Research Article | Open Access
Constantin Bota, Bogdan Căruntu, Olivia Bundău, "Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations", Mathematical Problems in Engineering, vol. 2014, Article ID 513473, 11 pages, 2014. https://doi.org/10.1155/2014/513473
Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations
We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.
Oscillatory phenomena are frequently encountered in various fields of science such as, for example, physics, molecular biology, and many branches of engineering. These oscillatory phenomena can be modelled using nonlinear differential equations. Nonlinear differential equations are one of the most important mathematical tools required for understanding these oscillatory phenomena present in everyday life. As is known, finding exact solutions of nonlinear differential equations is possible only in some particular cases. This justifies the need to resort to approximate methods for the computation of approximate periodic solutions, which in turn could provide important information about the phenomena studied.
In the present paper we apply the Fourier-least squares method (FLSM) for the computation of approximate periodic solutions for oscillatory phenomena modelled by nonlinear differential equations of the type with the initial conditions where is a nonlinear continuous function, , .
Equation (1) is a very general one, being able to model a large class of oscillatory phenomena. As a consequence, since our method can be applied for (1), it follows that it can be considered a powerful and useful method.
We remark that recently there has been much interest in finding approximate periodic solutions of nonlinear differential equations of type (1) and conditions (2) and many approximate methods were proposed for the computation of approximate periodic solutions of equations of this type.
Among the methods used to compute such approximate periodic solutions we mention the following.
The homotopy perturbation method (see, [1–5]), the variational formulation method (see, [5–7]), the harmonic balance methods (see, [8–13]), the quasilinearization technique (see, ), the reproducing kernel space method (see, ), the Adomian decomposition method (see, ), the parameter-expansion method (see, [17, 18]), the variational iteration methods (see, [19–21]), the energy balance method (see, [20, 22]), the amplitude-frequency formulation (see, [4, 18]), the homotopy analysis method (see, [23–25]), the max-min approach (see, ), the optimal homotopy asymptotic method (see, ), the residue harmonic balance method (see, ), the enhanced cubication method (see, ), the linearisation method (see, ), perturbation methods (see, ), and numerical methods (see, [31–35]).
If the problem consisting of (1) and conditions (2) admits a periodic solution, FLSM allows us to determine an accurate approximate solution of this problem. In order to test the accuracy of the method, we apply it to several well-known examples of nonlinear equations and compare the approximate solutions obtained with this method with approximate solutions obtained by other methods.
2. The Fourier-Least Squares Method
If is an approximate periodic solution of (1), we evaluate the error obtained by replacing the exact solution with the approximate one as the remainder:
Definition 1. One calls an Fourier-function a function of the form where , .
Definition 4. One calls a Fourier-sequence a sequence of Fourier-functions : where , .
Definition 5. One considers a Fourier-sequence satisfying the conditions
We call the Fourier-sequence convergent to the solution of the problem consisting of (1) and conditions (2) if
We will find a weak -approximate Fourier-solution of the type (8), where and the constants , , are calculated using the following steps.(i)By substituting the approximate solution (8) in (1) we obtain the following expression: (ii)We attach to the problem consisting of (1) and conditions (2) the following real functional: where , are computed as functions of , by using the initial conditions (7).(iii)We compute the values of , as the values which give the minimum of the functional (14) and the values of , again as functions of , by using the initial conditions (7).(iv)Using the constants , thus determined, we consider the Fourier-sequence with for , for , and (where ).
The following convergence theorem holds.
Theorem 6. If the problem consisting of (1) and conditions (2) admits a periodic solution, then the Fourier-sequence from (15) satisfies the property Moreover, for all , such that for all , it follows that is a weak -approximate Fourier-solution of the problem consisting of (1) and conditions (2).
Proof. From the fact that the problem consisting of (1) and conditions (2) admits a periodic solution it follows that the series exists and its sequence of partial sums converges to the solution of the problem consisting of (1) and conditions (2); that is, Based on the way the Fourier-function is computed and taking into account the relations (8)–(15), the following inequality holds: It follows that We obtain From this limit we obtain that for all , such that for all , it follows that is a weak -approximate Fourier-solution of the problem consisting of (1) and conditions (2).
