A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model

The evolution equations with fractional or variable order derivatives can deliver a proper mathematical modeling to define the transport dynamics and anomalous diffusion in complex dynamical structures. Herein, a hybrid method based on operational matrices of derivative is proposed and successfully applied to explore the solution of mobile–immobile advection–dispersion problem of variable order. The variable order of the model is considered as function of space and time. The operational matrices of derivative named exact and approximate are constructed with the aid of two different approaches and related theorems are available to support the mathematical justification. The error bound and convergence analysis is presented to validate the mathematical formulation of the computational algorithm. A comparative study is enclosed in our investigation which endorses the credibility of the exact operational matrix of derivative. The numerical simulations for various problems are encountered and set of graphs are presented. The numerical examples are endorsing that the proposed mathematical algorithm is computationally effective and efficient tool and one can extend it to other physical problems of fractional or variable order. •The computational scheme is based on two kinds of operational matrices and used to solved mobile–immobile advection–dispersion model.•The operational matrices of derivative are established via two different approaches.•Some new theorems are developed to propose the exact operational matrices of derivative.•Method converts the fractional problems into system of algebraic equations and solved via collocation method.•The numerical simulations are performed to examine the transport dynamics and anomalous diffusion.•The comparative analysis, error and convergence of the scheme is presented to justify its mathematical formulation.