Dynamical behavior of a time-fractional biological model via an efficient numerical method

This paper introduces a robust and efficient meshless collocation method based on Fibonacci and Lucas polynomials for the numerical solution of one- and two-dimensional partial differential equations related to the wound healing process. This method offers several advantages, including easy implementation for higher-dimensional problems and applicability to both regular and irregular computational domains, compared to mesh-based methods. The time derivatives are solved using the Caputo fractional derivative, coupled with the Strang splitting algorithm, while the spatial derivatives are approximated with a meshless technique using Lucas and Fibonacci polynomials. With wound healing models being so inherently complex in nature (due to the fact that they are generally composed of both convective and diffusive processes concerning cell migration and chemotaxis), it is crucial to model convection-dominated systems correctly in order to avoid numerical instability. This method has been thoroughly tested through numerical simulations involving various wound healing scenarios. A comparison with recent literature, utilizing exact solutions and established numerical methods, demonstrates that this approach provides significantly higher accuracy, greater efficiency, and enhanced computational performance. These results confirm the method’s robustness across various domain configurations.