About this Journal Submit a Manuscript Table of Contents
Advances in Mechanical Engineering
Volume 2014 (2014), Article ID 101230, 9 pages
http://dx.doi.org/10.1155/2014/101230
Research Article

Solution of an Integral-Differential Equation Arising in Oscillating Magnetic Fields Using Local Polynomial Regression

1Library of Chongqing University of Technology, Chongqing 400054, China
2School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China

Received 15 July 2013; Accepted 29 December 2013; Published 11 March 2014

Academic Editor: Hiroshi Yabuno

Copyright © 2014 Fenglan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An integrodifferential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields is considered. The local polynomial regression method (LPR) is used to solve this equation. The reliability of this method and the reduction in the size of computational domain give this method a wider applicability. Several representative examples are given to reconfirm the efficiency of these algorithms. The results of applying this theory to the integro-differential equation with time-periodic coefficients reveal that LPR method possesses very high accuracy, adaptability, and efficiency.

1. Introduction

Most scientific problems in engineering are inherently nonlinear. Except a few number of them, majority of nonlinear problems such as complex integro-differential equation do not have analytical solution. Therefore, these nonlinear equations should be solved using some other methods. Some mathematical methods have been employed to solve different physical differential equations, such as Homotopy perturbation technique by authors [14], Homotopy analysis method given by Dehghan and Shakeri [5], time periodic coefficients by Machado and Tsuchida [6], hysteretic damping used by Chen and You [7], rapidly vanishing convolution kernels by authors [8], Adomian decomposition method by Biazar et al. [9] and Biazar [10], operational tau approximation by Dehcheshmeh et al. [11], Modified Homotopy Perturbation Method by authors [12], the Tau approximation for the delayed Burgers equation by Khaksar Haghani et al. [13], a modified variable separated ordinary differential equation method solving mKdV sinh-Gordon equation by Xie [14], the projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics by Ma [15], and Lagrange-Noether method for solving second-order differential equations by Hui-Bin and Run-Heng [16]. They are all proved to have efficiency and utility widely. Recently, H. Caglar and N. Caglar [17] have used local polynomial regression (LPR) method for the numerical solution of linear and nonlinear Fredholm and Volterra integral equations [17]. Moreover, they manage to solve fifth-order boundary value problems by using LPR method [18]. Numerical results demonstrate that local polynomial fitting method is more accurate, simple, and efficient. In this study, we consider (1) with time-periodic coefficients: where , , and are given periodic functions of time that may be easily found in the charged particle dynamics for some field configurations. Here, we assume and consider the following initial conditions: The destination of this paper is to use local polynomial regression method for solving integro-differential equations arising in oscillating magnetic fields.

2. Local Polynomial Regression Method

Local polynomial regression technique was first proposed by Hui-Bin and Run-Heng [16] and H. Caglar and N. Caglar [17, 18]. And, this method was also mentioned and used by Su et al. [1923]. Recently, H. Caglar and N. Caglar [17, 18] firstly took advantage of local polynomial regression method to try to solve the numerical solution of linear and nonlinear Fredholm and Volterra integral equations successfully and evaluated the efficiency and convenience of the method. Meanwhile, they managed to resolve fifth-order boundary value problems by using local polynomial regression method [18] which also showed the efficiency of this technique. Moreover, by comparing with some methods, for an instance, B-spline method, LPR displayed more accuracy. In order to describe the basic ideas of the LPR, firstly, we introduce the mathematical thoughts of local polynomial regression. This idea was mentioned in detail in [1922, 2426]. Since the form of regression function is not specified, the data points with long distance from provide little information to . Therefore, we can only use the local data points around . Supposing that has derivative at , by the Taylor expansion, for point , located in the neighborhood of this point , we can use the -order multivariate polynomials to locally approximate and the surrounding local point of ; we model as where parameter depends on , so called local parameter. Obviously, the local parameter fits the local model with local data and it can be minimized as where controls the size of the bandwidth of local area. Using matrix notation to represent the local polynomial regression is more convenient. Below is the design matrix corresponding to (3) with and : The weighted least squares problem (3) can be written as Here, so the solution vector is

Furthermore, we can get the estimation : where is a column vector (the same size of ) with the first element equal to 1 and the rest equal to zero; that is, . The selection of does not influence the results much. We selected the quartic kernel as follows:

3. Parameters Selections

To implement the local polynomial regression and estimator, we need to choose the order , the kernel , and the bandwidth . These parameters are of course related to each other.

