Solving the Basset equation via Chebyshev collocation and LDG methods

Two different numerical methods are developed to find  approximate solutions of a class of linear fractional differential equations (LFDEs) appearing in the study of the generalized Basset force, when a sphere sinks in a viscous fluid. In the first one, using the Chebyshev bases, the collocation points, and the matrix operations, the given LFDE reduces to a matrix equation while in the second one, we employ the local discontinuous Galerkin (LDG) method, which uses the natural upwind flux yielding a stable discretization. Unlike the first method, in the latter method we are able to solve the problem element by element locally and there is no need to solve a full global matrix. The efficiency of the proposed algorithms are shown via some numerical examples.

Keywords:

Basset equation , Caputo fractional derivative , Chebyshev polynomials , collocation method , Local discontinuous Galerkin method , Numerical stability