Few-Body Systems

, 58:156

# Stable Numerical Approach for Fractional Delay Differential Equations

• Harendra Singh
• Rajesh K. Pandey
• D. Baleanu
Article

## Abstract

In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.

## References

1. 1.
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences (Springer, New York, 2011)
2. 2.
V. Lakshmikantham, S. Leela, Differential and Integral Inequalities (Academic Press, New York, 1969)
3. 3.
W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139 (1990)
4. 4.
A.R. Davis, A. Karageorghis, T.N. Phillips, Spectral Galerkin methods for the primary two-point bour problem in modelling viscoelastic flows. Int. J. Numer. Methods Eng. 26, 647 (1988)
5. 5.
S.A. Gourley, Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49, 188 (2004)
6. 6.
J. Li, Y. Kuang, C. Mason, Modelling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays. J. Theor. Biol. 242, 722 (2006)
7. 7.
A.D. Robinson, The use of control systems analysis in neurophysiology of eye movements. Ann. Rev. Neurosci. 4, 462 (1981)
8. 8.
R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201 (1983)
9. 9.
R.L. Bagley, P.J. Torvik, Fractional calculus a differential approach to the analysis of viscoelasticity damped structures. AIAA J. 21, 741 (1983)
10. 10.
R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelasticity damped structures. AIAA J. 23, 918 (1985)
11. 11.
R. Panda, M. Dash, Fractional generalized splines and signal processing. Signal Process 86, 2340 (2006)
12. 12.
R.L. Magin, Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1 (2004)
13. 13.
G.W. Bohannan, Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14, 1487 (2008)
14. 14.
C.E. Falbo, Some Elementary Methods for Solving Functional Differential Equations. http://www.mathfile.net/hicstatFDE.pdf
15. 15.
X. Lv, Y. Gao, The RKHSM for solving neutral functional differential equations with proportional delays. Math. Methods Appl. Sci. 36, 642 (2013)
16. 16.
W. Wang, Y. Zhang, S. Li, Stability of continuous Runge–Kutta type methods for nonlinear neutral delay-differential equations. Appl. Math. Model. Simul. Comput. Eng. Environ. Syst. 33(8), 3319 (2009)
17. 17.
U. Saeed, M. ur Rehman, Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. Article ID 359093, 8 (2014)Google Scholar
18. 18.
M.A. Iqbal, A. Ali, S.T. Mohyud-Din, Chebyshev wavelets method for fractional delay differential equations. Int. J. Mod. Appl. Phys. 4(1), 49 (2013)Google Scholar
19. 19.
D.J. Evans, K.R. Raslan, The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82(1), 49 (2005)
20. 20.
B.P. Mohaddam, Z.S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations. J. Taibah Univ. Sci. 7, 120 (2013)
21. 21.
Z. Wang, A numerical method for delayed fractional-order differential equations. J. Appl. Math. Article ID 256071, 7 (2013)Google Scholar
22. 22.
Z. Wang, X. Huang, J. Zhou, A numerical method for delayed fractional-order differential equations: based on GL definition. Appl. Math. 7(2), 525 (2013)
23. 23.
R.K. Pandey, N. Kumar, N. Mohaptra, An approximate method for solving fractional delay differential equations. Int. J. Appl. Comput. Math. (2016). doi: Google Scholar
24. 24.
M.A. Iqbal, U. Saeed, S.T. Mohyud-Din, Modified Laguerre wavelets method for delay differential equation of fractional-order. Egypt. J. Basic Appl. Sci. 2, 50 (2015)
25. 25.
E.H. Doha, A.H. Bhrawy, D. Baleanu, S.S. Ezz-Eldien, The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation. Adv. Differ. Equ. (2014). http://www.advancesindifferenceequations.com/content/2014/1/231
26. 26.
A. Ahmadian, M. Suleiman, S. Salahshour, D. Baleanu, A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv. Differ. Equ. (2014). http://www.advancesindifferenceequations.com/content/2013/1/104
27. 27.
A.H. Bhrawy, M.M. Tharwat, M.A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. (2) 37(4), 983 (2014)
28. 28.
S. Kazem, An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl. Math. Model. 37, 1126 (2013)
29. 29.
R.K. Pandey, S. Suman, K.K. Singh, O.P. Singh, Approximate solution of Abel inversion using Chebshev polynomials. Appl. Math. Comput. 237, 120 (2014)

## Authors and Affiliations

• Harendra Singh
• 1
• Rajesh K. Pandey
• 1
• D. Baleanu
• 1
• 2
• 3
1. 1.Department of Mathematical Sciences, Indian Institute of TechnologyBanaras Hindu UniversityVaranasiIndia
2. 2.Department of MathematicsCankaya UniversityAnkaraTurkey
3. 3.Institute of Space SciencesMăgurele, BucharestRomania