Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

Abstract

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.