doi: 10.3934/dcdsb.2022052
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

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

*Corresponding author: Zhongqing Wang

Received  October 2021 Revised  January 2022 Early access March 2022

Fund Project: 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)

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.

Citation: Yuling Guo, Zhongqing Wang. A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022052
References:
[1]

K. BalachandranS. 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. BhrawyA. A. Al-ZahraniY. 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. ChenJ. 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. LiF. 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. LiuL. 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. NematiP. 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. RahimkhaniY. 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. ShiX. DingZ. 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. WangZ. 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. WangZ. 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. WangY. ChenD. 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. WangY. 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. WangC. ShengH. 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.

show all references

References:
[1]

K. BalachandranS. 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. BhrawyA. A. Al-ZahraniY. 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. ChenJ. 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. LiF. 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. LiuL. 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. NematiP. 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. RahimkhaniY. 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. ShiX. DingZ. 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. WangZ. 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. WangZ. 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. WangY. ChenD. 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. WangY. 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. WangC. ShengH. 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.

Figure 1.  A simple mesh
Figure 2.  The errors of Example 1
Figure 3.  The errors of Example 2
Figure 4.  The errors of Example 3
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
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
[1]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685

[2]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 339-357. doi: 10.3934/dcdss.2021025

[3]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402

[4]

Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797

[5]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

[6]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667

[7]

Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064

[8]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299

[9]

Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223

[10]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

[11]

Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257

[12]

Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080

[13]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088

[14]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[15]

Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053

[16]

Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100

[17]

Hamid Reza Marzban, Hamid Reza Tabrizidooz. Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1379-1389. doi: 10.3934/cpaa.2010.9.1379

[18]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002

[19]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029

[20]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure and Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (185)
  • HTML views (98)
  • Cited by (0)

Other articles
by authors

[Back to Top]