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This research paper delves deeply into the intricate interplay betweenquintessence and a cloud of strings, exploring their influence on criticalpoints, behaviors, and fractional order phase transitions in AdS black holes.We conducted a meticulous analysis of the various thermodynamic propertiesassociated with AdS black holes surrounded by quintessence and a cloud ofstrings, with a specific focus on understanding how thermal fluctuations affecttheir thermodynamics. Using critical conditions, we systematically derived anapproximate analytical expressions for critical points that are applicable toall types of these black holes. Furthermore, we embarked on an exploration ofhow changes in the quintessence parameter and the intensity of the cloud ofstrings impact critical behaviors, using three-dimensional visualizations tovividly depict these effects across different values of omegaq. Our examinationilluminated how these factors shape the locations of phase transitions and theregions where both phases coexist. Additionally, we delved into the realm offractional order phase transitions, which indicate significant transformationswithin the black hole system. This work delves into examining how the behaviorsof both small and large black holes are influenced by the combined effects ofquintessence and a string cloud. The study employs two distinct approximationsof the horizon radius for Ads black holes, providing a nuanced analysis oftheir dynamic responses.

Responses of Small and Large Ads Black Holes to the Collective Influence of Quintessence and String Cloud