A Meshless Method for Solving Delay Differential Equation Using Radial Basis Functions with Error Analysis

Volume 11, Issue 2, pp 105-110 http://dx.doi.org/10.22436/jmcs.011.02.03

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2891 Downloads
• 4776 Views

Authors

F. Ghomanjani - Young Researchers and Elite Club,Mashhad Branch,Islamic Azad University,Mashhad,Iran. F. Akhavan Ghassabzade - Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

Abstract

In this paper, we present a numerical method for solving delay differential equations (DDEs). The method utilizes radial basis functions (RBFs). Error analysis is presented for this method. Finally, numerical examples are included to show the validity and efficiency of the new technique for solving DDEs.

Share and Cite

Keywords

• Radial basis functions (RBFs)
• Delay differential equations
• time delay systems.

MSC

•  65F10
•  49M37

References

• [1] J. V. Beltran, J. Monterde, Bezier solutions of the wave equation, Lecture notes in Computational Sciences, Computational Science and its Applications-ICCSA (A. Laganà, M. L. Gavrilova, V. Kumar, Y. Mun, C. J. K.Tan & O. Gervasi eds). Lecture Notes in Computer Science, Berlin: Springer, 3044 (2004), 631–640
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] A. Bemporad, A. Giua, C. Seatzu, Synthesis of state-feedback optimal controllers for continuous-time switched linear systems, In Proc. 41st IEEE conference on decision and control, Las Vegas, Nevada USA (2002)
  • View Article
  • Google Scholar


• [3] R. Cholewa, A. J. Nowak, R. A. Bialecki, L. C. Wrobel, Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems, Computational Mechanics, 28 (2002), 282-290
  • View Article
  • Google Scholar
  • MATH


• [4] C. H. Chu, C. C. L. Wang, C. R. Tsai, Computer aided geometric design of strip using developable Bezier patches, Computers in Industry, 59 (2008), 601-611
  • View Article
  • Google Scholar


• [5] M. Egerstedt, Y. Wardi, F. Delmotte, Optimal control of switching times in switched dynamical systems, In Proc. IEEE conference on decision and control, 3 (2003), 2138-2143
  • View Article
  • Google Scholar


• [6] M. Evrenosoglu, S. Somali, Least squares methods for solving singularity perturbed two-points boundary value problems using Bezier control point, Applied Mathematics Letters, 21 (2008), 1029-1032
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] G. E. Farin, Curve and surfaces for computer aided geometric design, First ed, Academic Press, New York (1988)
  • Google Scholar


• [8] B. Farhadinia, K. L. Teo, R. C. Loxton, A computational method for a class of non-standard time optimal control problems involving multiple time horizons, Mathematical Computer Modelling, 49 (2009), 1682-1691
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] M. Gachpazan, Solving of time varying quadratic optimal control problems by using Bezier control points, Computational and Applied Mathematics, 30 (2011), 367-379
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] J. Geromela, P. Colaneri, P. Bolzern, Passivity of switched linear systems: Analysis and control design, Systems and Control Letters, 61 (2012), 549-554
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] F. Ghomanjani, M. H. Farahi, The Bezier control points method for solving delay differential equation, Intelligent Control and Automation, 3 (2012), 188-196
  • View Article
  • Google Scholar


• [12] F. Ghomanjani, M. H. Farahi, M. Gachpazan, Bezier control points method to solve constrained quadratic optimal control of time varying linear systems, Computational and Applied Mathematics, 31 (2012), 433-456
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] A. Giua, C. Seatzu, C. Van Der Me, Optimal conrol of switched autonomous linear systems, In Proc. IEEE conference on decision and control, (2001), 2472-2477
  • View Article
  • Google Scholar


• [14] K. Harada, E. Nakamae, Application of the Bezier curve to data interpolation, Computer-Aided Design, International Journal of Computer Mathematics, 14 (1982), 55-59
  • View Article
  • Google Scholar


• [15] M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems, Applied Mathematics and Computation, 173 (2005), 169-198
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] I. Hwang, J. Li, D. Du, A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach, International Journal of Control, 81 (2008), 277-293
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] H. Juddu, Spectral method for constrained linear-quadratic optimal control, Mathematics Computers In simulation, 58 (2002), 159-169
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] B. Lang, The synthesis of wave forms using Bezier curves with control point modulation, In: The Second CEMS Research Student Conference, 1st edn. Morgan kaufamann, San Francisco (2004)
  • Google Scholar


• [19] A. T. Layton, M. Van de Panne, A numerically evident and stable algorithm for animating water waves, The visual Computer, 18 (2002), 41-53
  • View Article
  • Google Scholar


• [20] M. Kamgarpour, C. Tomlin, On optimal control of non-autonomous switched systems with a fixed mode sequence, Automatica, Doi:10.1016/j.automatica, 1177-1181 (2012)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] M. Margaliot, Stability analysis of switched systems using variational principles: An introduction, Automatica, 42 (2006), 2059-2077
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] B. Niua, J. Zhaoa, Stabilization and \(L_2\)-gain analysis for a class of cascade switched nonlinear systems: An average dwell-time method, Nonlinear Analysis: Hybrid Systems, 5 (2001), 671-680
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] G. Nürnberger, F. Zeilfelder, Developments in bivariate spline interpolation, Journal of Computational and Applied Mathematics, Computers and Mathematics with Applications, 121 (2000), 125-152
  • View Article
  • MathSciNet
  • Google Scholar


• [24] H. Prautzsch, W. Boehm, M. Paluszny , Bezier and B-Spline Techniques, Springer, (2001)
  • MathSciNet
  • Google Scholar


• [25] Y. Q. Shi, H. Sun, Image and video compression for multimedia engineering, CRC Press LLc, (2000)
  • Google Scholar


• [26] R. Winkel, Generalized Bernstein Polynomials and Bezier Curves: An Application of Umbral Calculus to Computer Aided Geometric Design, Advances in Applied Mathematics, 27 (2001), 51-81
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] C. Wu, K. L. Teo, R. Li, Y. Zhao, Optimal control of switched systems with time delay, Appl, Math. Letters, 19 (2006), 1062-1067
  • View Article
  • Google Scholar


• [28] X. Xu, G. Zhai, S. He, Some results on practical stabilizability of discrete-time switched affine systems, Nonlinear Analysis: Hybrid Systems, 4 (2010), 113-121
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] X. Xu, P. J. Antsakalis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] J. Zheng, T. W. Sederberg, R. W. Johnson, Least squares methods for solving differential equations using Bezier control points, Applied Numerical Mathematics, 48 (2004), 237-252
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH