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

www.nature.com/scientificreports OPEN Modeling of complex physical and biological problems using biunivalent function calculus Z. M. Saleh1,2, A. O. Mostafa1, M. A. Sohaly3 & S. M. Madian4 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. Keywords Simulation, Competitional complexity , Mathematical operator, Epidemics Faber polynomials are an important tool in complex analysis, especially in the study of conformal mappings and potential theory. They were first introduced by the German mathematician Georg Faber in the early 20th century1. These polynomials are closely linked to the theory of univalent (one-to-one analytic) functions, and they serve as a natural extension of the Taylor series for functions defined in more general domains. Instead of being limited to circular regions, Faber polynomials allow approximations on domains with smooth or irregular boundaries. Because of this, they have become a useful tool in many areas of applied mathematics. Their applications go beyond pure theory. For example, Faber polynomials are used in solving differential and integral equations that appear in potential theory and fluid dynamics2–4. They are particularly helpful when studying problems that involve conformal mapping techniques for simplifying complex geometries. In addition, they have been applied in areas such as electrostatics and computational geometry to analyze and approximate smooth domains. In recent years, there has been a growing interest in applying geometric function theory and fractional calculus to problems arising in science and engineering. Bi-univalent functions, in particular, offer a flexible analytic structure that has motivated extensive research on coefficient estimates and related geometric properties. When combined with the q-calculus, these functions gain additional generality and flexibility, allowing the introduction of deformation and memory parameters that extend many classical operators. In5, an overview of fundamental concepts and analytic properties related to meromorphic functions in the complex plane was presented, while recent studies have highlighted the importance of geometric structures in modern analytical and computational research directions6. Despite these developments, most existing studies in the literature are primarily concerned with theoretical coefficient bounds for subclasses of bi-univalent functions, often treating the associated operators in isolation. In particular, there is a noticeable lack of works that simultaneously employ Faber polynomial techniques and 1Mathematics Dept., Faculty of Science, Mansoura University, Mansoura, Egypt. 2Basic Science Dept., Higher Technological Institute, Tenth of Ramadan, Egypt. 3Department of Statistics and Operations Research, College of Science, Qassim University, P.O. Box 6644, 51452 Buraydah, Saudi Arabia. 4Basic Science Dept., Higher Institute of Engineering and Technology, New Damietta, Egypt. email: Scientific Reports | (2026) 16:16297 | https://doi.org/10.1038/s41598-026-41761-3 1 generalized q-fractional differential operators within a unified bi-univalent framework, while also providing a clear pathway toward meaningful applications. Motivated by this research gap, the present work introduces a new subclass of bi-univalent functions defined by a generalized q-fractional differential operator. By using the Faber polynomial expansion, we derive sharp 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. Furthermore, to demonstrate the relevance of the theoretical results, applications to ideal fluid flow modeling and epidemic dynamics are presented in a separate section of the paper. Methodology Let A be the class of functions. F (z) = z + ∞ ∑ aρ z ρ ,(2.1) ρ=2 ′ defined in D = {z ∈ C : |z| < 1} normalized by the conditions F (0) = F (0) − 1 = 0 for every z ∈ D and S be the subclass of A consisting of univalent functions in D. For every F ∈ S there exists an inverse function F −1 which is defined in some neighborhood of the origin, and satisfying the conditions F −1 (F (z)) = z, (z ∈ D), and F (F −1 (ω)) = ω, (|ω| < r0 (F ); r0 (F ) ≥ 1 ), 4 where g(ω) =F −1 (ω) = ω−a2 ω 2 + (2a22 − a3 )ω 3 + −(5a32 − 5a2 a3 + a4 )ω 4 + ... =ω + ∞ ∑ Aρ ω ρ . (2.2) ρ=2 If both F and F −1 are univalent in D, then F ∈ A is called bi-univalent in D and the class of these functions is denoted by Ψ. For more study, this class (see7–9). In1 Faber introduced a polynomial that bears his name and has a very important role in the theory of geometric functions. Using the expansion of this polynomial for F ∈ S , the coefficients of its inverse g = F −1 can be expressed (see10 and11) as g(ω) = F −1 (ω) =ω + ∞ ∑ 1 ρ=2 ρ ρ χ−ρ ρ−1 (a2 , a3 , ..., aρ )ω ,(2.3) where χ−ρ ρ−1 = (−ρ)! (−ρ)! aρ−1 + aρ−3 a3 (−2ρ + 1)!(ρ − 1)! 2 (2(−ρ + 1))!(ρ − 3)! 2 (−ρ)! (−ρ)! + aρ−4 a4 + aρ−5 (−2ρ + 3)!(ρ − 4)! 2 (2(−ρ + 2))!(ρ − 5)! 2 [ ] (−ρ)! × a5 + (−ρ + 2)a23 + aρ−6 × [a6 + (−2ρ + 5)a3 a4 ] (−2ρ + 5)!(ρ − 6)! 2 + ∞ ∑ aρ−j Vj , 2 j≥7 such that Vj with 7 ≤ j ≤ ρ is a homogeneous polynomial in the variables a2 , a3 , ..., aρ , (see11). The first three terms of χ−ρ ρ−1 are −3 2 −4 3 χ−2 1 = −2a2 , χ2 = 3(2a2 − a3 ), χ3 = −4(5a2 − 5a2 a3 + a4 ). In general, for any p ∈ Z= {0, ±1, ±2, ...}, an expansi (...truncated)