scholarly journals The exact solution of a linear integral equation with weakly singular kernel

2008 ◽  
Vol 344 (2) ◽  
pp. 726-734 ◽  
Author(s):  
Zhong Chen ◽  
YingZhen Lin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Singular Kernel ◽  
Linear Integral Equation ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Integral

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.

scholarly journals An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Rezvan Ghoochani-Shirvan ◽  
Jafar Saberi-Nadjafi ◽  
Morteza Gachpazan
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Transform Method ◽  
Weakly Singular ◽  
Differential Transform ◽  
Fractional Differential

An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.

scholarly journals The properties of solutions of weakly singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 419-430 ◽  
Author(s):  
G. Vainikko ◽  
A. Pedas
Keyword(s):  
Integral Equation ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Linear Integral Equation ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
The Difference ◽  
Linear Integral ◽  
Differential Properties ◽  
Derivatives Of

AbstractWe examine the differential properties of the solution of the linear integral equation of the second kind, whose kernel depends on the difference of arguments and has an integrable singularity at the point zero. The derivatives of the solution of the equation have singularities at the end points of the domain of integration, and we derive precise estimates for these singularities.

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