A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.