Document Type : Research Articles

Authors

1 Faculty of Mathematics, University of Gonabad, Gonabad, Iran

2 Faculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran

Abstract

Due to the easy adaption of radial basis functions (RBFs), a direct
RBF collocation method is considered to develop an approximate scheme to solve
fractional delay differential equations (FDDEs). The method of RBFs is a method of scattered data interpolation that has many application in different fields. In spite of easy implementation of other high-order methods and finite difference schemes for solving a problem of fractional order derivatives, the challenge of these methods is their limited accuracy, locality, complexity and high cost of computing in discretization of the fractional terms, which suggest that global scheme such as RBFs that are more accurate way for discretizing fractional calculus and would allow us to remove the ill-conditioning of the system of discrete equations. Applications to a variety of
problems confirm that the proposed method is slightly more efficient than those
introduced in other literature and the convergence rate of our approach is high.

Keywords

Main Subjects

[1] H. L. Smith, “An introduction to delay differential
equations with applications to the life sciences,” Vol. 57,
New York, Springer. 2011.

[2] G.A. Walter, H.I. Freedman, J. Wu , “Analysis of a
model representing stage-structured population growth with
state-dependent time delay,”
SIAM Journal Appl Math,
Vol.
52, No.3,: pp. 855-869 , 1992.
[3] Y. Kuang, Delay differential equations: with applications
in population dynamics”,
Academic Press, New York, 1993.
[4] M. Malek-Zavarei, M .Jamshidi, “Time-delay systems:
analysis optimization and applications”,
Elsevier Science
Ltd
, New York, 1987.
[5] P. Muthukumar, and B. Ganesh Priya, Numerical
solution of fractional delay differential equation by shifted
Jacobi polynomials,”
International Journal of Computer
Mathematics
, Vol. 94, No. 3, pp.471-492, 2017.
[6] V. Daftardar-Gejji, Y.Sukale, and S. Bhalekar, Solving
fractional delay differential equations: a new approach,”

Fractional Calculus and Applied Analysis,
Vol. 18, No. 2,
pp.
400-418, 2015.
[7] D. Baleanu, R. L.Magin, S. Bhalekar, and V.
Daftardar-Gejji, Chaos in the fractional order nonlinear
Bloch equation with delay,”
Communications in Nonlinear
Science and Numerical Simulation
, Vol. 25, No. 1-3,
pp.
41-49, 2015.
[8] C. Ravichandran, and D. Baleanu, Existence results for
fractional neutral functional integro-differential evolution
equations with infinite delay in Banach spaces,”
Advances
in Difference Equations
, Vol. 1 , pp.215, 2013.
[9] E. Kaslik, and S. Sivasundaram, Analytical and numerical
methods for the stability analysis of linear fractional delay
differential equations,”
Journal of Computational and
Applied Mathematics
, Vol. 236, No.16, pp. 4027-4041,
2012
.
[10] J. Cermak, Z. Dosla, and T. Kisela, Fractional differential
equations with a constant delay: stability and asymptotics of
solutions,”
Applied Mathematics and Computation, Vol.298,
pp.
336-350, 2017.
[11] A. Saadatmandi, and M. Dehghan, “A new operational
matrix for solving fractional-order differential
equations,”Computers & mathematics with applications,
Vol.59, No.3, pp.1326-1336. , 2010.

[12] M.L. Morgado, N.J. Ford, and P.M. Lima, Analysis and
numerical methods for fractional differential equations with
delay,” Journal of computational and applied mathematics,
Vol.252, pp.159-168, 2013.
[13] U. Saeed, M. ur Rehman, and M.A Iqbal, Modified
Chebyshev wavelet methods for fractional delay-type
equations,” Applied Mathematics and Computation, 264,
431-442. 2015

[14] M. Dehghan, and R. Salehi, “ Solution of a nonlinear
time-delay model in biology via semi-analytical approaches,”
Computer Physics Communications, Vol.181, No.7,
pp.1255-1265, 2010.