Remark 7. Any -approximate Fourier-solution of the problem consisting of (1) and conditions (2) is also a weak approximate Fourier-solution, but the opposite is not always true. It follows that the set of weak approximate Fourier-solutions of the problem consisting of (1) and conditions (2) also contains the approximate Fourier-solutions of the problem.
Taking into account the above remark, in order to find -approximate Fourier-solutions of the problem consisting of (1) and conditions (2) by the Fourier-Least Squares Method, we will first determine weak approximate Fourier-solutions, . If , then is also an -approximate Fourier-solution of the problem.
The test problems included this section are the Duffing oscillator (two cases, an autonomous one and one involving integral forcing terms) and the Jerk equations.
These problems were extensively studied over the years, and various solutions, both approximate analytical ones and numerical ones, were proposed.
The qualitative properties of these oscillators were also extensively studied. Stability and bifurcation studies for the Duffing oscillators include [36–38] among many others. For Jerk-type equations a comprehensive bifurcation study can be found in  and a study of the limit cycles can be found in . Following the computations presented in these papers, corresponding conclusions can be drawn for the problems studied in the Sections 3.1–3.3. For example, in the case of the autonomous Duffing oscillator (23), for the values of the parameters considered in the computations ( and ), a quick computation similar to the one in  indicates that the only equilibrium point is the origin, which is a center; similar computations can be performed for the other problems.
In the following we compute approximate solutions for the Duffing oscillator and the Jerk equations and compare our results with similar analytical approximations previously computed by using other methods.
3.1. Application 1: The Autonomous Duffing Oscillator
Our first test problem is the autonomous Duffing oscillator:
The Duffing oscillator is extensively studied in literature and some relatively recent results are presented in [12, 19]. In  approximate solutions are computed using the rational harmonic balance method (RHB) and in  approximate solutions of (23) are computed using a variational iteration procedure (VI). The approximate frequency of the oscillations obtained by using these methods is compared with the exact period known in the literature.
In the following, in order to obtain our approximation of the frequency and of the solution, we will perform the steps described in the previous section. The computations were performed using the SAGE open source software (version 5.5, available at http://www.sagemath.org/).
We will perform in detail our computations for the values and .
Since the computations of the minimum of the functional (14) are relatively difficult for large values of in (8), we will actually use an iterative procedure, starting with and increasing the value until we achieve the desired precision.
3.1.1. Approximate Solution for
Taking into account these considerations, first we choose the approximate solution (8) of the form
In Step 1, the expression (13) becomes
Taking into account the initial conditions , we obtain the relations
Replacing these values, the corresponding functional (14) from Step 2 is
In Step 3 we must compute the minimum of with respect to and . For relatively simple problems such as this it is possible to compute directly the critical points of and subsequently select the value corresponding to the minimum.
In general, the critical points corresponding to the functional (14) are the solution of the system:
For the system becomes
This system can be solved directly. We used the “solve” command in SAGE and, after we excluded the complex solutions, we found the critical points
In order to find the minimum, we use the second partial derivative test, which is easy enough to implement in SAGE, and find that , is the local minima. Using again the relations , we replace all the values in the expression of the approximate solution (8) and we obtain our first approximation:
In Figure 1 we present the comparison between our approximate solution (solid line) and the numerical solution obtained by using a fourth order Runge-Kutta method (dotted line).
As we already saw, the approximate value of the frequency obtained here is and it is already close to the exact value which in this case is .
We observe that, generally speaking, if the value of in the expression of is larger than , then the solution of the corresponding system of equations which gives the critical points (28) cannot be found directly. In the particular case of SAGE, the command “solve” fails to find the solutions, exiting with some kind of error message.
In this situation it is still possible to find good approximations of the solutions of the problem solving the system (28) by means of a numerical method. More precisely, we can find approximate solutions for the given problem (23) solving (28) by means of a SAGE implementation of the well-known Newton method.
3.1.2. Approximate Solution for
For the approximate solution (8) has the form
After we compute the corresponding expressions of and and of the system (28) (all too large to insert here), we apply Newton’s method taking as the starting point of the iteration , where and are the values of and computed for the previous approximation , namely, and . In order for the sequence of approximations given by Newton’s method to converge to the solution(s) of the system (28), and will take successively values on a given grid of the type , where is a division of a symmetric interval centered in zero.
