An $ hp $-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

Chuanli Wang; Biyun Chen; Chuanli Wang; Biyun Chen

doi:10.3934/math.20231010

AIMS Mathematics

2023, Volume 8, Issue 8: 19816-19841. doi: 10.3934/math.20231010

Previous Article Next Article

Research article

An $ hp $-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

Chuanli Wang ^{1
,
,},
Biyun Chen ²

1.
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224007, China
2.
School of Information Engineering, Yancheng Teachers University, Yancheng 224007, China

Received: 07 February 2023 Revised: 30 May 2023 Accepted: 05 June 2023 Published: 13 June 2023
MSC : 41A05, 41A10, 41A25, 45D05, 65N35

We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $ hp $-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented.
- spectral collocation method,
- Caputo fractional derivative,
- fractional integro-differential equations,
- Volterra integral equations
Citation: Chuanli Wang, Biyun Chen. An $ hp $-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels[J]. AIMS Mathematics, 2023, 8(8): 19816-19841. doi: 10.3934/math.20231010

Related Papers:

Abstract

We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $ hp $-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented.

References

[1]	F. Mainardi, Fractional calculus and waves in linear viscoleasticity, Word Scientific, 2010. http://doi.org/10.1142/p614
[2]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 1999. https://doi.org/10.1142/3779
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. https://doi.org/10.1016/s0304-0208(06)x8001-5
[4]	I. Podlubny, Fractional differential equations, Academic Process, 1999.
[5]	K. Diethelm, The analysis of fractional differential equations, Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[6]	M. M. Khader, N. H. Sweilam, On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method, Appl. Math. Modell., 37 (2013), 9819–9828. https://doi.org/10.1016/j.apm.2013.06.010 doi: 10.1016/j.apm.2013.06.010
[7]	J. Zhao, J. Xiao, N. J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms, 65 (2014), 723–743. https://doi.org/10.1007/s11075-013-9710-2 doi: 10.1007/s11075-013-9710-2
[8]	C. Wang, Z. Wang, L. Wang, A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput., 76 (2018), 166–188. https://doi.org/10.1007/s10915-017-0616-3 doi: 10.1007/s10915-017-0616-3
[9]	H. Dehestani, Y. Ordokhani, M. Razzaghi, Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations, Eng. Comput., 37 (2020), 1791–1806. https://doi.org/10.1007/s00366-019-00912-z doi: 10.1007/s00366-019-00912-z
[10]	K. Sadri, K. Hosseini, D. Baleanu, S. Salahshour, A high-accuracy Vieta-Fibonacci collocation scheme to solve linear time-fractional telegraph equations, Waves Random Complex Media, https://doi.org/10.1080/17455030.2022.2135789 doi: 10.1080/17455030.2022.2135789
[11]	K. Sadri, K. Hosseini, E. Hincal, D. Baleanu, S. Salahshour, A pseudo-operational collocation method for variable-order time-space fractional KdV-Burgers-Kuramoto equation, Math. Meth. Appl. Sci., 46 (2023), 8759–8778. https://doi.org/10.1002/mma.9015 doi: 10.1002/mma.9015
[12]	H. Dehestania, Y. Ordokhania, M. Razzaghib, Numerical solution of Variable-order time fractional weakly singular partial integro-difffferential equations with error estimation, Math. Modell. Anal., 25 (2020), 680–701. https://doi.org/10.3846/mma.2020.11692 doi: 10.3846/mma.2020.11692
[13]	H. Cai, A fractional spectral collocation for solving second kind nonlinear Volterra integral equations with weakly singular kernels, J. Sci. Comput., 80 (2019), 1529–1548. https://doi.org/10.1007/s10915-019-00987-2 doi: 10.1007/s10915-019-00987-2
[14]	S. Chen, J. Shen, L. L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85 (2016), 1603–1638. https://doi.org/10.1090/mcom3035 doi: 10.1090/mcom3035
[15]	X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[16]	S. Esmaeili, 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. https://doi.org/10.1016/j.cnsns.2010.12.008 doi: 10.1016/j.cnsns.2010.12.008
[17]	P. Mokhtary, F. Ghoreishi, The $L^2$-convergence of the Legendre spectral tau matrix formulation for nonlinear fractional integro-differential equations, Numer. Algorithms, 58 (2011), 475–496. https://doi.org/10.1007/s11075-011-9465-6 doi: 10.1007/s11075-011-9465-6
[18]	C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calculus Appl. Anal., 15 (2012), 383–406. https://doi.org/10.2478/s13540-012-0028-x doi: 10.2478/s13540-012-0028-x
[19]	C. T. Sheng, Z. Q. Wang, B. Y. Guo, A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal., 52 (2014), 1953–1980. https://doi.org/10.1137/130915200 doi: 10.1137/130915200
[20]	C. L. Wang, Z. Q. Wang, H. L. Jia, An $hp$-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels, J. Sci. Comput., 72 (2017), 647–678. https://doi.org/10.1007/s10915-017-0373-3 doi: 10.1007/s10915-017-0373-3
[21]	Z. Wang, Y. Guo, L. Yi, An $hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comput., 86 (2017), 2285–2324. https://doi.org/10.1090/mcom/3183 doi: 10.1090/mcom/3183
[22]	Z. Wang, C. Sheng, An $hp$-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays, Math. Comput., 85 (2016), 635–666. https://doi.org/10.1090/mcom/3023 doi: 10.1090/mcom/3023
[23]	Y. Guo, Z. Wang, An $hp$-version Chebyshev collocation method for nonlinear fractional differential equations, Appl. Numer. Math., 158 (2020), 194–211. https://doi.org/10.1016/j.apnum.2020.08.003 doi: 10.1016/j.apnum.2020.08.003
[24]	Y. Guo, Z. Wang, An $hp$-version Legendre spectral collocation method for multi-order fractional differential equations, Adv. Comput. Math., 47 (2021), 37. https://doi.org/10.1007/s10444-021-09858-7 doi: 10.1007/s10444-021-09858-7
[25]	G. Yao, D. Tao, C. Zhang, A hybrid spectal method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays, Appl. Math. Comput., 417 (2022), 126780. https://doi.org/10.1016/j.amc.2021.126780 doi: 10.1016/j.amc.2021.126780
[26]	H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge University Press, 2004. https://doi.org/10.1017/cbo9780511543234

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)