Mapping techniques for collocation method of time-fractional convection–diffusion equations in domains with cracks

Abstract

This paper proposes numerical methods that effectively deal with time-fractional convection–diffusion equations containing crack singularities. To deal with singularities, we design the geometrical mapping whose push-forward from the parameter space into the physical space generates point singularity functions based on the parametrization of the circular arc and NURBS (non-uniform rational B-spline). We adopt the collocation method with B-spline basis functions to approximate the solution in the spatial direction and enrich the approximation space by k-refinements in IGA (Isogeometric Analysis). For the discretization along the temporal direction, we employ the explicit Predictor-Corrector (PC) scheme that has the order and of the truncation error for the linear and quadratic interpolation, respectively. Taking advantage of the NURBS geometrical mapping, we demonstrate the performance of the proposed methods applying to time-fractional convection–diffusion equations with nonlinear terms on curved domains containing crack singularities.