Abstract
Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.
1. Introduction
Oscillation phenomena are very common in nature and industrial production [1–3], and they are of great interest to scientists and engineers. Most oscillation systems are inherently nonlinear, and the superposition principle is invalid, so they are more difficult to handle than linear ones [2, 3].
In this paper, we focus on the initial value problem of the free nonlinear oscillator with cyclic motion, governed by where the dot denotes the derivative with respect to the time and is a physical variable, such as displacement. Usually the free nonlinear oscillator with cyclic motion has a limit cycle, which is independent of initial conditions. Then without loss of generality, the following initial value condition is considered:
Thus far, there are two branches on handling the aforementioned nonlinear oscillation equations. The first one is pure analytical, among which the well-known perturbation technique is widely applied to investigate the nonlinear oscillation equations [3–5]. Besides, the homotopy analysis method [6, 7] and variational iteration method [8] are used to investigate the nonlinear oscillation equations recently.
On the other hand, numerical methods are adopted to solve the nonlinear oscillations, for example, the Runge-Kutta method and the spectral method. When the aforementioned problem is numerically integrated by the Runge-Kutta method, the solution in discrete time is obtained, where the time step is restricted by the stability condition. It is well known that a function with the period can be expressed by a Fourier series. To acquire some spectral characteristics of , such as the frequency, amplitude-frequency distribution, the discrete Fourier analysis should be carried on the discrete solution . When the number becomes larger, the Fast Fourier Transform (FFT) algorithm, which is somewhat complex, should be adopted for the computational efficiency [9].
In the recent decade, the spectral method [9–12] is prevailing due to its high accuracy. The function is approximated by a sum of the orthogonal functions, for example, the complex exponential function for the periodic function: Since the function is assumed to be real, the two Fourier coefficients with an opposite value of are complex conjugates; that is, , where the bar denotes complex conjugate operation. The nonlinear terms, such as and , are handled by the pseudospectral technique, which is involved in the FFT algorithm for the computational efficiency, especially for large . Meantime, the aliasing removal technique is needed to alleviate the aliasing error [9]. The accuracy of the approximate solution is mainly decided by the number , and usually the larger , the more accurate of is. The idea of the spectral methods is clear and straightforward, but the appropriate is problem dependent and cannot be determined beforehand. Usually some different , such as , , and should be chosen to find the appropriate to satisfy the accuracy demand. What is the relationship between , , and ? It is expected that the computational cost is economical if a method has the succession property, which means the more accurate (higher order) approximation can be further acquired from the less accurate (lower order) approximations and/or by adding some more correction terms and without discarding the existing less accurate ones. Unfortunately, the traditional spectral method does not have this succession property, and the valuable information provided by the less accurate approximation is not utilized sufficiently when we seek the more accurate ones.
Recently, the fixed point method [13–15], which is based on the fixed point concept in functional analysis, is adopted to acquire the series solution of the differential equation. In this paper, the idea of the fixed point method and the traditional spectral method are combined; therefore, the spectral fixed point method (SFPM) is proposed. By SFPM, we could directly obtain an explicit Fouries series solution of the nonlinear oscillation with cyclic motion. Moreover, the high accuracy property of the traditional spectral method is inherited by SFPM, and the spectral characteristic of the solution is simultaneously obtained without resorting to the FFT algorithm. Furthermore, it is notable that SFPM possesses the succession property, which means the computational cost of SFPM is economical.
The organization of the rest of this paper is as follows. In Section 2 the idea of the spectral fixed point method is elaborated, and the steepest descent seeking algorithm is proposed to improve computational efficiency. In Section 3, two examples are investigated by SFPM in detail. Finally, Section 4 is devoted to concluding remarks.
2. The Spectral Fixed Point Method
2.1. The Key Idea of the Spectral Fixed Point Method
The fixed point is a basic concept in functional analysis [16, 17]. The famous Newton’s method for nonlinear algebra equations is just based on the Banach fixed point theorem. In [13], the fixed point concept is extended to solve nonlinear differential equations and the fixed point method (FPM) is proposed.
In the present work, the idea of FPM and the spectral method are combined; therefore, the spectral fixed point method (SFPM) is brought forward to investigate the nonlinear oscillation problem.
Let denote the frequency of governed by (1) and (2). When we introduce the transformation: the original governing equation is rearranged as follows, where the prime denotes the derivative with respect to and is a nonlinear operator. According to the above transformation (4), it is clear that is a function with the period that is, .
