regularization of the generalized auto-convolution volterra integral equation of the first kind

In this paper,  a generalized version of the auto-convolution volterra integral equation of the first kind as an ill-posed problem is studied. we apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. according to the proposed numerical method, for $n=0$ and  $n=1,ldots, n-1$, we obtain a  nonlinear and linear system, respectively. we have to distinguish between two cases, nonlinear and linear systems of algebraic equations. a double iteration process based on the modified tikhonov regularization method is considered to solve the nonlinear algebraic equations. in this process, the outer iteration controls the evolution path of the unknown vector $u_0^{delta}$ in the selected direction $tilde{u}_0$, which is determined from the inner iteration process. for the linear case, we apply the lavrentiev $tilde{m}$ times iterated regularization method to deal with the ill-posed linear system. the validity and efficiency of the proposed method are demonstrated by several numerical experiments.