Numerical solution of the time-variable fractional order mobile-immobile advection-dispersion model using a hybrid method

Abstract

This paper introduces a hybrid numerical method combining the cubic B-spline collocation method with the implicit Euler approximation to solve the timevariable fractional order mobile-immobile advection-dispersion model (MIM-ADM). The model includes the Coimbra variable-order (VO) time fractional derivative, which is wellsuited for dynamic system modeling. Cubic B-spline functions are employed for spatial discretization, offering both flexibility and efficiency in approximating solutions. The implicit Euler method accurately approximates the Coimbra VO time fractional derivative. Moreover, we establish that the proposed method achieves a convergence order of O(h2−λ(x,t)t + h4x). Numerical simulations confirm the accuracy of the method by comparing its results with analytical solutions. Evaluation metrics such as L2 and L∞ error norms demonstrate the method’s efficacy in solving the MIM-ADM. This study highlights the effectiveness of the proposed approach in modeling and solving complex transport phenomena with high accuracy.