First of all, the choice of the bandwidth parameter is considered, which plays a rather crucial role. A too large bandwidth underparametrizes the regression function [18, 24], causing a large modeling bias, while a too small bandwidth overparametrizes the unknown function and results in noisy estimates. The basic idea is to find a bandwidth that minimizes the estimated mean integrated square error (MISE): where . We give the asymptotic bias and the asymptotic variance regression function . Assume that and that , , and are continuous in a neighborhood of . Further, assume that and . Then the asymptotic conditional variance of is given by The asymptotic conditional bias for odd is given by The asymptotic conditional bias for even is given by provided that and are continuous in a neighborhood of and , where The moments of and are denoted, respectively, by (Proof see Rupper and Wand [27]), where is a constant which relates of the kernel function and the order of the local polynomial is the density function of . However, this ideal bandwidth is not directly usable since it depends on unknown functions.

Another issue in local polynomial fitting is the choice of the order of the polynomial. Since the modeling bias is primarily controlled by the bandwidth, the issue is less crucial. For a given bandwidth , a large value of would expectedly reduce the modeling bias, but would cause a large variance and a considerable computational cost. It is shown in [25] that there is a general pattern of increasing variability; for estimate , there is no increase in variability when passing from an even (i.e., even) order fit to an odd order fit, but when passing from an odd order fit to the consecutive even , order fit, there is a price to be paid in terms of increased variability. Therefore, even order fits are not recommended. Since the bandwidth is used to control the modeling complexity, we recommend the use of lowest odd order; that is, or occasionally .

Another question concerns the choice of the kernel function . Since the estimation is based on the local regression equation (4), no negative weight should be used. As shown in [25], the optimal weight function is , the Epanechnikov kernel, which minimizes the asymptotic mean square error (MSE) of the resulting local polynomial estimators.

4. Solution of the Integrodifferential Equation

In this section, the LPR method for solving (1)-(2) is outlined. Letting (3) be an approximate solution of (1)-(2), where . We can adjust the value of parameter flexibly, and it is required that the approximate solution (19) satisfies (1)-(2) at the point . Therefore, we can transform (1) to (19): Putting (18) in (19), then we can get the following: In order to satisfy unique solution theorem of integral-differential equation, we should take initial conditions (2) into account. Consequently, putting (18) to initial conditions (2), then we can acquire the approximate expression (21) as follows: where . Consequently, we can deduce the minimization function of (1)-(2) from (4); the minimization function can be written as where and . Combining (20), (21) with (22), we can acquire the system which can be expressed as then, the matrix form (5) can be written as follows by using (23)-(24): where , , , , , , ,

Putting (24)-(25) to (9), then estimated set of coefficients can be obtained by solving matrix system solution; therefore, approximate-value at point can be obtained.

5. Illustrative Tests

In this section, to illustrate the description above and to show the efficiency of the mentioned method for solving (1)-(2), we include some examples with known analytical solutions.

Test 1. Considering (1) with is the exact solution of this equation.
Then we have To apply the local polynomial regression method to this equation, based on (9), (24), and (25), we calculate the numerical results and absolute errors at point by setting up different parameters , , and . Numerical results obtained by these approximations are summarized in Table 1. Figure 1 also reveals the results detailed and explicitly.
Considering Table 1, we solve the example with by choosing given various values of parameters presented in Table 1. The absolute errors in the solutions are computed. Given , , it can be seen that the magnitude of absolute errors is between 10 to the power of −4 and 10 to the power of −8. Nevertheless, by setting up , , it is obvious that the magnitude of absolute errors is between 10 to the power of −7 and 10 to the power of −11. We conclude that absolute errors decrease with the increase of the value of and the value of order has no much influence on absolute errors. Considering only parameter , specifically, absolute errors get up to minimum for , when , for , when . We can find that the optimal bandwidth locates in district by using LPR for .

tab1
Table 1: The numerical results and absolute errors at point given different parameters , , and , Test 1, where stands for .
101230.fig.001
Figure 1: (a1) The exact solution; (a2) the LPR solution for , , : the error function for (b1) , , : (b2) , , : (b3) , , : (c1) , , : (c2) , , : (c3) , , in Test 1.

