A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays

2022 ◽  
Vol 417 ◽  
pp. 126780
Author(s):  
Guoqing Yao ◽  
DongYa Tao ◽  
Chao Zhang
Keyword(s):  
Integral Equations ◽  
Spectral Method ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular

Convergence Analysis of the Spectral Method for Second-Kind Volterra Integral Equations with a Weakly Singular Kernel

2012 ◽  
Vol 263-266 ◽  
pp. 3313-3316
Author(s):  
Yiao Yong Zhang ◽  
Hua Feng Wu
Keyword(s):  
Integral Equations ◽  
Source Function ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
L2 Norm ◽  
Spectral Galerkin Method ◽  
Rigorous Error ◽  
Theoretical Results

The Legendre spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 -norm and the L∞ -norm ) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

Solving the Volterra Integral Equations with Weakly Singular Kernel by Taylor Expansion Methods

2012 ◽  
Vol 220-223 ◽  
pp. 2129-2132
Author(s):  
Li Huang ◽  
Yu Lin Zhao ◽  
Liang Tang
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Taylor Expansion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we propose a Taylor expansion method for solving (approximately) linear Volterra integral equations with weakly singular kernel. By means of the nth-order Taylor expansion of the unknown function at an arbitrary point, the Volterra integral equation can be converted approximately to a system of equations for the unknown function itself and its n derivatives. This method gives a simple and closed form solution for the integral equation. In addition, some illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

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