Design and convergence analysis of a new efficient iterative method for solving systems of nonlinear equations with applications in economic dynamics and physical modeling

The Riccati, Duffing, and Black-Scholes equations are central to modeling nonlinear phenomena in dynamical, physical modeling, and financial systems. The Riccati equation is instrumental in chaos theory and soliton dynamics, providing key insights into wave propagation and stability analysis. The Duffing equation, a classical model for nonlinear oscillators, enables the study of chaotic dynamics, bifurcations, and fractal structures. Meanwhile, the Black-Scholes equation underpins modern option pricing theory, with its extensions offering powerful tools for analyzing financial fractals, market chaos, and stochastic volatility. In this study, we introduce a novel three-step iterative method whose local convergence order is established using Taylor series expansions. Unlike conventional approaches, our method requires only a single Jacobian evaluation and inversion per iteration, leveraging LU decomposition to efficiently solve the associated linear systems at each step. By circumventing direct Jacobian inversion, the method enhances computational efficiency without compromising precision. The total computational cost (CC) of the proposed method is 13n3+8n2+203n. Therefore, the efficiency index of the method is very favorable compared to earlier competing methods that utilize similar information. Finally, we perform extensive numerical simulations on large-scale systems, including those arising from the discretization of differential equations such as the Riccati, Duffing, and Black-Scholes equations. The results demonstrate that the proposed method provides significant advantages in solving stiff systems and high-dimensional Jacobian problems, where conventional approaches frequently encounter prohibitive computational costs or numerical instability. Across all cases, the numerical outcomes consistently align with theoretical predictions, showcasing faster convergence, improved accuracy, and greater reliability compared to existing methods.