Essential criteria for existence of solution of a modified-ABC fractional order smoking model

Drug usage has always been a top concern for parents and government officials, harming younger people's lives in circumstances that cannot be undone. This paper considers a modified ABC-fractional-order Ice-smoking dynamical system for the theoretical and numerical results. In the theoretical aspect, we study the solution existence, uniqueness work, and Hyers-Ulam (HU) stability of the presumed extended fractional order dynamical system of Ice smoking. Recursive sequences that follow one another are designed to verify the existence of a solution to the provided Ice-smoking model. Functional analysis principles and findings are applied to showcase the uniqueness of the solution and the stability under the Hyers-Ulam (HU) framework. We offer simulations and a comparison by graphs by using Lagrange polynomials. The simulations demonstrate the scheme's applicability, future forecasting, and validations of the results of modified ABC-fractional orders.