Modeling of complex physical and biological problems using bi-univalent function calculus

In recent years, bi-univalent functions and fractional calculus have attracted considerable attention due to their strong theoretical foundations and potential applications in applied sciences. However, most existing studies focus primarily on coefficient estimates from a purely theoretical perspective, with limited attention given to practical physical or biological interpretations. Moreover, the integration of bi-univalent function theory with q-fractional calculus in applied modeling remains largely unexplored. In this paper, we introduce a new subclass of bi-univalent functions defined in the open unit disk using a generalized q-fractional differential operator. By employing the Faber polynomial expansion, we derive upper bounds for the Taylor-Maclaurin coefficients, including explicit estimates for the second and third coefficients as well as a general bound for higher-order terms. To demonstrate the applicability of the proposed analytic framework, we present two illustrative applications. First, we model ideal fluid flow boundaries using conformal mappings generated by bi-univalent functions, highlighting the influence of the function coefficients on boundary geometry. Second, we incorporate the developed theory into a modified SIR epidemic model by representing the cumulative number of infections through a bi-univalent function and its q-fractional derivative. Numerical simulations and graphical results illustrate how fractional parameters capture memory effects and nonlinear transmission dynamics. These applications confirm that the proposed class provides both theoretical significance and practical modeling potential.