Numerical Solution of Integro-Differential Equations with Local Polynomial Regression

Affiliation(s)

School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China.

Library, Chongqing University of Technology, Chongqing, China.

School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China.

Library, Chongqing University of Technology, Chongqing, China.

ABSTRACT

In this paper, we try to find numerical solution of y'(x)= p(x)y(x)+g(x)+λ∫^{b}_{a} K(x, t)y(t)dt, y(a)=α. a≤x≤b, a≤t≤b
or
y'(x)= p(x)y(x)+g(x)+λ∫^{x}_{a} K(x, t)y(t)dt, y(a)=α. a≤x≤b, a≤t≤b
by using Local polynomial regression (LPR) method. The numerical solution shows that this method is powerful in solving integro-differential equations. The method will be tested on three model problems in order to demonstrate its usefulness and accuracy.

In this paper, we try to find numerical solution of y'(x)= p(x)y(x)+g(x)+λ∫

Cite this paper

L. Su, T. Yan, Y. Zhao, F. Li and R. Liu, "Numerical Solution of Integro-Differential Equations with Local Polynomial Regression,"*Open Journal of Statistics*, Vol. 2 No. 3, 2012, pp. 352-355. doi: 10.4236/ojs.2012.23043.

L. Su, T. Yan, Y. Zhao, F. Li and R. Liu, "Numerical Solution of Integro-Differential Equations with Local Polynomial Regression,"

References

[1] B. Asady and M. T. Kajani, “Direct Method for Solving Integro Differential Equations Using Hybrid Fourier and Block-Pluse Functions,” International Journal of Computer Mathematics, Vol. 82, No. 7, 2007, pp. 889-895. doi:10.1080/00207160412331336044

[2] D. F. Han and X. F. Shang, “Numerical Solution of Integro-Differential Equations by Using CAS Wavelet Operational Matrix of Integration,” Vol. 156, No. 2, Applied Mathematics Computation, 2007, pp. 460-466. doi:10.1016/j.amc.2007.04.048

[3] A. Golbabai and M. Javidi, “Application of He’s Homotopy Perturbation Method for nth-Order Integro-Differential Equations,” Applied Mathematics Computation, Vol. 190, No. 2, 2007, pp. 1409-1416. doi: 10.1016/j.amc.2007.02.018

[4] A. Karamete and M. Sezer, “A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations,” International Journal of Computer Mathematics, Vol. 79, No. 9, 2002, pp. 987-1000. doi:10.1080/00207160216116

[5] H. Jaradat, O. Alsayyed and S. Al-Shara, “Numerical Solution of Linear Integro-Differential Equations 1,” Journal of Mathematics Statistics, Vol. 4, No. 4, 2008, pp. 250-254. doi: 10.1.1.152.8509

[6] K. Maleknejad and F. Mizaee, “Numerical Solution of Integro-Differential Equations by Using Rationalized Haar Functions Method,” International Journal of System Mathematics, Vol. 35, No. 10, 2006, pp. 1735-1744. doi 10.1108/03684920610688694

[7] J. Pour-Mahmoud, M. Y. Rahimi-Ardabili and S. Shahmorad, “Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method,” Applied Mathematics Computation, Vol. 168, No. 1, 2005, pp. 465-478. doi:10.1016/j.amc.2004.09.026

[8] M. T. Rashed, “Numerical Solution of Functional DifferenTial, Integral and Integro-Differential Equations,” Applied Numerical Mathematics, Vol. 156, No. 2, 2004, pp. 485-492. doi:10.1016/j.amc.2003.08.021

[9] 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, 2008, pp. 187-195.

