Application of the homotopy perturbation method for weakly singular Volterra integral equations

In this paper, we study a weakly singular Volterra integral equation of the second kind with the kernel $\displaystyle K(x,t) = \left (\frac{t}{x}\right )^\nu\frac{1}{t}$, for some $\nu >0$ and $x\in[0,X]$. The powerful homotopy perturbation method (HPM) is initially applied to find a solution to the integral equation for $\nu > 1$. We then consider the interesting case where $0< \nu < 1$. Applying the homotopy perturbation method constructed by a convex homotopy or other series-related methods produces unwanted results for this case. In this study, we propose conditions to be imposed to overcome this issue. In addition, for completeness, we investigate all cases where $\nu\in \mathbb{R}$. Some numerical examples are provided to confirm the simplicity and applicability of the applied methods.