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

Area of Science:

• Complex Analysis
• Fractional Calculus
• Mathematical Modeling

Background:

• Bi-univalent functions and fractional calculus are areas of significant theoretical and applied interest.
• Existing research often lacks practical physical or biological interpretations and unexplored applications in applied modeling.
• The integration of bi-univalent function theory with q-fractional calculus is a nascent field.

Purpose of the Study:

• Introduce a new subclass of bi-univalent functions using a generalized q-fractional differential operator.
• Derive upper bounds for Taylor-Maclaurin coefficients using Faber polynomial expansion.
• Demonstrate the practical applicability of the new framework in fluid dynamics and epidemic modeling.

Main Methods:

• Definition of a new subclass of bi-univalent functions.
• Application of Faber polynomial expansion for coefficient estimation.
• Development of a generalized q-fractional differential operator.

Main Results:

• Obtained upper bounds for Taylor-Maclaurin coefficients, including specific estimates for the second and third coefficients and a general bound.
• Successfully modeled ideal fluid flow boundaries using conformal mappings derived from the bi-univalent functions.
• Incorporated the theory into a modified SIR epidemic model, showing the role of fractional parameters in capturing memory effects and nonlinear dynamics.

Conclusions:

• The proposed class of bi-univalent functions offers both theoretical depth and practical utility in applied sciences.
• The study bridges theoretical advancements in complex analysis with tangible applications in physical and biological systems.
• The q-fractional calculus approach provides a powerful tool for modeling complex phenomena with memory effects.