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An $ hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels


Authors: Zhong-qing Wang, Yu-ling Guo and Li-jun Yi
Journal: Math. Comp. 86 (2017), 2285-2324
MSC (2010): Primary 65L60, 45D05, 41A10, 65L70
DOI: https://doi.org/10.1090/mcom/3183
Published electronically: February 15, 2017
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Abstract: In this paper, we present an $ hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. We establish several new approximation results of the Legendre/Jacobi polynomial interpolations for both smooth and singular functions. As applications of these approximation results, we derive $ hp$-version error bounds of the Legendre-Jacobi collocation method under the $ H^1$-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. We also show the exponential rates of convergence for singular solutions by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are included to illustrate the theoretical results.


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Additional Information

Zhong-qing Wang
Affiliation: College of Science, University of Shanghai for Science and Technology, Shanghai 200093, People’s Republic of China
Email: zqwang@usst.edu.cn

Yu-ling Guo
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

Li-jun Yi
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China

DOI: https://doi.org/10.1090/mcom/3183
Keywords: $hp$-version Legendre-Jacobi spectral collocation method, Volterra integro-differential equations, smooth and weakly singular kernels, convergence analysis
Received by editor(s): March 16, 2015
Received by editor(s) in revised form: March 28, 2016
Published electronically: February 15, 2017
Additional Notes: This work was supported in part by the National Natural Science Foundation of China (Nos. 11571238 and 11301343), the Research Fund for Doctoral Program of Higher Education of China (Nos. 20133127110006 and 20113127120002), and the Natural Science Foundation of Shanghai (No. 15ZR1430900).
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