Matrix approach to the fractional calculus

In this paper, we introduce the 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 correspondings 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.

💡 Research Summary

This paper introduces a novel and unified framework for fractional calculus, termed the “matrix approach,” which bridges operator theory and numerical analysis. The core idea is to construct fractional derivatives and integrals with respect to a function, d/dg, by approximating the underlying differential operator with a finite-difference matrix and then computing fractional powers of that matrix.

The work begins with the operator A = d/dg, whose discretization on a uniform grid leads to a specific two-band, lower triangular Toeplitz matrix A_n. A key insight is that this matrix approximation not only represents the operator A but also serves as an approximation to the semigroup T_t = exp(tA) generated by A. This connection is crucial because it allows the application of Balakrishnan’s seminal formula for fractional powers of operators, which is defined through an integral involving the semigroup.

The paper provides an illustrative example using the standard derivative d/dx. It explicitly computes the real power A_n^α of the finite-difference matrix, showing that for natural α, it recovers classical finite-difference formulas, and for non-integer α, it converges to the Grünwald-Letnikov fractional derivative/integral formulas. This validates the matrix approach as a generalization of classical methods.

A significant technical contribution is the derivation of an explicit formula for computing any natural power of a general two-band matrix, utilizing complete homogeneous symmetric polynomials. This groundwork is essential for handling more complex operators.

The main theoretical result is a convergence rate estimate for the approximation of fractional powers. The paper establishes bounds for the error ||(-A)^α - (-A_n)^α|| in terms of two quantities: the error in approximating the semigroup (sup_t ||T(t) - T_n(t)||) and the error in approximating the generator itself (||A - A_n||). This estimate provides a rigorous quantitative guarantee for the numerical scheme, linking the discretization step size to the accuracy of the fractional power approximation.

The theory is then applied to the target operator d/dg. Under conditions where g is strictly increasing with a derivative bounded above and away from zero, it is shown that the matrix discretization A_n provides a good approximation for both the operator and its semigroup. Consequently, the fractional powers A_n^α converge to the true fractional operator (d/dg)^α with a quantifiable rate.

Finally, the paper highlights the practical utility of the explicit formula for A_n^α, which enables the direct numerical construction of fractional differential and integral operators. This approach offers an alternative to existing methods for discretizing fractional calculus, unifying analytical definitions with computationally feasible matrix operations. The matrix approach thus presents a powerful tool for both the theoretical study and the numerical solution of fractional differential and integral equations involving derivatives with respect to functions.