Convergence domains and dynamical study of third-order Ulm’s method for nonlinear models

In this paper, we investigate the convergence domains of an inverse-free third-order Ulm’s method under weaker continuity conditions for solving nonlinear equations. Recurrence relations are used to derive the convergence estimates and establish the existence and uniqueness theorem. The present approach provides sufficient conditions for cubic convergence, where the existing results involving the Lipschitz condition are not applicable. A comparative study of the basins of attractions of the family shows that a third-order Ulm’s method offers broader stability and accessibility. Theoretical results are validated through various numerical examples involving Hammerstein-type integral equations, a boundary value ordinary differential equation, and an elliptic partial differential equation with nonlinear boundary conditions. Further, we apply third-order inverse-free Ulm’s method to nonlinear equations to obtain the numerical solution and to demonstrate that the numerical solution remains within the predicted convergence domains.