SOLUTION OF ABEL’S INTEGRAL EQUATION USING ABAOUB-SHKHEAM TRANSFORM

Abstract

In this paper, the authors determine the exact solution of a special case of Volterra integral equations of the first kind when the kernel k(x,t)=(x−t)−αk(x,t) = (x-t)^{-\alpha}k(x,t)=(x−t)−α is not defined at certain values within its domain. Using the Abaoub-Shkheam transform, this equation, known as Abel's integral equation, is specifically studied for α=1/2\alpha = 1/2α=1/2. Several numerical problems have been analyzed and solved with the assistance of the Abaoub-Shkheam transform. The results of these numerical problems demonstrate that the Abaoub-Shkheam transform is a highly effective integral transform for determining the exact solution of this equation.