B ¼ 1:5 and p = 1, a typographical error in Table 4 of Ref. [15] for T 2 was found; T 2 should read 4.807901 instead of 4.808475 as reported in that reference. Moreover, the errors corresponding to Ma et al. [13] second-order approximation for B ¼ 0:1, 0.2 and 0.5 should read 0.00000586, 0.00001886 and 0.00525736, respectively, instead of 0.00000006, 0.00000213 and 0.000526, respectively, as reported by Ma et al. [13] in their Table 2. The new errors reported here were obtained by employing Eqs. (34) and (36) of Ma et al. [13]. For Eq. (2) with p = 1, it was found that the series for x (cf. Eq. (11)) converges for B 6 1 by means of Mathematica 7 [27]. Note, however, that the approximations provided by a first-order harmonic balance [6], parameter–perturbation-Linstetd– Poincaré [13] and artificial parameter-Linstetd–Poincaré methods [15] to the frequency of oscillation of Eq. (2) with p = 1 are not valid for B P 2, even though Eq. (2) has an analytical solution for B P 2. The failure of these techniques for Eq. (2) with B P 2 is associated with the fact that x and x_ exhibit very steep gradients _ plane is nearly a [15] for B greater than about 1.2 and the projection of the three-dimensional phase diagram onto the ðx; xÞ circumference for B 1 and exhibits very steep gradients of x_ at the locations where jxj is largest and very flat regions where _ is largest for B larger than about 1.2. As a consequence, the first-order approximation employed in harmonic balance projxj cedures and that resulting from both the Volterra integral formulation presented here and parameter perturbation methods, _ is very inaccurate for B greater than about 1.2. This has been verified both which correspond to a circumference in the ðx; xÞ,
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4 2 z 0 −2 −4 −2
−2 −1
−1 0
0 1
1 2
2
x
y
Fig. 1. Three-dimensional phase diagram for Eq. (1) with p = 1. (Short, medium and long curves correspond to B ¼ 0:1, 0.5 and 2.0, respectively. B ¼ 0:1, 0.5 and 2.0 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
6.5
6
5.5
T 5
4.5
4
0
0.5
1
1.5
2
2.5
B Fig. 2. Period T as a function of B for Eq. (1) with p = 1. Only the periods marked with circles have been determined.
numerically by means of a fourth-order accurate Runge–Kutta method and a Galerkin approximation for Eq. (2). When using the Runge–Kutta technique, it was observed that the time step must be reduced as B is increased in order to obtain very accurate trajectories; for example, the time step for B ¼ 2 was 0.001, whereas that for B ¼ 10 was 0.00001. In the Galerkin P approximation, Eq. (2) was first written in terms of h and xðhÞ was then approximated by xðhÞ Nn¼0 C ð2nþ1Þ cosðð2n þ 1ÞhÞ in the resulting equation; the residual of that equation was then required to be orthogonal in ½0; 2p to cosðð2n þ 1ÞhÞ and this provided a set of nonlinear equations for C ð2nþ1Þ which was solved iteratively by means of a Newton–Raphson procedure until jC kþ1 C kð2nþ1Þ j ð2nþ1Þ jC kð2nþ1Þ j
k 8 6 105 if jC kð2nþ1Þ j P 103 or jC kþ1 if jC kð2nþ1Þ j < 103 , for 0 6 n 6 N. For B 6 1, very accurate freð2nþ1Þ C ð2nþ1Þ j 6 10
quencies of oscillation where obtained with the Galerkin procedure and N ¼ 1; however, in order to obtain accurate results for larger values of B, the number of harmonics, N, had to be increased as B was increased. For example, for B ¼ 2; N ¼ 12, whereas, for B ¼ 5; N ¼ 400.
