Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods

In this paper to approximate the Heydari–Hosseininia non-singular fractional derivative, we construct the L1-2 discretization by providing the error estimate. The error estimation of the L1 formula is also presented. The scheme uses the local discontinuous Galerkin method combing with the L1/L1-2 formula as spatial and time discretizations, respectively. To investigate the efficiency and accuracy of our scheme, variable-order fractional ordinary differential and 2-dimensional Sobolev equations are proposed. The scheme is second/third-order accurate in time for the L1/L1-2 formula, respectively. Utilizing k, the approximation degree, the rates of convergence in space are reported k+1 when time step chosen τ=hk+12 and τ=hk+13. Our argument is that new approximation L1-2 has less computational cost than the L1 discretization and numerical results would be given to confirm this reduction.