Block-by-block method for solving non-linear Volterra integral equation of the first kind

In this paper, we focus on the numerical solution of the non-linear Volterra integral equation of the first kind. We start by converting its form to a second-kind Volterra equation and we construct some assumptions to give a sufficient condition to ensure the solution’s existence and uniqueness, this is on the one hand. On the other hand, to use it in the numerical and analytical analysis computation which put the two analyses compatible. Then, by applying block-by-block method, we transform our integral equation to a non-linear system. This system gives us a discrete numerical solution converging to our analytical solution. The approximate solution is computed using a Newton’s method and the numerical examples clearly demonstrate the efficiency of our proposed method compared with the Nystöm method.