Here, a contractive map is constructed as follows: where is chosen as a linear continuous bijective operator, named as the linear characteristic operator. The operator is the inverse operator of . is a real nonzero free parameter, named as relaxation factor, which can improve the convergence and stability of iteration procedure. The optimal value of relaxation factor usually is dependent on the problem to be solved. From (6), an iteration procedure is built up as follows: where is a sequence of relaxation factors. According to (8), we can obtain a solution sequence . If the convergence of is ensured and we take the limit operation on both sides of (8), it is found that the limit value exactly is the zero point of the nonlinear operator , that is, Then is called as a fixed point of the contractive map .
2.2. The Linear Characteristic Operator
Just as mentioned in Section 1, a -periodic function can be approximated by a sum of complex exponential functions in the spectral method: The basis function satisfies the following 2nd-order differential equation: and has the following orthogonality relation: The more relevant properties of can be found in the literature [9].
In consideration of the differential equation (11), here let us choose the linear characteristic operator in (8) as follows: It is clear that is much simpler than the original nonlinear operator . The common solution of linear equation, , is where and are named as the kernels of and is a complex constant of integration.
When the basis functions system and the linear characteristic operator are determined, the member of the solution sequence should be expressed by a sum of the basis functions: which is a so-called principle of completeness in SFPM. In Section 2.4, we will point out that this fundamental principle can heuristically provide a solvability condition to determine the unknown frequency .
On account of the well-known Euler’s formula, (15) is rearranged as where the coefficients and are real numbers.
2.3. The Initial Guess
The iteration procedure (8) can start with some arbitrary initial value , and usually the more closer to the , the more rapidly the solution sequence converges to . Here, the initial guess could be conveniently chosen as follows: Hence,
2.4. Solvability Condition and Succession Property
Here, let us investigate the relationship between the lower order approximation and the higher order ones. If in (8) is expanded by a sum of basis functions, we can obtain the solution of the inhomogeneous linear equation (8) as follows: The appearance of terms such as and will disobey the principle of completeness; that is to say, these terms cannot be expressed by a linear combination of the basis functions , so the coefficients and should vanish: which is named as the solvability condition. The idea to avoid the appearance of and is actually widely applied in perturbation methods, and its physical meaning is to render the solution uniformly valid.
The aforementioned derivation of the solvability condition is heuristic. In fact, it can be obtained in a strict manner. Given the expression in (13), the iteration procedure (8) can be rewritten as follows: where is the -order approximation to the exact frequency . Equation (22) is linear and inhomogeneous, so it has a solution if and only if the inhomogeneous parts satisfy some solvability conditions. Let us multiply both sides of (22) by and integrate (22) over the range . By integration by parts on the left-hand side, we obtain on account of the periodic property of ; that is, . Hence, the right-hand side is In consideration of the orthogonality relation (12), we thus obtain Taking complex conjugates operation on (25), an equivalent expression is Once again we deduce the solvability condition rigorously, which just provides an equation to determine the -order approximate frequency .
Then, it follows from (20) and (21) that where the complex constants are determined by the initial condition: From (27), it is clear that the higher order approximation is the sum of the lower order approximation and the correction terms + . In other words, SFPM has the succession property, which means that the valuable information provided by the lower order approximation is preserved and utilized sufficiently, and the accuracy of the approximation could be improved step-by-step to any possibility.
2.5. The Steepest Descent Seeking Algorithm
As mentioned in Section 2.1, the relaxation factors can improve the convergence and stability of iteration procedure, and usually the optimal value of relaxation factors is dependent on the problem to be solved. In the present work, an algorithm, named as the steepest descent seeking algorithm (SDS), is adopted to determine the optimal value of the relaxation factor.
Let denote the square residual error of the iteration procedure (8): which is a kind of global residual error and can simultaneously evaluate the accuracy of and . Then it is suggested that the optimal value of relaxation factor corresponds to the value at which the square residual error obtains the minimum value . For example, when , the square residual error is a function of only and thus the optimal value can be obtained by solving the nonlinear algebraic equation:
When , the square residual error is dependent on and . Because the optimal value has been acquired from the previous step, the optimal value is governed by the following nonlinear algebraic equation: Similarly, for the -higher order, the square residual error actually contains an unknown relaxation factor only, so the optimal value is determined by the following nonlinear algebraic equation: The name of the steepest descent seeking algorithm just comes from the aforementioned approach; that is, every optimal value is sought to minimize the corresponding square residual error . According to this approach, only one nonlinear algebraic equation should be solved in every iteration step, and the elements of the sequence are obtained sequentially and separately. For convenience, the spectral fixed point method with the steepest descent seeking algorithm is abbreviated to SFPM-SDS in this paper.
