In this paper the authors combine the quantum calculus applications regarding the theories of differential subordination and superordination with fuzzy theory. These results are established by means of an operator defined as the $q$-analogue of the multiplier transformation. Interesting fuzzy differential subordination and superordination results are derived by the authors involving the functions belonging to a new class of normalized analytic functions in the open unit disc $U$ which is defined and investigated here by using this $q$-operator.

The development of certain aspects of geometric function theory after incorporating fractional calculus and $q$-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann-Liouville fractional integral of the $q$-hypergeometric function. Numerous theorems are stated and proved involving the fractional $q$-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional $q$-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional $q$-operator, several sandwich-type theorems combine the subordination and superordiantion established results.

This research introduced a new method, the Aboodh Tamimi Ansari transform method ( $(AT{)}^2$ method), for solving systems of linear and nonlinear fractional partial differential equations. The method combined the Aboodh transform method and the Tamimi Ansari method, allowing for the simultaneous solution of linear and nonlinear terms without restrictions. The Caputo sense was considered for fractional derivatives. The effectiveness of the proposed method was demonstrated through numerical solutions, graphical representations, and tabular data, showing strong agreement with exact solutions. The approach was deemed precise, easy to apply, and could be extended to address further challenges in fractional-order problems. Computational tasks were carried out using Mathematica 13.

In this paper, the research discussed involves fractional calculus applied to a $q$-operator. Fractional integrals applied to the $q$-analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research.