A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations

Yuling Guo; Zhongqing Wang

doi:10.3934/dcdsb.2022052

Article Contents

2022, Volume 27, Issue 12: 7521-7545. Doi: 10.3934/dcdsb.2022052

This issue Previous Article Existence of compact

$ \varphi $

-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems Next Article Inheritance of

$ {\mathscr F}- $

chaos and

$ {\mathscr F}- $

sensitivities under an iteration for non-autonomous discrete systems

A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations

Yuling Guo and
Zhongqing Wang^,

University of Shanghai for Science and Technology, Shanghai, 200093, China

^*Corresponding author: Zhongqing Wang
^*Corresponding author: Zhongqing Wang

Received: October 2021

Revised: January 2022

Early access: March 2022

Published: December 2022

The work is supported in part by National Natural Science Foundation of China (Grant Nos. 12101409 and 12071294), Shanghai Natural Science Foundation (No. 22ZR1443800) and China Postdoctoral Science Foundation (No. 2020M681345)

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Abstract

In this paper, we propose a multi-domain Chebyshev collocation method for the nonlinear fractional pantograph differential equations. We analyze the existence and uniqueness, and present the $ hp $-version error bounds under the $ L^2 $-norm and the $ L^\infty $-norm. Numerical experiments are included to illustrate the theoretical results.

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Mathematics Subject Classification: 65N35, 45B99, 34K10, 41A10.

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Figure 1. A simple mesh

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Figure 2. The errors of Example 1

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Figure 3. The errors of Example 2

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Figure 4. The errors of Example 3

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Table 1.

Degree of freedom	$ L^2 $ error		$ L^\infty $ error
	Method in [26]	Our method	Method in [26]	Our method
8	1.1927e-04	6.1516e-06	2.0572e-04	1.3595e-05
16	5.4557e-07	3.1165e-07	9.3294e-06	2.4136e-06
32	2.9693e-07	1.2661e-08	5.0821e-07	7.5616e-08
64	1.7394e-08	6.4313e-11	2.9749e-08	1.3804e-10

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References

[1]	K. Balachandran, S. Kiruthika and J. J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci., 33 (2013), 712-720. doi: 10.1016/S0252-9602(13)60032-6.
[2]	A. H. Bhrawy, A. A. Al-Zahrani, Y. A. Alhamed and D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations, Rom. J. Phys., 59 (2014), 646-657.
[3]	H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.
[4]	S. Chen, J. Shen and L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85 (2016), 1603-1638. doi: 10.1090/mcom3035.
[5]	K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lect. Notes Math., Springer, Berlin, 2004. doi: 10.1007/978-3-642-14574-2.
[6]	S. Esmaeili and M. Shamsi, A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3646-3654. doi: 10.1016/j.cnsns.2010.12.008.
[7]	R. M. Hafez and Y. H. Youssri, Legendre-collocation spectral solver for variable-order fractional functional differential equations, Comput. Methods Differ. Equ., 8 (2020), 99-110. doi: 10.22034/cmde.2019.9465.
[8]	A. A. Keller, Contribution of the delay differential equations to the complex economic macrodynamics, Wseas Trans. Syst., 9 (2010), 358-371.
[9]	N. Kopteva and M. Stynes, An efficient collocation method for a Caputo two-point boundary value problem, BIT, 55 (2015), 1105-1123. doi: 10.1007/s10543-014-0539-4.
[10]	V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337-3343. doi: 10.1016/j.na.2007.09.025.
[11]	C. Li, F. Zeng and F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15 (2012), 383-406. doi: 10.2478/s13540-012-0028-x.
[12]	D. Li and C. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simulation, 172 (2020), 244-257. doi: 10.1016/j.matcom.2019.12.004.
[13]	W. Liu, L. Wang and S. Xiang, A new spectral method using nonstandard singular basis functions for time-fractional differential equations, Commun. Appl. Math. Comput., 1 (2019), 207-230. doi: 10.1007/s42967-019-00012-1.
[14]	P. Mokhtary and F. Ghoreishi, The $L^2$-convergence of the Legendre spectral tau matrix formulation for nonlinear fractional integro-differential equations, Numer. Algor., 58 (2011), 475-496. doi: 10.1007/s11075-011-9465-6.
[15]	S. Nemati, P. Lima and S. Sedaghat, An effective numerical method for solving fractional pantograph differential equations using modification of hat functions, Appl. Num. Math., 131 (2018), 174-189. doi: 10.1016/j.apnum.2018.05.005.
[16]	P. Rahimkhani, Y. Ordokhania and E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309 (2017), 493-510. doi: 10.1016/j.cam.2016.06.005.
[17]	L. Shi, X. Ding, Z. Chen and Q. Ma, A new class of operational matrices method for solving fractional neutral pantograph differential equations, Adv. Diff. Eq., 2018 (2018), 1-17. doi: 10.1186/s13662-018-1536-8.
[18]	G. Szegö, Orthogonal Polynomials, 4th edition, AMS Coll. Publ. 23, Providence, 1975.
[19]	C. Wang, Z. Wang and H. Jia, An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels, J. Sci. Comput., 72 (2017), 647-678. doi: 10.1007/s10915-017-0373-3.
[20]	C. Wang, Z. Wang and L. Wang, A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput., 76 (2018), 166-188. doi: 10.1007/s10915-017-0616-3.
[21]	L. Wang, Y. Chen, D. Liu and D. Boutat, Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials, Int. J. Comput. Math., 96 (2019), 2487-2510. doi: 10.1080/00207160.2019.1573992.
[22]	Z. Wang, Y. Guo and L. Yi, An $hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comp., 86 (2017), 2285-2324. doi: 10.1090/mcom/3183.
[23]	Z. Wang and C. Sheng, An $hp$-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays, Math. Comp., 85 (2016), 635-666. doi: 10.1090/mcom/3023.
[24]	Z. Wang, C. Sheng, H. Jia and D. Li, A Chebyshev spectral collocation method for nonlinear Volterra integral equations with vanishing delays, East Asian J. Appl. Math., 8 (2018), 233-260. doi: 10.4208/eajam.130416.071217a.
[25]	J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.
[26]	C. Yang and J. Hou, Jacobi spectral approximation for boundary value problems of nonlinear fractional pantograph differential equations, Numer. Algorithms, 86 (2021), 1089-1108. doi: 10.1007/s11075-020-00924-7.
[27]	Y. Yang and Y. Huang, Spectral-collocation methods for fractional pantograph delay-integrodifferential equations, Adv. Math. Phys., 2013 (2013), Art. ID 821327, 14 pp. doi: 10.1155/2013/821327.