High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an r th-order time stepping Lagrange interpolation polynomial, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>3</m:mn> <m:mo>≤</m:mo> <m:mi>r</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math> 3≤ r∈ N . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O ( τ r +1− α , h 2 ), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.