scholarly journals Numerical solution of Volterra integral equations with weakly singular kernel based on the DE-sinc method

2008 ◽  
Vol 25 (2) ◽  
pp. 165-183 ◽  
Author(s):  
Masatake Mori ◽  
Ahniyaz Nurmuhammad ◽  
Takefumi Murai
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Sinc Method ◽  
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.

scholarly journals Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Mohamed R. Ali ◽  
Mohamed M. Mousa ◽  
Wen-Xiu Ma
Keyword(s):  
Integral Equations ◽  
Numerical Technique ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Wavelet Method ◽  
Block Pulse Functions ◽  
Weakly Singular ◽  
The Way

We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). The considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. The explained technique is delineated with two numerical cases to demonstrate the benefit of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.

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