An $hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels
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- by Zhong-qing Wang, Yu-ling Guo and Li-jun Yi PDF
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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.References
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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
- MR Author ID: 678011
- 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
- MR Author ID: 883098
- 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).
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 2285-2324
- MSC (2010): Primary 65L60, 45D05, 41A10, 65L70
- DOI: https://doi.org/10.1090/mcom/3183
- MathSciNet review: 3647959