3. Some Examples
In this section, SFPM-SDS is used to investigate some examples. All the calculations are implemented on a laptop PC with 2 GB RAM and Intel Core2 Duo 1.80 GHz CPU.
3.1. Example 1
Let us start from the first example [3–5]: Introducing the transformation (4), we obtain Hence, the iteration procedure is built as follows: and the square residual error is
The initial guess is chosen as mentioned in Section 2.3,
According to SFPM-SDS, the equation of the first order approximation is governed by with the associated initial condition , . Then the solvability condition is and is The corresponding is Hence, we obtain
For the higher-order and , the procedure is similar and can be deduced by the symbolic computation software, such as MAXIMA, MAPLE, and MATHEMATICA. The first lower order approximations by SFPM-SDS are succinctly listed here,
Further insight into the first lower order approximations (42)~(43), it is found that the absolute magnitude of the coefficients occurred in (16) decreasing very rapidly when becomes larger, so it is not necessary to retain all the coefficients during the iteration evaluation. It is expected that the effect of the higher-order wave component on the accuracy of is relatively negligible to the lower-order ones. Then a threshold value is introduced to decide which would be omitted, and the computational cost is shrunken if we only preserve the coefficients which satisfy . In the following calculation, we set . The detail of , , and CPU time (seconds) is shown in Table 1. From Table 1, it is also found that corresponding to the higher order approximation deviates a little from the lower order ones, so the value corresponding to the higher order approximation can simply inherit from the lower order ones. It is expected that further diminishment of the computational cost will be gained in this manner. For convenience, let denote the executing times of the SDS algorithm. Here we consider the case , which means only the first five relaxation factors ~ are determined by the SDS algorithm, and the succeeding other relaxation factors are set as follows: The result of the above approach is also shown in Table 1 for comparison. Meantime, some of three different order approximations are given in Figure 1, which shows that actually decrease very rapidly when becomes larger.
The original equation (33) is also numerically integrated by the high-order Runge-Kutta method with high accuracy. Let and denote the solution and the frequency calculated by the Runge-Kutta method, respectively. The comparison of with is given in Table 2, which shows that is the same as within 7 significant digits when . The comparison of with is given in Figures 2 and 3, which shows that the maximum relative error between and is about % and % for , , respectively. The accuracy of can also be justified by the square residual error . As shown in Table 1 and Figure 4, it is clear that the solution sequence converges to the exact solution very rapidly.
| (a) 5th approximation |
| (b) 10th approximation |
3.2. Example 2
The second example [4, 5] is governed by According to SFPM-SDS, the iteration procedure has the form: The solution procedure is similar to Example 1, so only the results are provided. To investigate the effect of SDS algorithm on the convergence of the approximate solution, the results corresponding to two different values are shown in Table 3. It is found that has a little effect on the convergence of the approximate solution, so usually can take a small value to decrease the computational cost, especially when the first few are close to each other. The comparison of the frequency with is given in Table 4, which shows that the value is the same as within 7 significant digits when .
4. Conclusion
In this paper, based on the fixed point concept in functional analysis, the spectral fixed point method (SFPM) is proposed for the nonlinear oscillation equation with periodic solution, and the steepest descent seeking (SDS) algorithm is brought forward in the framework of SFPM to improve the computational efficiency. Two typical examples are discussed in detail as the application of SFPM. The result shows the following.(a)The SFPM behaves with a high accuracy as the traditional spectral method.(b)The SFPM possesses the succession property, so the accuracy of the approximation can be improved step-by-step to any possibility with a low computational cost.(c)In the framework of SFPM, the spectral characteristic of oscillation equation is the byproduct without resorting to FFT algorithm.(d)SDS algorithm can greatly improve the computational efficiency, and therefore it is practicable to apply SFPM-SDS to handle some types of nonlinear oscillation equations with periodic solution.
So far, only the nonlinear ordinary differential equations are investigated by SFPM in this paper, but SFPM has the capability to handle the nonlinear system and partial differential equation, and it will be discussed in the future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work is supported by National Natural Science Foundation of China (Approval nos. 11102150 and 11302165) and the Fundamental Research Funds for the Central Universities.