On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Abstract

Several numerical techniques have been developed to approximate Riemann-Liouville (R- L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R- L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and H & ouml;lder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.