Close Search Advanced
Journals
Resources
About Us
Open Access

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels
Preview
Add to basket

Year:    2016

Author:    Xiulian Shi, Yanping Chen

Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 4 : pp. 648–669

Abstract

A spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Finally, two numerical examples are presented to demonstrate our error analysis.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.2015.m1088

Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 4 : pp. 648–669

Published online:    2016-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    Volterra integro-differential equations spectral Jacobi-collocation method pantograph delay weakly singular kernel.

Author Details

Xiulian Shi

Yanping Chen

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

    Ma, Zheng | Huang, Chengming

    Numerical Algorithms, Vol. 92 (2023), Iss. 4 P.2377

    https://doi.org/10.1007/s11075-022-01394-9 [Citations: 4]

  2. Spectral methods to solve nonlinear problems: A review

    Rai, Nischay | Mondal, Sabyasachi

    Partial Differential Equations in Applied Mathematics, Vol. 4 (2021), Iss. P.100043

    https://doi.org/10.1016/j.padiff.2021.100043 [Citations: 22]

  3. A new Tau-collocation method with fractional basis for solving weakly singular delay Volterra integro-differential equations

    Azizipour, G. | Shahmorad, S.

    Journal of Applied Mathematics and Computing, Vol. 68 (2022), Iss. 4 P.2435

    https://doi.org/10.1007/s12190-021-01626-6 [Citations: 10]

  4. An hp-version error estimate of spectral collocation methods for weakly singular Volterra integro-differential equations with vanishing delays

    Qin, Yu | Huang, Chengming

    Computational and Applied Mathematics, Vol. 43 (2024), Iss. 5

    https://doi.org/10.1007/s40314-024-02818-z [Citations: 0]

  5. Jacobi collocation method and smoothing transformation for numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations

    Rezazadeh, Tohid | Najafi, Esmaeil

    Applied Numerical Mathematics, Vol. 181 (2022), Iss. P.135

    https://doi.org/10.1016/j.apnum.2022.05.019 [Citations: 6]

  6. Convergence analysis of the spectral collocation methods for two-dimensional nonlinear weakly singular Volterra integral equations*

    Shi, Xiulian | Wei, Yunxia

    Lithuanian Mathematical Journal, Vol. 58 (2018), Iss. 1 P.75

    https://doi.org/10.1007/s10986-018-9387-2 [Citations: 8]

  7. Spectral Collocation Methods for Fractional Integro‐Differential Equations with Weakly Singular Kernels

    Shi, Xiulian | Zhou, Ding-Xuan

    Journal of Mathematics, Vol. 2022 (2022), Iss. 1

    https://doi.org/10.1155/2022/3767559 [Citations: 1]

  8. Convergence analysis of spectral methods for high-order nonlinear Volterra integro-differential equations

    Shi, Xiulian | Huang, Fenglin | Hu, Hanzhang

    Computational and Applied Mathematics, Vol. 38 (2019), Iss. 2

    https://doi.org/10.1007/s40314-019-0827-3 [Citations: 4]

  9. An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

    Wang, Lina | Yi, Lijun

    Advances in Computational Mathematics, Vol. 43 (2017), Iss. 6 P.1437

    https://doi.org/10.1007/s10444-017-9531-2 [Citations: 6]

  10. Spectral collocation methods for fractional multipantograph delay differential equations*

    Shi, Xiulian | Wang, Keyan | Sun, Hui

    Lithuanian Mathematical Journal, Vol. 63 (2023), Iss. 4 P.505

    https://doi.org/10.1007/s10986-023-09614-y [Citations: 0]

  11. Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

    Liu, Xiong | Chen, Yanping

    Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 6 P.1506

    https://doi.org/10.4208/aamm.OA-2016-0049 [Citations: 4]

  12. High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels

    Deng, Guoting | Yang, Yin | Tohidi, Emran

    Applied Mathematics and Computation, Vol. 396 (2021), Iss. P.125866

    https://doi.org/10.1016/j.amc.2020.125866 [Citations: 6]

  13. Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

    Shi, Xiulian | Wei, Yunxia | Huang, Fenglin

    Numerical Methods for Partial Differential Equations, Vol. 35 (2019), Iss. 2 P.576

    https://doi.org/10.1002/num.22314 [Citations: 15]

  14. Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels

    Tang, Zhuyan | Tohidi, Emran | He, Fuli

    Computational and Applied Mathematics, Vol. 39 (2020), Iss. 4

    https://doi.org/10.1007/s40314-020-01352-y [Citations: 11]

  15. Numerical simulation of the time-delay optoelectronic oscillator model using locally supported radial basis functions

    Hosseinian, Alireza | Assari, Pouria | Dehghan, Mehdi

    The European Physical Journal Plus, Vol. 139 (2024), Iss. 5

    https://doi.org/10.1140/epjp/s13360-024-05158-3 [Citations: 1]