For all the test problems included in this paper and for all the values of tested, a grid of the form (i.e., from −1 to 1 with step size 0.1) is large enough in the sense that if the starting point scans we can obtain using Newton's method the desired solutions of (28).
In the particular case of the problem (23) it was actually sufficient to choose and and the sequence converged to the minimum of .
For we repeat the same procedure: we compute the corresponding expressions of and and of the system (28) and we apply Newton's method taking as the starting point of the iteration , where are the minimum values computed for the previous approximation and .
The process can be carried on for increased values of until the desired accuracy is reached.
While at a first look the computations may seem long and tedious, by using a software program such as SAGE, we were actually able to perform them easy and quick, obtaining a very good accuracy. Thus, for , we obtained for the approximate frequency the value and the corresponding approximate solution (8) has the form
In Figure 2 we present the comparison between the approximate solution (solid line) and the numerical solution obtained by using a fourth order Runge-Kutta method (dotted line).
For the case , , the approximate solution from  is
Table 1 presents the comparison of the absolute errors (computed as the difference in absolute value between the approximate solution and the corresponding numerical solution given by the Runge-Kutta method) corresponding to the approximate solutions (see, ) and for the case , .
For the case , , the approximate solution from  is
For this case, the approximate solution computed using our method is
Table 2 presents the comparison of the absolute errors (computed as the difference in absolute value between the approximate solution and the corresponding numerical solution given by the Runge-Kutta method) corresponding to the approximate solutions (see, ) and for the case , .
Finally, in Table 3, we compare the approximate values for the frequency computed using our method with approximate values computed in  () and  (). The comparison is made by means of the percentage error, which for a given approximate error is defined as , where is the corresponding exact error. It is easy to see that our approximations are far better than the ones previously computed and they remain accurate even for the case of a very strong nonlinearity.
3.2. Application 2: The Duffing Equation Involving Integral Forcing Terms
Our second test problem is where .
The problem (38) (see, [14, 15]) is a version of the well-known Duffing equation involving both integral and nonintegral forcing terms with separated boundary conditions. This equation has been studied in a series of recent papers including [14, 15].
In , the authors applied a generalized quasilinearization technique to prove the existence and uniqueness of the solution of Duffing equation involving both integral and nonintegral forcing terms. They showed that there are sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem.
In , the authors gave a representation of exact solution and approximate solution of Duffing equation involving both integral and nonintegral forcing terms in the reproducing kernel space (RKS). They represented the exact solution in the form of a series and they showed that the n-term approximation of the exact solution converges to the exact solution.
Thus, for , we obtained the fact that the approximate periodic solution (8) has the form
Since in  only the numerical results are presented while the expression of approximate solution expression is not, we cannot perform a direct graphical comparison of our approximate solution with the corresponding solution from .
Therefore, in Table 4, we present the comparison of several values of the absolute errors (computed as the difference in absolute value between the exact solution and the approximate solution) corresponding to the approximate solutions from  and to our approximate solutions for , as given in .
3.3. Application 3: The Jerk Equation Containing Velocity-Cubed and Velocity Times Displacement-Squared
Our last test is a Jerk nonlinear equation, which describes several physical problems using mechanical oscillations of the third order. The most general form of the Jerk nonlinear equations, which contains the third temporal derivative of displacement, is where the parameters , and are constants.
We consider the following Jerk equation:
Recently, Ma et al. in , using the Homotopy Perturbation Method, obtained high-order analytic approximate periods and periodic solutions of the Jerk Equation (41). In [8, 9], Gottlieb used the lowest-order harmonic balance method to determine analytical approximations to the periodic solution of the Jerk equations. Also, Leung and Guo in  obtained approximations for the angular frequency and the limit cycle for (41) based on the residue harmonic balance approach.
For , the approximate periodic solution from  is
Applying FLSM we computed an approximate periodic solution of the problem (41) of the same order (containing terms up to ): The approximate frequency and period are , with an error of .
In Figure 3, we can visualize and compare our approximate solution (solid line), the approximate solution from  (dashed line), and the numerical solution obtained by using a fourth order Runge-Kutta method (dotted line).
Table 5 presents the comparison of the absolute errors (computed as the difference in absolute value between the approximate solution and the corresponding numerical solution given by the Runge-Kutta method) corresponding to the approximate solution from  and our approximate solution for the case .