Enhancing the Convergence Order from p to p + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from p to p+3 units, in the order of convergence. This is constructed from any Jacobian-free scheme of order p. We conduct comprehensive numerical tests first in academical examples to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes. Then, we apply them on the non-differentiable partial differential equations that models the nutrient diffusion in a biological substrate.