This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of the results highlights the method’s convergence and accuracy, comparing the numerical solution with the exact solution derived from linear parabolic partial differential transformations. This innovative fifth-order method provides an efficient numerical solution to the nonlinear convection–diffusion equation, addressing the problem through a systematic methodology that combines discretization and nonlinear equation solving. The study underscores the importance of advanced numerical techniques in tackling complex problems in physics and mathematics.

La implementación de estrategias digitales en la educación ecuatoriana se enfrenta a desafíos y oportunidades significativas. La persistente brecha digital entre diferentes estratos socio-culturales requiere una atención prioritaria para garantizar la equidad en el acceso y desarrollo de competencias digitales. La formación docente emerge como un pilar fundamental, capacitando a educadores para integrar de manera efectiva la tecnología en el aula y guiar a los estudiantes hacia la competencia digital. La inclusión y equidad en la educación digital se posicionan como objetivos cruciales, empoderando a todos los estudiantes para participar plenamente en la sociedad digital actual y futura. La competencia digital no se limita al manejo de herramientas tecnológicas, sino que también implica habilidades críticas como el pensamiento analítico y la evaluación de la información en línea. La tecnología, en armonía con enfoques pedagógicos innovadores, facilita experiencias de aprendizaje significativas y contextualizadas. La promoción de la ciudadanía digital responsable se vuelve esencial, enseñando a los estudiantes sobre la ética en línea y la seguridad digital. En conjunto, abordar estos aspectos de manera integral promueve un sistema educativo que prepara a los estudiantes para triunfar en la era digital actual y venidera.

Nonlinear systems of equations arise in a wide range of applications in engineering, physics, and biological modeling; however, classical Newton-type methods may fail when the initial approximation lies outside the basin of attraction of the desired solution. This work proposes a two-stage hybrid framework that couples Particle Swarm Optimization (PSO) for global exploration with the fifth-order Newton-Jarratt (NJN) iterative method for local refinement. The fifth-order convergence of the NJN phase, established through a complete Fréchet-derivative Taylor expansion with explicitly computed error constants, guarantees rapid local convergence once PSO delivers a sufficiently close starting point. The framework is validated on four test problems of increasing dimension (n=2,5,20,40): a two-dimensional benchmark algebraic system, a five-dimensional metabolic network model for ethanol production in Saccharomyces cerevisiae, and two large-scale systems arising from the discretization of a Hammerstein nonlinear integral equation. Over 30 independent runs per method, PSO-NJN achieves a 100% convergence rate in all four problems, with mean final residuals on the order 10⁻¹⁴-10⁻¹⁶. In comparison, pure PSO fails completely (0% success) on the high-dimensional Hammerstein cases (n=20,40) and achieves only 10% success on the metabolic model. These results confirm that combining global metaheuristic search with high-order local refinement yields a robust, scalable solver suitable for complex biological and engineering nonlinear systems.