Efficient numerical treatment of time fractional advection diffusion equations for modeling heat, pollutant and particle transport using subdivision collocation

In numerical methods, fine mesh sizes are often necessary to obtain highly accurate solutions of the differential equations and this increases the memory consumption and decreases the efficiency of the calculation. This paper presents a subdivision collocation algorithm of time-fractional advection diffusion equation that is a model that is used to characterize anomalous diffusion in scientific and engineering systems. The fractional time derivative is discretized in the Caputo sense to model memory effects, whereas spatial approximation is done using subdivision schemes. The approach converts the problem into an effective and steady system of algebraic equations by collocating at spatial nodes. The consistency and error analysis indicate that the methodology is reliable, and the numerical tests indicate that the suggested method is highly accurate and requires less computational tools than the current methods. Other than its numerical performance, the technique aids in real world simulation like the transportation of pollutants, heat conduction, and wave propagation in non-homogeneous media. The results highlight the subdivision collocation method as a promising tool for efficiently solving fractional partial differential equations while contributing to global sustainability challenges.