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For Eq. (2) with p = 1, it was found that the two-term approximation to the frequency of oscillation obtained with the Volterra integral formulation presented here is more accurate than that corresponding to a parameter perturbation approach [13] and also more accurate than a one-term approximation (cf. Eq. (11)). Figs. 3 and 4 illustrate the three-dimensional phase diagram and the period of Eq. (2), respectively, with p = 1 as functions of B and indicate that the period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. Although not shown here, the projection of the three-dimensional phase diagram for Eq. (2) with p = 1 and B ¼ 2 onto the (x, y) plane is characterized by flat regions near the locations where jxj and jyj are largest; the projection onto the ðx; zÞ plane has a more pronounced elongated (and rotated counter-clockwise) S shape than that for Eq. (1) with B ¼ 1, whereas the projection onto the ðy; zÞ plane has very steep gradients where jyj is largest. The projections of the phase diagram onto the (x, y) and (y, z) planes for Eq. (2) with p = 1 and B P 1:2 clearly indicate the reasons for the failure of the one-term approximation to the frequency of oscillation predicted by harmonic balance and parameter perturbation methods and the Volterra integral formulation presented in this paper for this equation. As indicated in Section 2, the Volterra formulation as well as the other techniques mentioned previously predict a one-term approximation to the periodic solution consisting of only one frequency which results in an ellipsoidal phase diagram in the (x, y) plane, whereas the projection of the phase diagrams corresponding to Eq. (2) with p = 1 and B P 1:2 are not ellipses at all and their accurate prediction requires the use of several harmonics. Moreover, as indicated previously, the time step in Runge–Kutta methods and the number of harmonics in the Galerkin procedure for the numerical solution of Eq. (2) had to be decreased and increased, respectively, as B was increased, because the (x, y)-phase diagrams became steeper at the locations where jxj is largest as B was increased. In Table 1, T i ¼ P2i p denotes the period corresponding to i-terms of the series approximation to the frequency of oscilj¼0
xj
lation (cf. Eq. (11)) for Eq. (3) with p = 1. T 1 and its corresponding error are not reproduced because they are identical to those presented in Table 2 of Ref. [16]. Table 1 indicates that the two-term approximation to the frequency of oscillation calculated with the method presented in this paper is more accurate than Hu [14] parameter perturbation approach and Wu et al.’s [8] second-order linearized harmonic balance procedure for 0:1 6 B 6 20; it is also slightly more (respectively, slightly less) accurate than Ma et al.’s [13] parameter perturbation method for 0:1 6 B < 1 (respectively, 1 < B 6 20). Table 1 also shows that the parameter perturbation method of Ma et al.’s [13] predicts different frequencies of oscillation than that of Hu [14]. It must be pointed out that the errors of T 2 corresponding to the parameter perturbation method of Ma et al. [13] presented in Table 1 do not coincide with those reported by these authors in their Table 3; neither do they coincide with those reported in Table 2 of Ref. [16] which are identical to those of Table 3 of Ref. [13]. The errors corresponding to Ma et al.’s parameter perturbation method [13] presented in Table 1 of this paper have been obtained from Eqs. (50)–(52) of Ref. [13] and do not coincide with those reported by these authors in their Table 3. The results presented in Table 1 differ from and are more accurate than those presented in Table 1 of Ref. [16] which were obtained by means of an artificial parameter-Linstedt–Poincaré method applied to the corresponding third-order nonlinear
5
z 0
−5 −2
−2 −1
−1 0
0 1 y
1 2
2
x
Fig. 3. Three-dimensional phase diagram for Eq. (2) with p = 1. (Short, medium and long curves correspond to B ¼ 0:1, 0.5 and 1.5, respectively. B ¼ 0:1, 0.5 and 1.5 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
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6.5
6
5.5
T
5
4.5
4
3.5
0
0.5
1
1.5
2
2.5
B Fig. 4. Period T as a function of B for Eq. (2) with p = 1. Only the periods marked with circles have been determined.