[10] L. Y. Su, Y. Y. Zhao and T. S. Yan, “Two-Stage Method Based on local Polynomial Fitting for a Linear Hetero- scedastic Regression Model and Its Application in Economics,” Discrete Dynamics in Nature and Society, Vol. 2012, 2012, Article ID 696927. doi:10.1155/2012/696927

[11] L. Y. Su, “Prediction of Multivariate Chaotic Time Series with Local Polynomial Fitting,” Computers Mathematics with Applications, Vol. 59, No. 2, 2010, pp. 737-744. doi:10.1016/j.camwa.2009.10.019

[12] L. Y. Su, Y. J. Ma and J. J. Li, “Application of Local Polynomial Estimation in Suppressing Strong Chaotic Noise,” Chinese Physics B, 2012, Vol. 21, No. 2, 2012. doi:10.1088/1674-1056/21/2/020508

[13] J. Fan and I. Gijbels, “Local Polynomial Modelling and Its Applications, Vol. 66 of Monographs on Statistics and Applied Probability,” Chapman Hall, London, 1996.

[14] J. Fan and Q. Yao, “Nonlinear Time Series: Nonparametric and Parametric Methods, Springer Series in Statistics,” Springer, Berlin, 2003.

[1] B. Asady and M. T. Kajani, “Direct Method for Solving Integro Differential Equations Using Hybrid Fourier and Block-Pluse Functions,” International Journal of Computer Mathematics, Vol. 82, No. 7, 2007, pp. 889-895. doi:10.1080/00207160412331336044

[2] D. F. Han and X. F. Shang, “Numerical Solution of Integro-Differential Equations by Using CAS Wavelet Operational Matrix of Integration,” Vol. 156, No. 2, Applied Mathematics Computation, 2007, pp. 460-466. doi:10.1016/j.amc.2007.04.048

[3] A. Golbabai and M. Javidi, “Application of He’s Homotopy Perturbation Method for nth-Order Integro-Differential Equations,” Applied Mathematics Computation, Vol. 190, No. 2, 2007, pp. 1409-1416. doi: 10.1016/j.amc.2007.02.018

[4] A. Karamete and M. Sezer, “A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations,” International Journal of Computer Mathematics, Vol. 79, No. 9, 2002, pp. 987-1000. doi:10.1080/00207160216116

[5] H. Jaradat, O. Alsayyed and S. Al-Shara, “Numerical Solution of Linear Integro-Differential Equations 1,” Journal of Mathematics Statistics, Vol. 4, No. 4, 2008, pp. 250-254. doi: 10.1.1.152.8509

[6] K. Maleknejad and F. Mizaee, “Numerical Solution of Integro-Differential Equations by Using Rationalized Haar Functions Method,” International Journal of System Mathematics, Vol. 35, No. 10, 2006, pp. 1735-1744. doi 10.1108/03684920610688694

[7] J. Pour-Mahmoud, M. Y. Rahimi-Ardabili and S. Shahmorad, “Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method,” Applied Mathematics Computation, Vol. 168, No. 1, 2005, pp. 465-478. doi:10.1016/j.amc.2004.09.026

[8] M. T. Rashed, “Numerical Solution of Functional DifferenTial, Integral and Integro-Differential Equations,” Applied Numerical Mathematics, Vol. 156, No. 2, 2004, pp. 485-492. doi:10.1016/j.amc.2003.08.021

[9] 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, 2008, pp. 187-195.

[10] L. Y. Su, Y. Y. Zhao and T. S. Yan, “Two-Stage Method Based on local Polynomial Fitting for a Linear Hetero- scedastic Regression Model and Its Application in Economics,” Discrete Dynamics in Nature and Society, Vol. 2012, 2012, Article ID 696927. doi:10.1155/2012/696927

[11] L. Y. Su, “Prediction of Multivariate Chaotic Time Series with Local Polynomial Fitting,” Computers Mathematics with Applications, Vol. 59, No. 2, 2010, pp. 737-744. doi:10.1016/j.camwa.2009.10.019

[12] L. Y. Su, Y. J. Ma and J. J. Li, “Application of Local Polynomial Estimation in Suppressing Strong Chaotic Noise,” Chinese Physics B, 2012, Vol. 21, No. 2, 2012. doi:10.1088/1674-1056/21/2/020508

[13] J. Fan and I. Gijbels, “Local Polynomial Modelling and Its Applications, Vol. 66 of Monographs on Statistics and Applied Probability,” Chapman Hall, London, 1996.

[14] J. Fan and Q. Yao, “Nonlinear Time Series: Nonparametric and Parametric Methods, Springer Series in Statistics,” Springer, Berlin, 2003.