Test 2. As for the second example, consider (1) with is the exact solution of this problem. Then we have similar to Test 1. We take advantage of the local polynomial regression method to solve this equation, based on (9), (24) and (25) which we deduced above. Then we can calculate the numerical results and absolute errors at point by given different parameters , , and . Numerical results are concluded in Table 2 and Figure 2.
Considering Table 2, we solve the example with by choosing given various values of parameters presented in Table 1. The absolute errors in the solutions are computed. Given , , it can be seen that the magnitude of absolute errors is between 10 to the power of −4 and 10 to the power of −8. Nevertheless, by setting up , , it is obvious that the magnitude of absolute errors is between 10 to the power of −9 and 10 to the power of −13. We conclude that absolute errors decrease with the increase of the value of and the value of order has no much influence on absolute errors. Again, considering parameter , specifically, absolute errors get up to minimum for , when , for , when . We can find that the optimal bandwidth locates approximately in district by using LPR for .

tab2
Table 2: The numerical results and absolute errors at point by setting up different parameters , , and , Test 2, where stands for .
101230.fig.002
Figure 2: (a1) The exact solution; (a2) the LPR solution for , , : the error function for (b1) , , : (b2) , , : (b3) , , : (c1) , , : (c2) , , : (c3) , , in Test 2.

Test 3. In this example, consider (1) with , , is the exact solution of this equation. Consequently, we have
Similar to Tests 1 and 2, appling the local polynomial regression method to this equation, based on the (9), (24), and (25), we calculate the numerical results and absolute errors at point by setting up different parameters , , and . Numerical results obtained by these approximations are also explained in Table 3. Figure 3 provides exact solution, LPR solution, and different error functions given by kinds of parameters.
Considering Table 3, we resolve example 3 with by choosing given various values of parameters presented in Table 1. The absolute errors in the solutions are computed. Given , , it can be seen that the magnitude of absolute errors is between 10 to the power of −4 and 10 to the power of −8. By setting up , , it is obvious that the magnitude of absolute errors is between 10 to the power of −7 and 10 to the power of −11. Nevertheless, By setting up , , it is obvious that the magnitude of absolute errors is between 10 to the power of −10 and 10 to the power of −13. We conclude that absolute errors decrease with the increase of the value of and the value of order has no much influence on absolute errors. Again, considering parameter , specifically, absolute errors get up to minimum for , when , for , when . We can find that the optimal bandwidth also locates approximately in district by using LPR for .

tab3
Table 3: The numerical results and absolute errors at point given different parameters , , and , Test 3, where stands for .
101230.fig.003
Figure 3: (a1) The exact solution; (a2) the LPR solution for , , : the error function for (b1) , , : (b2) , , : (b3) , , : (c1) , , : (c2) , , : (c3) , , in Test 3.

6. Conclusions

This paper deals with the numerical solution of an integro-differential equation with time-periodic coefficients which describe the charged particle motion for certain configurations of oscillating magnetic fields by using local polynomial regression method. The kind of technique was tested on three examples and was seen to produce satisfactory results. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. Furthermore, this technique, in contrast to the traditional methods, does not require a small parameter and the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters. The numerical results obtained in this research are indistinguishable due to the fact that this approach justifies its efficiency and presents quite promising results and provides a very high degree of accuracy and efficiency which can be adjused by changing three kinds of parameters , , and without any restrictive assumptions.

Conflict of Interests

The authors declare that they have no possible conflict of interests with any trademark mentioned in the paper.

Acknowledgments

This project was supported by the Natural Science Foundation Project of CQ CSTC of China (Grant no. CSTC2011jjA40033 and Grant no. CSTC2012jjA00037) and was supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ120829).

