Non-Linear Force Dynamics in Black Hole Accretion Disks Using Logarithmic and Trigonometric Approximations

Accretion disks are material structures that form when gas, dust or plasma is pulled towards a black hole by its strong gravity. In rotating (Kerr) black holes, the dynamics of the accretion disk becomes more complex due to frame-dragging effects that affect particle trajectories and energy release patterns. This study aims to develop an algorithm for calculating the force F(x) on particles in the accretion disk by considering the non-linear interaction between the black hole spin parameter and the relative position of the particles. The developed algorithm uses logarithmic and trigonometric approaches to improve the accuracy of the force calculation. The results show that variations in the spin parameter and relative position significantly affect the force distribution in the accretion disk. Visualization of the force interaction reveals the existence of non-linear patterns that contribute to the system dynamics. The main contribution of this research is the refinement of force calculation models that previously did not fully incorporate the combined effects of logarithmic and trigonometry in particle interactions. The proposed approach offers a more accurate predictive tool to explore the physical processes around black holes, and supports the interpretation of observational data from telescopes and gravitational wave detectors