New discretization of [formula omitted]-Caputo fractional derivative and applications

Abstract

In the present paper, two approximations to evaluate the ψ-Caputo fractional derivative are developed using the linear and the quadratic polynomial interpolations. We present a study of the pointwise error for each approximation and illustrate some particular cases that correspond to approximations of the well known fractional derivatives, such as: Caputo, Katugampola and Hadamard fractional derivatives. In order to elucidate the investigated results, we present some examples for each approximation. For concreteness, we show some applications where we solve initial value problems and problems involving fractional sub-diffusion equations. Finally, some concluding remark are presented.