scholarly journals Kneser’s theorem for weak solutions of an integral equation with weakly singular kernel

2005 ◽  
Vol 77 (91) ◽  
pp. 87-92 ◽  
Author(s):  
Aldona Dutkiewicz ◽  
Stanislaw Szufla
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Weak Solutions ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Nonempty Compact

We prove that the set of all weak solutions of the Volterra integral equation (1) is nonempty, compact and connected.

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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