The Bezier Control Points Method for Solving Delay Differential Equation

ABSTRACT

In this paper, Bezier surface form is used to find the approximate solution of delay differential equations (DDE’s). By using a recurrence relation and the traditional least square minimization method, the best control points of residual function can be found where those control points determine the approximate solution of DDE. Some examples are given to show efficiency of the proposed method.

In this paper, Bezier surface form is used to find the approximate solution of delay differential equations (DDE’s). By using a recurrence relation and the traditional least square minimization method, the best control points of residual function can be found where those control points determine the approximate solution of DDE. Some examples are given to show efficiency of the proposed method.

Cite this paper

F. Ghomanjani and M. Hadi Farahi, "The Bezier Control Points Method for Solving Delay Differential Equation,"*Intelligent Control and Automation*, Vol. 3 No. 2, 2012, pp. 188-196. doi: 10.4236/ica.2012.32021.

F. Ghomanjani and M. Hadi Farahi, "The Bezier Control Points Method for Solving Delay Differential Equation,"

References

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[2] F. M. Asl and A. G. Ulsoy, “Analysis of a System of Linear Delay Differential Equations,” Journal of Dynamic Systems, Measurement and Control, Vol. 125, No. 2, 2003, pp. 215-223. doi:10.1115/1.1568121

[3] J. K. Hale and S. M. V. Lunel, “Introduction to Functional Differential Equations,” Springer-Verlag, Berlin, 1993.

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[5] A. K. Alomari, M. S. M. Noorani and R. Nazar, “Solution of Delay Differential Equation by Means of Homotopy Analysis Method,” Acta Applicandae Mathematicae, Vol. 108, No. 2, 2009, pp. 395-412. doi:10.1007/s10440-008-9318-z

[6] D. J. Evans and K. R. Raslan, “The Adomian Decomposition Method for Solving Delay Differential Equation,” International Journal of Computer Mathematics, Vol. 82, No. 1, 2005, pp. 49-54. doi:10.1080/00207160412331286815

[7] S. J. Liao, “Series Solutions of Unsteady Boundary-Layer Flows over Plate,” Mathematical Analysis and Applications, Vol. 117, No. 3, 2006, pp. 239-263. doi:10.1111/j.1467-9590.2006.00354.x

[8] F. Shakeri and M. Dehghan, “Solution of Delay Diffrential Equation via a Homotopy Perturbation Method,” Mathematical and Computer Modelling, Vol. 48, No. 3-4, 2008, pp. 486-498. doi:10.1016/j.mcm.2007.09.016

[9] H. Gorecki, S. Fuksa, P. Grabowski and A. Korytowski, “Analysis and Synthesis of Time Delay Systems,” John Wiley and Sons, New York, 1989.

[10] X. Chen and L. Wang, “The Variational Iteration Method for Solving a Neutral Functional-Differential Equation with Proportional Delays,” Computers and Mathematics with Applications, Vol. 59, No. 8, 2010, pp. 2696-2702. doi:10.1016/j.camwa.2010.01.037

[11] Z. Fan, M. Liu and W. Cao, “Existence and Uniqueness of the Solutions and Convergence of Semi-Implicit Euler Methods for Stochastic Pantograph Equations,” Mathematical Analysis and Applications, Vol. 325, No. 2, 2007, pp. 1142-1159. doi:10.1016/j.jmaa.2006.02.063

[12] R. Bellman and K. L. Cooke, “Differential-Difference Equations,” Academic Press, London, 1963.

[13] W. Wang, T. Qin and S. Li, “Stability of One-Leg θMethods for Nonlinear Neutral Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 213, No. 1, 2009, pp. 177-183. doi:10.1016/j.amc.2009.03.010

[14] A. Bellen and M. Zennaro, “A Reviw of DDE Methods,” In: G. H. Golub, C. H. Schwab, W. A. Light and E. Suli, Eds., Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Clarendon Press, New York, 2003, pp. 36-60.