[15] A. Debbouche, and D.F. Torres, Approximate
controllability of fractional delay dynamic inclusions with
nonlocal control conditions,” Applied Mathematics and
Computation, Vol.243, pp.161-175, 2014.

[16] B.P. Mohaddam, Z.S Mostaghim, “ A numerical method
based on finite difference for solving fractional delay
differential equations,” Journal of Taibah University of
Science7, pp.120-127, 2013.

[17] Z. Wang, “A numerical method for delayed fractional-order
differential equations,” Journal of Applied Mathematics,
2013.

[18] Z. Wang, X. Huang, and J. Zhou, “A numerical method for
delayed fractional-order differential equations: based on GL
definition,” Appl. Math. Inf. Sci, Vol. 7, No.2, pp.525-529,
2013.

[19] R. K. Pandey, N. Kumar, and R. N. Mohaptra, “An
approximate method for solving fractional delay differential
equations,” International Journal of Applied and
Computational Mathematics, Vol. 3, No. 2, pp.1395-1405,
2017.

[20] F. Mohammadi, Numerical solution of systems of
fractional delay differential equations using a new kind of
wavelet basis,” Computational and Applied Mathematics,
pp.1-23, 2018.

[21] M. Li, and J. Wang, “Finite time stability of fractional delay
differential equations,” Applied Mathematics Letters, Vol.
64, pp.170-176, 2017.

[22] S. S. Zeid, Approximation methods for solving fractional
equations,” Chaos, Solitons & Fractals, Vol. 125,
pp.171-193, 2019.

[23] D. Baleanu, J. A. T. Machado, and A. C. Luo,
(Eds.),”Fractional dynamics and control. Springer Science
and Business Media,” 2011.

[24] C.A Monje, B.M Vinagre, V. Feliu, and Y Chen, Tuning
and auto-tuning of fractional order controllers for industry
applications,” Control engineering practice, Vol. 16, No.7,
pp.798-812, 2008.

[25] R. L. Bagley and R.A Calico, “ Fractional order state
equations for the control of viscoelastically damped
structures,” Journal of Guidance, Control, and Dynamics,
Vol.14, No.2, pp. 304-311, 1991.

[26] D. Xue, and Y. Chen, “ A comparative introduction of
four fractional order controllers, In Proceedings of the 4th
World Congress on Intelligent Control and Automation
IEEE2002 , Vol. 4, Cat. No. 02EX527, pp. 3228-3235, June,
2001.

[27] B.M Vinagre, C.A Monje, A.J Calderón, and J.I. Suarez,
Fractional PID controllers for industry application. A brief
introduction,” Journal of Vibration and Control, Vol.13, No.
9-10, pp.1419-1429, 2007.

[28] J. Nakagawa, K. Sakamoto and M.Yamamoto, “Overview
to mathematical analysis for fractional diffusion
equations-new mathematical aspects motivated by industrial
collaboration,” Journal of Math-for-Industry, Vol. 2, No.
10, pp.99-108, 2010.

[29] A. Razminia, and D. Baleanu, Fractional order models of
industrial pneumatic controllers,” In Abstract and Applied
Analysis Hindawi ,Vol. 2014, 2014.

[30] M. O. Efe, Fractional order systems in industrial
automation-a survey,” IEEE Transactions on Industrial
Informatics, Vol.7, No.4, pp.582-591, 2011.

[31] Q. Liu, S.Mu, B. Bi, X. Zhuang, and J. Gao, An RBF
based meshless method for the distributed order time
fractional advection-diffusion equation,” Engineering
Analysis with Boundary Elements, Vol. 96, pp.55-63, 2018.

[32] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, An
implicit RBF meshless approach for solving the time
fractional nonlinear sine-Gordon and Klein-Gordon
equations,” Engineering Analysis with Boundary Elements,
Vol. 50, pp.412-434, 2015.

[33] N. Ahmadi, A.R Vahidi, and T. Allahviranloo, An efficient
approach based on radial basis functions for solving
stochastic fractional differential equations,” Mathematical
Sciences, Vol. 11, No. 2, pp.113-118, 2017.

