An efficient spectral method for nonlinear Volterra integro-differential equations with weakly singular kernels
Abstract
For Volterra integro-differential equations (VIDEs) with weakly singular kernels, their solutions are singular at the initial time. This property brings a great challenge to traditional numerical methods. Here, we investigate the numerical approximation for the solution of the nonlinear model with weakly singular kernels. Due to its characteristic, we split the interval and focus on the first one to save operation. We employ the corresponding singular functions as basis functions in the first interval to simulate its singular behavior, and take the Legendre polynomials as basis functions in the other one. Then the corresponding hp-version spectral method is proposed, the existence and uniqueness of solution to the numerical scheme are proved, the hp-version optimal convergence is derived. Numerical experiments verify the effectiveness of the proposed method.
Keyword : spectral element method, Volterra integro-differential equation, weak singularity, exponential convergence
How to Cite
Liu, Z., Tao, D., & Zhang, C. (2024). An efficient spectral method for nonlinear Volterra integro-differential equations with weakly singular kernels. Mathematical Modelling and Analysis, 29(3), 387–405. https://doi.org/10.3846/mma.2024.18354
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Z.-Q. Wang, Y.-L. Guo and L.-J. Yi. An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Mathematics of Computation, 86(307):2285–2324, 2017. https://doi.org/10.1090/mcom/3183
Z.-Q. Wang and C.-T. Sheng. An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Mathematics of Computation, 85(298):635–666, 2016. https://doi.org/10.1090/mcom/3023
Y. Wei and Y. Chen. Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Advances in Applied Mathematics and Mechanics, 4(1):1–20, 2012. https://doi.org/10.4208/aamm.10-m1055
Y. Yang and Y. Chen. Spectral collocation methods for nonlinear Volterra integro-differential equations with weakly singular kernels. Bulletin of the Malaysian Mathematical Sciences Society, 42(1):297–314, 2017. https://doi.org/10.1007/s40840-017-0487-7
L. Yi and B. Guo. An h-p version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM Journal on Numerical Analysis, 53(6):2677–2704, 2015. https://doi.org/10.1137/15m1006489
W. Yuan and T. Tang. The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation.Mathematics of Computation, 54(189):155–168,1990. https://doi.org/10.1090/s0025-5718-1990-0979942-6
C. Zhang, Z. Liu, S. Chen and D.Y. Tao. New spectral element method for Volterra integral equations with weakly singular kernel. Journal of Computational and Applied Mathematics, 404:113902, 2022. https://doi.org/10.1016/j.cam.2021.113902
W. Zhen and Y. Chen. A spectral method for a weakly singular Volterra integro-differential equation with pantograph delay. Acta Mathematica Scientia, 42(1):387–402, 2022. https://doi.org/10.1007/s10473-022-0121-0
H. Brunner. Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels. IMA Journal of Numerical Analysis, 6(2):221–239, 1986. https://doi.org/10.1093/imanum/6.2.221
H. Brunner. Collocation methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511543234
Y. P. Chen and T. Tang. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Mathematics of Computation, 79(269):147–167, 2010. https://doi.org/10.1090/s0025-5718-09-02269-8
D. Hou, Y. Lin, M. Azaiez and C. Xu. A Müntz-collocation spectral method for weakly singular Volterra integral equations. Journal of scientific computing, 81(3):2162–2187, 2019. https://doi.org/10.48550/arXiv.1904.09594
Q.Y. Hu. Stieltjes derivatives and β-polynomial spline collocation for Volterra integrodifferential equations with singularities. SIAM Journal on Numerical Analysis, 33(1):208–220, 1996. https://doi.org/10.1137/0733012
Y.J. Jiang and J.T. Ma. Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. Journal of Computational and Applied Mathematics, 244:115–124, 2013. https://doi.org/10.1016/j.cam.2012.10.033
J. Shen, T. Tang and L.L. Wang. Spectral methods. Algorithms, analysis and applications. Springer Series in Computational Mathematics. Springer, Heidelberg, 2011.
J. Shen and Y.W. Wang. Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems. SIAM Journal on Scientific Computing, 38(4):A2357–A2381, 2016. https://doi.org/10.1137/15m1052391
C.-T. Sheng, Z.-Q. Wang and B.-Y. Guo. A multistep LegendreGauss spectral collocation method for nonlinear Volterra integral equations. SIAM Journal on Numerical Analysis, 52(4):1953–1980, 2014. https://doi.org/10.1137/130915200
X.L. Shi, Y.X. Wei and F.L. Huang. Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations. Numerical Methods for Partial Differential Equations, 35(2):576–596, 2019. https://doi.org/10.1002/num.22314
C.L. Wang, Z.Q. Wang and H.L. Jia. An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels. Journal of Scientific Computing, 72(2):647–678, 2017. https://doi.org/10.1007/s10915-017-0373-3
Z.-Q. Wang, Y.-L. Guo and L.-J. Yi. An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Mathematics of Computation, 86(307):2285–2324, 2017. https://doi.org/10.1090/mcom/3183
Z.-Q. Wang and C.-T. Sheng. An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Mathematics of Computation, 85(298):635–666, 2016. https://doi.org/10.1090/mcom/3023
Y. Wei and Y. Chen. Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Advances in Applied Mathematics and Mechanics, 4(1):1–20, 2012. https://doi.org/10.4208/aamm.10-m1055
Y. Yang and Y. Chen. Spectral collocation methods for nonlinear Volterra integro-differential equations with weakly singular kernels. Bulletin of the Malaysian Mathematical Sciences Society, 42(1):297–314, 2017. https://doi.org/10.1007/s40840-017-0487-7
L. Yi and B. Guo. An h-p version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM Journal on Numerical Analysis, 53(6):2677–2704, 2015. https://doi.org/10.1137/15m1006489
W. Yuan and T. Tang. The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation.Mathematics of Computation, 54(189):155–168,1990. https://doi.org/10.1090/s0025-5718-1990-0979942-6
C. Zhang, Z. Liu, S. Chen and D.Y. Tao. New spectral element method for Volterra integral equations with weakly singular kernel. Journal of Computational and Applied Mathematics, 404:113902, 2022. https://doi.org/10.1016/j.cam.2021.113902
W. Zhen and Y. Chen. A spectral method for a weakly singular Volterra integro-differential equation with pantograph delay. Acta Mathematica Scientia, 42(1):387–402, 2022. https://doi.org/10.1007/s10473-022-0121-0