Derivatives and integrals: matrix order operators as an extension of the fractional calculus

Abstract: A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function applied to the fractional differintegration definition. This work focuses on some results applied to the Riemann-Liouville version of the fractional calculus extended to its matrix-order concept. This extension also may apply to other versions of fractional calculus. Some examples of possible ordinary and partial matrix-order differential equations and related exact solutions are shown.