Poisson Equation vs. Hamzah Equation: Superiority Analysis.

The Poisson equation, a cornerstone of classical physics, is widely used in the modeling of electrostatic, gravitational, and fluid fields. However, it operates under the assumption of static fields and fails to account for memory effects, non-local behaviors, and the fractal-like nature of complex systems. The Hamzah equation, a novel extension of traditional frameworks, overcomes these limitations by incorporating memory, non-local fractional derivatives, and a multi-dimensional fractal structure. This paper provides a comprehensive comparison between the classical Poisson equation and the advanced Hamzah equation, highlighting the superior capabilities of the latter in capturing dynamic, memory-dependent behaviors in physical fields. Through a series of numerical and analytical comparisons, we demonstrate that the Hamzah equation exhibits a higher level of precision, stability under chaotic conditions, and a deeper understanding of field behaviors in complex systems. The Hamzah equation also offers groundbreaking potential in various scientific and technological domains, such as quantum gravity, artificial intelligence, and cognitive neuroscience, making it a key advancement in the evolution of physical and informational theories.