[34] S. Wei, W. Chen, Y. Zhang, H. Wei, and R. M. Garrard, A
local radial basis function collocation method to solve the
variable-order time fractional diffusion equation in a
two-dimensional irregular domain,” Numerical Methods for
Partial Differential Equations, Vol. 34, No. 4, pp.
1209-1223, 2018.

[35] M. Dehghanand A. Shokri, A numerical method for KdV
equation using collocation and radial basis functions,”
Nonlinear Dynamics, Vol. 50, No. 1-2, pp.111-120, 2007.

[36] I. Dag, Y. Dereli , “ Numerical solutions of KdV equation
using radial basis functions,” Appl Math Model 2008, Vol.
32, pp. 535-46, 2008.

[37] B. Sarler, R. Vertnik, and G. Kosec, Radial basis function
collocation method for the numerical solution of the
two-dimensional transient nonlinear coupled Burgers
equations,” Applied Mathematical Modelling, Vol. 36, No.
3, pp. 1148-1160, 2012.

[38] M. Kumar, and N. Yadav, Multilayer perceptrons and
radial basis function neural network methods for the
solution of differential equations: a survey,”Computers and
Mathematics with Applications, Vol. 62, No. 10,
pp.3796-3811, 2011.

[39] B. F. Kazemi, and F. Ghoreishi, “ Error estimate in
fractional differential equations using multiquadratic radial
basis functions,” Journal of Computational and Applied
Mathematics, Vol. 245, pp. 133-147, 2013.

[40] M. Dehghan, M. Tatari , Determination of control
parameter in a one-dimensional parabolic equation using the
method of radial basis functions,” Math Comput Model, Vol.
44, pp. 1160-1168, 2006.
[41] C. Franke, R. Schaback , Convergence order estimates of
meshless collocation methods using radial basis functions,”
Adv Comput Math, Vol. 8, pp. 381-99, 1998.

[42] A. Khalid, J. Huey, W.Singhose, J. Lawrence, and D.
Frakes, Human operator performance testing using an
input-shaped bridge crane,”
Journal of dynamic systems,
measurement, and control
, Vol. 128, No. 4, pp.835-841,
2006.

[43] S. G. Lee, Sliding mode controls of double-pendulum
crane systems,”
Journal of Mechanical Science and
Technology
, Vol. 27, No. 6, pp.1863-1873, 2013.
[44] M. R. Dastranj, Design optimal Fractional PID Controller
for Inverted Pendulum with Genetic Algorithm,”
Majlesi
Journal of Mechatronic Syste
ms, Vol. 2, No.2, 2013.
[45] R. Schaback, MATLAB Programming for Kernel-Based
Methods, Preprint Gottingen, 2009.

[46] H. Wendland, Scattered data approximation Cambridge
university press, Vol. 17, 2004.

[47] G. E. Fasshauer, Meshfree methods,” Handbook of
theoretical and computational nanotechnology, Vol. 27, pp.
33-97, 2005.

[48] M. Mohammadi, and R. Schaback, On the fractional
derivatives of radial basis functions,”arXiv preprint arXiv,
1612.07563, 2016.

[49] R. Schaback, Native Hilbert spaces for radial basis
functions I. In New Developments in Approximation
Theory,” Birkhäuser, Basel. pp. 255-282, 1999.

[50] S. Hosseinpour, A. Nazemi, and E. Tohidi, A New
Approach for Solving a Class of Delay Fractional Partial
Differential Equation,” Mediterranean Journal of
Mathematics, Vol. 15, No. 6, pp.218, 2018.

[51] B.P. Moghaddam, and Z. S. A. Mostaghim, “Numerical
method based on finite difference for solving fractional
delay differential equations,” Journal of Taibah University
for Science, Vol. 7, No. 3, pp.120-127, 2013.

[52] Xu, Min-Qiang, and Ying-Zhen Lin. Simplified reproducing
kernel method for fractional differential equations with
delay. Applied Mathematics Letters 52 (2016): 156-161.