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Research Papers

# Jacobi Spectral Galerkin and Iterated Methods for Nonlinear Volterra Integral Equation

[+] Author and Article Information
Yin Yang

Department of Mathematics,
Xiangtan University,
Xiangtan, Hunan 411105, China
e-mail: yangyinxtu@xtu.edu.cn

Yanping Chen

School of Mathematical Sciences,
South China Normal University,
Guangzhou, Guangdong 510631, China
e-mail: yanpingchen@scnu.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 23, 2015; final manuscript received April 11, 2016; published online May 24, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 11(4), 041027 (May 24, 2016) (8 pages) Paper No: CND-15-1073; doi: 10.1115/1.4033439 History: Received March 23, 2015; Revised April 11, 2016

## Abstract

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the $L∞$-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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## Figures

Fig. 1

Example 1: Comparison between approximate solution u¯N(x) and exact solution u(x) (left). The errors in ‖·‖∞ and weighted ‖·‖ω norms (right) versus N.

Fig. 2

Example 2: Comparison between approximate solution u¯N(x) and exact solution u(x) (left). The errors in ‖·‖∞ and weighted ‖·‖ω norms (right) versus N.

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