Quadrature solution of fractional coupled burgers and plankton-oxygen dynamics under climate change.

This research aims to provide a precise and computationally efficient solution for two important systems in mathematical modeling and environmental science: the fractional nonlinear coupled Burgers' system and the fractional dynamics of the coupled plankton-oxygen model in (1 + 1) dimensions. Despite the increasing complexity of these systems, existing numerical methods often struggle with the high computational costs of non-local operators. To address this, we propose a robust hybrid framework integrating the Fractional Differential Quadrature Method (FDQM) with a Newton-Raphson (NR) iterative procedure.These systems are critical for understanding various physical processes, including turbulent fluid dynamics and the biological interactions of oxygen and plankton in aquatic environments. The models use fractional derivatives, which provide greater flexibility in capturing memory and hereditary features, making them more suitable for correctly simulating real-world processes than traditional integer-order models. The proposed work solves these problems using a generalized Liouville-Caputo fractional-order model mixed with versions of the differential quadrature technique (DQM), which allows effective handling of complex boundary conditions and spatial derivatives. The nonlinearity in these equations is handled using Newton-Raphson's iterative approach, which ensures the solutions' stability and convergence. The proposed system was implemented in MATLAB, and a complete parametric analysis was carried out to investigate how various parameters, including the fractional-order derivative, the oxygen generation rate, and the maximum per capita growth rate of phytoplankton, influence model outputs. This study not only proves the suggested techniques' accuracy, convergence, and efficiency but also sheds light on their sensitivity to important parameters, making them more applicable to real-world circumstances. The findings of this work are expected to contribute to better modeling methodologies for complex systems, ultimately benefiting academics in domains ranging from environmental science to fluid dynamics.