A Numerical Scheme Utilizing Trigonometric Cubic B-Splines for Quasilinear Fractional Mobile/Immobile Transport Model in the Caputo-Fabrizio Sense

Fractional calculus has become an effective tool in the modeling of complicated physical processes with memory effects and anomalous diffusion that are not well described by integer-order models. Of these, the mobile/immobile transport model is important in modeling solute transport in heterogeneous systems where standard diffusion breaks down. This work proposes a novel semi-implicit scheme that use trigonometric cubic B-splines (TCuBS) to find solutions for a quasilinear time fractional mobile/immobile transport equation in the Caputo-Fabrizio context. The innovation is in using the smooth, piecewise continuity of TCuBS to its advantage compared to traditional finite difference methods. Whereas finite differences estimate the solution only at discrete nodes, our TCuBS-based method builds a continuous solution throughout the whole spatial domain, which results in a physically more realistic and accurate representation. The rigorous analysis demonstrates that the proposed scheme is unconditionally stable. A convergence analysis is also conducted. Various numerical experiments for multiple test problems validate these theoretical results and illustrate that the proposed scheme provides superior accuracy and smaller errors in comparison to other methods available in the literature.