[15] E. Ishiwata and Y. Muroya, “Rational Approximation Method for Delay Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp.741-747. doi:10.1016/j.amc.2006.08.086

[16] E. Ishiwata, Y. Muroya and H. Brunner, “A SuperAttainable Order in Collocation Methods for Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 198, No. 1, 2008, pp. 227-236. doi:10.1016/j.amc.2007.08.078

[17] P. Hu, C. Huang and S. Wu, “Asymptotic Stability of Linear Multistep Methods for Nonlinear Neutral Delay Differential Equations,” Applied Mathematics and Computation, Vol. 211, No. 1, 2009, pp. 95-101. doi:10.1016/j.amc.2009.01.028

[18] W. Wang, Y. Zhang and S. Li, “Stability of Continuous Runge-Kutta-Type Methods for Nonlinear Neutral DelayDifferential Equations,” Applied Mathematical Modelling, Vol. 33, No. 8, 2009, pp. 3319-3329. doi:10.1016/j.apm.2008.10.038

[19] W. Wang and S. Li, “On the One-Leg θ-Methods for Solving Nonlinear Neutral Functional Differential Equations,” Applied Mathematics and Computation, Vol. 193, No. 1, 2007, pp. 285-301. doi:10.1016/j.amc.2007.03.064

[20] G. Farin, “Curves and Surfaces for CAGO: A Practical Guide,” 1st Edition, Morgan Kaufmann, Waltham, 2001.

[21] G. Farin, “Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide,” 4th Edition, Academic Press, London, 1997.

[22] S. Mann, “A Blossoming Development of Spliness,” 1st Edition, Morgan Claypool, San Rafael, 2004.

[23] S. Biswa and B. Lovell, “Bezier and Splines in Image Processing and Machine Vision,” Springer-Verlag, Berlin, 2008.

[24] J. Zheng, T. Sedberg and R. Johansons, “Least Squares Methods for Solving Differential Equation Using Bezier Control Points,” Applied Numerical Mathematics, Vol. 48, No. 2, 2004, pp. 137-152. doi:10.1016/j.apnum.2002.01.001

[25] B. Egerstedt and F. Martin, “A Note on the Connection between Bezier Curves and Linear Optimal Control,” IEEE Transactions on Automatic Control, Vol. 49, No. 10, 2004, pp. 1728-1731. doi:10.1109/TAC.2004.835393

[26] M. Mahmoud and P. Shi, “Methodologies for Control of Jump Time-Delay Systems,” Kluwer Academic Publishers, London, 2004.

[1] G. Adomian and R. Rach, “Nonlinear Stochastic Differential Delay Equation,” Journal of Mathematical Analysis and Applications, Vol. 91, No. 1, 1983, pp. 94-101. doi:10.1016/0022-247X(83)90094-X

[2] F. M. Asl and A. G. Ulsoy, “Analysis of a System of Linear Delay Differential Equations,” Journal of Dynamic Systems, Measurement and Control, Vol. 125, No. 2, 2003, pp. 215-223. doi:10.1115/1.1568121

[3] J. K. Hale and S. M. V. Lunel, “Introduction to Functional Differential Equations,” Springer-Verlag, Berlin, 1993.

[4] S. I. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Springer, Berlin, 2001.

[5] A. K. Alomari, M. S. M. Noorani and R. Nazar, “Solution of Delay Differential Equation by Means of Homotopy Analysis Method,” Acta Applicandae Mathematicae, Vol. 108, No. 2, 2009, pp. 395-412. doi:10.1007/s10440-008-9318-z

[6] D. J. Evans and K. R. Raslan, “The Adomian Decomposition Method for Solving Delay Differential Equation,” International Journal of Computer Mathematics, Vol. 82, No. 1, 2005, pp. 49-54. doi:10.1080/00207160412331286815

[7] S. J. Liao, “Series Solutions of Unsteady Boundary-Layer Flows over Plate,” Mathematical Analysis and Applications, Vol. 117, No. 3, 2006, pp. 239-263. doi:10.1111/j.1467-9590.2006.00354.x

[8] F. Shakeri and M. Dehghan, “Solution of Delay Diffrential Equation via a Homotopy Perturbation Method,” Mathematical and Computer Modelling, Vol. 48, No. 3-4, 2008, pp. 486-498. doi:10.1016/j.mcm.2007.09.016

[9] H. Gorecki, S. Fuksa, P. Grabowski and A. Korytowski, “Analysis and Synthesis of Time Delay Systems,” John Wiley and Sons, New York, 1989.

