Matrix Approach To The Fractional Calculus

Journal of Theoretical Probability. 2026. Vol. 39. Article 83.

V. N. Kolokoltsov, E. L. Shishkina

In this paper, we introduce a new construction of fractional derivatives and integrals
with respect to a function, based on a matrix approach. We believe that this is a powerful
tool in both analytical and numerical calculations.We begin with the differential
operator with respect to a function that generates a semigroup. By discretizing this
operator, we obtain a matrix approximation. Importantly, this discretization provides
not only an approximating operator but also an approximating semigroup. This point
motivates our approach, as we then apply Balakrishnan’s representations of fractional
powers of operators, which are based on semigroups.Using estimates of the semigroup
norm and the norm of the difference between the operator and its matrix approximation,
we derive the convergence rate for the approximation of the fractional power of
operators with the fractional power of corresponding matrix operators. In addition, an
explicit formula for calculating an arbitrary power of a two-band matrix is obtained,
which is indispensable in the numerical solution of fractional differential and integral
equations.

Keywords: Feller semigroups, Fractional derivatives with respect to a function, Fractional power of operator, Matrix powers

Publication based on the results of: New directions in applied stochastic analysis (2025)