References

  1. H. Saberi-Nik, S. Effati, and R. Buzhabad, “Analytic-approximate solutionfor an integro-differential equation arising in oscillating magnetic fields using homotopy analysis method,” Iranian Journal of Optimization, vol. 3, no. 10, pp. 518–535, 2010.
  2. J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. J.-H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Dehghan and F. Shakeri, “Solution of an integro-differential equation arising in oscillating magnetic fields using he's homotopy perturbation method,” Progress in Electromagnetics Research, vol. 78, pp. 361–376, 2008. View at Scopus
  6. J. M. Machado and M. Tsuchida, “Solutions for a class of integro-differential equations with time periodic coefficients,” Applied Mathematics E, vol. 2, pp. 66–71, 2002.
  7. J. T. Chen and D. W. You, “An integral-differential equation approach for the free vibration of a SDOF system with hysteretic damping,” Advances in Engineering Software, vol. 30, no. 1, pp. 43–48, 1999. View at Scopus
  8. F. C. Hoppensteadt, Z. Jackiewicz, and B. Zubik-Kowal, “Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels,” BIT Numerical Mathematics, vol. 47, no. 2, pp. 325–350, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. J. Biazar, E. Babolian, and R. Islam, “Solution of the system of ordinary differential equations by Adomian decomposition method,” Applied Mathematics and Computation, vol. 147, no. 3, pp. 713–719, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. J. Biazar, “Solution of systems of integral-differential equations by Adomian decomposition method,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1232–1238, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. S. S. Dehcheshmeh, S. K. Vanani, and J. S. Hafshejani, “Operational Tau approximation for the Fokker-Planck equation,” Chinese Physics Letters, vol. 29, no. 4, Article ID 045201, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. N. Yao-Bin, W. Zhong-Wei, and D. Si-Wei, “Modified homotopy perturbation method for certain strongly nonlinear oscillators,” Chinese Physics Letters, vol. 29, no. 6, Article ID 060502, 2012.
  13. F. Khaksar Haghani, S. Karimi Vanani, and J. Sedighi Hafshejani, “Numerical computation of the tau approximation for the delayed burgers equation the tau approximation,” Chinese Physics Letters, vol. 30, no. 2, Article ID 020201, 2013.
  14. Y.-X. Xie, “Solving mKdV-sinh-Gordon equation by a modified variable separated ordinary differential equation method,” Chinese Physics B, vol. 18, no. 12, pp. 5123–5132, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. Z.-Y. Ma, “The projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics,” Chinese Physics, vol. 16, no. 7, pp. 1848–1854, 2007. View at Publisher · View at Google Scholar · View at Scopus
  16. W. Hui-Bin and W. Run-Heng, “Lagrange-Noether method for solving second-order differential equations,” Chinese Physics B, vol. 18, no. 9, pp. 3647–3650, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. H. Caglar and N. Caglar, “Solution of fifth order boundary value problems by using local polynomial regression,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 952–956, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Caglar and N. Caglar, “Numerical solution of integral equations by using local polynomial regression,” Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 187–195, 2008. View at Scopus
  19. L. Y. Su and F. Li, “Deconvolution of defocused image with multivariate local polynomial regression and iterative wiener filtering in DWT domain,” Mathematical Problems in Engineering, vol. 2010, Article ID 605241, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. L.-Y. Su, “Prediction of multivariate chaotic time series with local polynomial fitting,” Computers and Mathematics with Applications, vol. 59, no. 2, pp. 737–744, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. L. Su, “Multivariate local polynomial regression with application to Shenzhen component index,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 930958, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  22. L. Y. Su, Y. Zhao, and T. Yan, “Two-stage method based on local polynomial fitting for a linear heteroscedastic regression model and its application in economics,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 696927, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. L.-Y. Su, Y.-J. Ma, and J.-J. Li, “Application of local polynomial estimation in suppressing strong chaotic noise,” Chinese Physics B, vol. 21, no. 2, Article ID 020508, 2012. View at Publisher · View at Google Scholar · View at Scopus
  24. J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York, NY, USA, 2003.
  25. J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman Hall, London, UK, 1996.
  26. J. Fan and I. Gijbels, “Data-driven bandwidth selection in local polynomialfitting: variable bandwidth and spatial adaptation,” Journal of the Royal Statistical Society, vol. 57, pp. 371–394, 1995.
  27. D. Rupper and M. P. Wand, “Multivariate weighted least squares regression,” Annals of Statistics, vol. 6, pp. 1122–1137, 1994.