[10] X. Chen and L. Wang, “The Variational Iteration Method for Solving a Neutral Functional-Differential Equation with Proportional Delays,” Computers and Mathematics with Applications, Vol. 59, No. 8, 2010, pp. 2696-2702. doi:10.1016/j.camwa.2010.01.037

[11] Z. Fan, M. Liu and W. Cao, “Existence and Uniqueness of the Solutions and Convergence of Semi-Implicit Euler Methods for Stochastic Pantograph Equations,” Mathematical Analysis and Applications, Vol. 325, No. 2, 2007, pp. 1142-1159. doi:10.1016/j.jmaa.2006.02.063

[12] R. Bellman and K. L. Cooke, “Differential-Difference Equations,” Academic Press, London, 1963.

[13] W. Wang, T. Qin and S. Li, “Stability of One-Leg θMethods for Nonlinear Neutral Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 213, No. 1, 2009, pp. 177-183. doi:10.1016/j.amc.2009.03.010

[14] A. Bellen and M. Zennaro, “A Reviw of DDE Methods,” In: G. H. Golub, C. H. Schwab, W. A. Light and E. Suli, Eds., Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Clarendon Press, New York, 2003, pp. 36-60.

[15] E. Ishiwata and Y. Muroya, “Rational Approximation Method for Delay Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp.741-747. doi:10.1016/j.amc.2006.08.086

[16] E. Ishiwata, Y. Muroya and H. Brunner, “A SuperAttainable Order in Collocation Methods for Differential Equations with Proportional Delay,” Applied Mathematics and Computation, Vol. 198, No. 1, 2008, pp. 227-236. doi:10.1016/j.amc.2007.08.078

[17] P. Hu, C. Huang and S. Wu, “Asymptotic Stability of Linear Multistep Methods for Nonlinear Neutral Delay Differential Equations,” Applied Mathematics and Computation, Vol. 211, No. 1, 2009, pp. 95-101. doi:10.1016/j.amc.2009.01.028

[18] W. Wang, Y. Zhang and S. Li, “Stability of Continuous Runge-Kutta-Type Methods for Nonlinear Neutral DelayDifferential Equations,” Applied Mathematical Modelling, Vol. 33, No. 8, 2009, pp. 3319-3329. doi:10.1016/j.apm.2008.10.038

[19] W. Wang and S. Li, “On the One-Leg θ-Methods for Solving Nonlinear Neutral Functional Differential Equations,” Applied Mathematics and Computation, Vol. 193, No. 1, 2007, pp. 285-301. doi:10.1016/j.amc.2007.03.064

[20] G. Farin, “Curves and Surfaces for CAGO: A Practical Guide,” 1st Edition, Morgan Kaufmann, Waltham, 2001.

[21] G. Farin, “Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide,” 4th Edition, Academic Press, London, 1997.

[22] S. Mann, “A Blossoming Development of Spliness,” 1st Edition, Morgan Claypool, San Rafael, 2004.

[23] S. Biswa and B. Lovell, “Bezier and Splines in Image Processing and Machine Vision,” Springer-Verlag, Berlin, 2008.

[24] J. Zheng, T. Sedberg and R. Johansons, “Least Squares Methods for Solving Differential Equation Using Bezier Control Points,” Applied Numerical Mathematics, Vol. 48, No. 2, 2004, pp. 137-152. doi:10.1016/j.apnum.2002.01.001

[25] B. Egerstedt and F. Martin, “A Note on the Connection between Bezier Curves and Linear Optimal Control,” IEEE Transactions on Automatic Control, Vol. 49, No. 10, 2004, pp. 1728-1731. doi:10.1109/TAC.2004.835393

[26] M. Mahmoud and P. Shi, “Methodologies for Control of Jump Time-Delay Systems,” Kluwer Academic Publishers, London, 2004.