Generalized Taylor formulas involving generalized fractional derivatives

TL;DR

This paper develops a generalized Taylor expansion using fractional derivatives, extending classical formulas, with applications in function approximation and solving fractional differential equations.

Contribution

It introduces a new generalized Taylor formula involving fractional derivatives, providing explicit coefficients and error estimates, and demonstrates applications in approximation and differential equations.

Findings

Derived a generalized Taylor expansion involving fractional derivatives.

Provided explicit formulas for coefficients and error bounds.

Applied the expansion to approximate functions and solve fractional differential equations.

Abstract

In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$ \noindent with $m\in \mathbb{N}$, $c_j^{\alpha,\rho}\in \mathbb{R}$, $x>a$ and $0< \alpha \leq 1$. In case $\rho = \alpha = 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{\alpha,\rho}$, $j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the generalized Caputo-type fractional derivatives. Some applications of these results for approximation of functions and for solving some fractional differential equations in series form are given in illustration.