# Characteristic roots of a second order retarded functional differential equation via spectral-tau method

Volume 13, Issue 3, pp 147--153
Publication Date: November 15, 2019 Submission Date: July 28, 2019 Revision Date: September 17, 2019 Accteptance Date: October 29, 2019
• 254 Downloads
• 514 Views

### Authors

Habeeb Kareem Abdullah - Department of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, Iraq. Amal Khalaf Haydar - Department of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, Iraq. Kawther Reyadh Obead - Department of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, Iraq.

### Abstract

In this paper, we have found the solution of second-order delay differential equations of retarded type with multiple delays. As well as developing an approximation for finding characteristic roots for such delay differential equations via the method of spectral tau which depends on the basis mixed Fourier basis or shifted Chebyshev polynomials.

### Share and Cite

##### ISRP Style

Habeeb Kareem Abdullah, Amal Khalaf Haydar, Kawther Reyadh Obead, Characteristic roots of a second order retarded functional differential equation via spectral-tau method, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 3, 147--153

##### AMA Style

Abdullah Habeeb Kareem, Haydar Amal Khalaf, Obead Kawther Reyadh, Characteristic roots of a second order retarded functional differential equation via spectral-tau method. J. Nonlinear Sci. Appl. (2020); 13(3):147--153

##### Chicago/Turabian Style

Abdullah, Habeeb Kareem, Haydar, Amal Khalaf, Obead, Kawther Reyadh. "Characteristic roots of a second order retarded functional differential equation via spectral-tau method." Journal of Nonlinear Sciences and Applications, 13, no. 3 (2020): 147--153

### Keywords

• Linear functional-differential equations
• IBVPs for linear higher-order equations
• spectral theory of functional-differential operators

•  34K06
•  34K08

### References

• [1] H. K. Abdullah, A. K. Haydar, K. R. Obead, Solving of third order retarded dynamical system via lambert W function and stability analysis, World Wide J. Multidisciplin. Res. Develop., 4 (2018), 105--112

• [2] H. K. Abdullah, A. K. Haydar, K. R. Obead, Solving retarded dynamical system of nth- order and stability analysis via lambert W function, Int. J. Pure Appl. Math., 118 (2018), 2567--2584

• [3] O. Arino, M. L. Hbid, E. A. Dads, Delay differential equations and application, Springer, The Nethelands (2006)

• [4] A. Bellen, S. Maset, Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems, Numer. Math., 84 (2000), 351--374

• [5] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publ. Inc., Mineola (2001)

• [6] S.-T. Chen, S.-P. Hsu, H.-N. Huang, B.-Y. Yang, Time response of a scalar dynamical system with multiple delays via Lambert W functions, arXiv, 2016 (2016), 24 pages

• [7] L. E. Èlʹsgolʹts, S. B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press, New York-London (1973)

• [8] T. Koto, Method of lines approximations of delay differential equations, Comput. Math. Appl., 48 (2004), 45--59

• [9] M. S. Lee, C. S. Hsu, On the $\tau$-decomposition method of stability analysis for retarded dynamical systems, SIAM J. Control, 7 (1969), 242--259

• [10] T. X. Li, B. Baculíková, J. Džurina, C. H. Zhang, Oscillation of fourth-order neutral differential equations with $p$--Laplacian like operators, Bound. Value Probl., 2014 (2014), 9 pages

• [11] Q. Li, R. Wang, F. Chen, T. X. Li, Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients, Adv. Difference Equ., 2015 (2015), 7 pages

• [12] I. Mezo, G. Keady, Some physical applications of generalized Lambert functions, Eur. J. Phys., 37 (2016), 10 pages

• [13] K. Schmitt, Delay and functional differential equations and their applications, Academic Press, New York-London (1972)

• [14] S. S. Sujitha, D. Piriadarshani, A Lambert W function a approach for solution of second order delay differential equation as a special case of the one-mass system controlled over the network, Int. J. Mech. Eng. Tech., 8 (2017), 502--511

• [15] C. P. Vyasarayani, S. Subhash, T. Kalmar-Nagy, Spectral approximations for characteristic roots of delay differential equations, Int. J. Dynam. Control., 2 (2014), 126--132

• [16] P. Wahi, A. Chatterjee, Galerkin projections for delay differential equations, J. Dyn. Syst. Meas. Control., 127 (2005), 80--87