Table 1 The period ðT i Þ and relative error ðEi ¼ 100ðT i T e Þ=T e Þ of the i-term approximation (cf. Eq. (11)) for Eq. (3) with p = 1, where T e denotes the exact period obtained by integrating numerically Eq. (3) with p = 1 by means of a fourth-order accurate Runge–Kutta method with a time step equal to 0.001. The two-term approximations to the period of oscillation obtained by Hu et al. [14] and Ma et al. [13] by means of a parameter perturbation method are also presented; these methods predict the same T 1 as the technique presented in this paper and a first-order harmonic balance procedure [6]. The errors corresponding to Ma et al. parameter perturbation method [13] have been obtained from Eqs. (50)–(52) of Ref. [13] and do not coincide with those reported by these authors in their Table 3. The periods obtained with a second-order linearized harmonic balance procedure [8] are also shown. B
Te
T 2 ðE2 Þ
T 2 ðE2 Þ [14]
T 2 ðE2 Þ [13]
T 2 ðE2 Þ [8]
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0
25.359725 17.495410 10.210761 6.283185 3.508793 1.468638 0.739762 0.370580
25.56407 (0.806) 17.61452 (0.681) 10.24328 (0.318) 6.28319 (0.000) 3.50486 (0.112) 1.46715 (0.101) 0.73907 (0.094) 0.37024 (0.092)
25.60228 (0.956) 17.63225 (0.782) 10.24768 (0.362) 6.28319 (0.000) 3.50239 (0.182) 1.46528 (0.229) 0.73803 (0.234) 0.36971 (0.235)
24.51794 (-3.319) 17.09799 (-2.272) 10.14660 (-0.628) 6.28319 (0.000) 3.50968 (0.025) 1.46796 (0.046) 0.73930 (0.063) 0.37033 (0.067)
24.6677 (2.729) 17.2251 (1.545) 10.1884 (0.2196) 6.28319 (0.000) 3.50554 (0.0927) 1.46650 (0.1454) 0.738632 (0.1527) 0.370008 (0.1544)
ordinary differential equation in the original independent variable, i.e., t, after introducing the term x2 x_ in the left and right sides of the original equation. As indicated in Section 3.1 of Ref. [16], a Linstedt–Poincaré method applied to the third-order ordinary differential equation in the original independent variable does not provide approximations to the frequency of oscillations; it provides approximations to the differences between the squares of successive approximations to the frequency of oscillations because that method uses an equation analogous to Eq. (11) for x2 rather than for x. The differences in the values of T 2 reported in Table 1 obtained with different methods are to be expected because, as stated in the Introduction, the parameter perturbation methods of Ma et al. [13] and Hu [14] are based on the expansion of some constants that appear or are introduced in the governing equations, whereas the method presented in this paper is based on the expansion of the unknown frequency of oscillation. Moreover, the parameter perturbation methods of Ma et al. [13] and Hu [14] use the original independent variable (t), whereas the technique presented here uses a new independent variable ðhÞ. Figs. 5 and 6 illustrate the three-dimensional phase diagram and the period of Eq. (3), respectively, with p = 1 as functions of B and indicate that period of oscillation decreases whereas the space occupied by the phase diagram increases as B is increased. For B ¼ 1, the projection of the three-dimensional phase diagram onto the (x, y) plane has an elliptical shape, whereas that onto the (y, z) plane is characterized by flat regions of large (respectively, small) slope where jyj (respectively, jzj) is largest. On the other hand, the projection onto the (x, z) plane has a rotated closed S shape. It is worth noting that, for Eqs. (1) and (2) with p = 1, the absolute value of the errors of the two-term approximation to the frequency of oscillation determined by the formulation presented here increases as the initial velocity, i.e., B, is increased; a similar behavior is observed in parameter perturbation methods [13,14]. However, for Eq. (3) with p = 1, the absolute value of the errors of the two-term approximation to the frequency of oscillation provided by parameter perturbation
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300 200 100 z
0 −100 −200 −300 −40
−2 −20
−1 0
0 20
1 40
2
y
x
Fig. 5. Three-dimensional phase diagram for Eq. (3) with p = 1. (Short, medium and long curves correspond to B ¼ 1, 5 and 20, respectively. B ¼ 1, 5 and 20 correspond to blue, red and green, respectively). (For interpretation of colour in figures, the reader is referred to the Web version of this article.)
30
25
20
T 15
10
5
0
0
5
10
15
20
B Fig. 6. Period T as a function of B for Eq. (3) with p = 1. Only the periods marked with circles have been determined.
methods first decreases as B is increased from B ¼ 0:1, then reaches a minimum value at B ¼ 1 for which the exact solution of Eq. (3) with p = 1 (in the absence of round-off errors) is obtained, and then increases as B is increased, although it tends to an almost constant value for B > 20. On the other hand, for Eq. (3) with p = 1, the absolute value of the errors corresponding to the two-term approximation presented here, first decreases as B is increased for B 6 1, then increases (cf. the periods for B ¼ 1 and B ¼ 2 in Table 1), and then decreases for B > 2, although it also tends to a constant value for B > 20. Eqs. (1)–(3) with p = 1 and Eq. (4) with l ¼ 1 clearly indicate that the parameter perturbation methods employed by both Ma et al. [13] and Hu [14], respectively, are based on the introduction of artificial parameters in the governing equations and the expansion of both the solution and some constants that appear in these equations in terms of this artificial parameter; in addition, these authors worked with the original independent variable, i.e., t, and differential equations. By way of contrast, the method proposed here is based on the introduction of a linear term, i.e., x2 x_ on both sides of Eq. (5), a new independent variable, i.e., h ¼ xt, and an integral formulation (cf. Eqs. (7)–(9)) and does not make use at all of an artificial parameter. Furthermore, as the results shown in Table 1 clearly indicate, the two-term approximation of the power series expansion to the
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frequency of oscillation given by the parameter perturbation methods of Ma et al. [13] and Hu [14] depends on the constant being expanded in terms of the artificial parameter (cf. Eqs. (3) and (4)). This is not the case for the method presented in this paper because it does not make use of artificial parameters. 4. Conclusions A Volterra integral formulation based on the introduction of a term equal to the velocity times the square of the unknown frequency of oscillation in both sides of the equation, a new independent variable linearly proportional to the unknown frequency of oscillation and the method of variation of parameters, is presented and used to determine, in series expansions, both the periodic solution and the frequency of oscillation of three autonomous, nonlinear, third-order nonlinear ordinary differential equations which have analytical solutions, and its results are compared with those of a first-order harmonic balance and two parameter perturbation techniques. It has been shown that the method presented in this paper provides a one-term approximation to the frequency of oscillation which is identical to those of a first-order harmonic balance procedure and the first approximation of parameter perturbation procedures, and, for the second equation considered here, the two-term approximation results in a more accurate frequency of oscillation than a parameter perturbation method based on the expansion of the coefficient that multiplies the linear velocity term in power series of an artificial parameter for all initial velocities analyzed here. The technique presented here also predicts a more accurate frequency of oscillation for the third nonlinear third-order ordinary differential equation considered in this paper than a parameter perturbation method that expands zero in terms of power series of an artificial parameter for all the initial conditions analyzed in this paper; it also predicts slightly more (respectively, slightly less) accurate frequencies of oscillation than a parameter perturbation method that introduces the velocity in both sides of the equation and expands the coefficient, i.e., one, that multiplies this term in power series of an artificial parameter, for initial velocities less (respectively, greater) than one. The method presented here has also been found to be more accurate than the second approximation of a linearized harmonic balance procedure for all the initial conditions considered in this paper. It has also be shown that the Volterra integral formulation and the series expansion techniques presented here provide series solutions to both the periodic solution and frequency of oscillation but do not make use of artificial parameters, whereas parameter perturbation methods employ artificial parameters, provide power series expansions in the artificial parameter for both the solution and the frequency of oscillation and predict a two-term approximation to the frequency of oscillation which depends on the constant which is expanded in terms of the artificial parameter. Acknowledgements The research reported in this paper was supported by Projects FIS2005–03191 and FIS2009–12894 from the Ministerio de Ciencia y Educación y Ministerio de Ciencia e Innovación, respectively, of Spain and fondos FEDER. The authors are grateful to the reviewers for their helpful and critical comments which have resulted in an improved manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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[23] J.I. Ramos, C.M. García–López, Periodic solutions of some nonlinear jerk equations, in: J.F. Silva Gomes, Shaker A. Meguid (Ed.), In IRF’2009, Integrity, Reliability and Failure (Challenges and Opportunities), Ediçoes INEGI, Porto, Portugal, 2009, pp. 311–312. [24] T.M. Apostol, Calculus, One-Variable Calculus with an Introduction to Linear Algebra, second ed., vol. 1, John Wiley & Sons, New Yok, 1967. [25] W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill Book Company, New Yok, 1986. [26] D.V. Widder, Advanced Calculus, second ed., Dover Publications, Inc., New Yok, 1989. [27] Mathematica 7, Wolfram Research, Inc., Champaign, Illinois, USA, 2009.