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Generalized Taylor's formula for power fractional derivatives

Hanaa Zitane, Delfim F. M. Torres

Abstract

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor's expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.

Generalized Taylor's formula for power fractional derivatives

Abstract

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor's expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.
Paper Structure (5 sections, 8 theorems, 42 equations, 1 figure)

This paper contains 5 sections, 8 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary definitions
  3. Taylor's formula via power fractional derivatives
  4. Application: Approximation of functions
  5. Conclusion

Key Result

Lemma 1

The power fractional derivative ${}^p\!D_{a,\omega}^{\alpha,\beta}$ can be expressed as follows: This series converges locally and uniformly in $t$ for any $a$, $\alpha$, $\beta$, $p$, $\omega$ and $f$, verifying the conditions laid out in Definition PowerDerivative.

Figures (1)

  • Figure 1: The function $f(t)=\sin(t)$ and the corresponding Taylor polynomials ${}^p\!A^{\alpha,\beta}_{n}(t)$ of order $n=1,2,3$, centered at $t = 0$, for $\alpha=0.1$, $\beta=1.5$ and different values of $p$.

Theorems & Definitions (28)

  • Definition 1: See PowerDerivative
  • Remark 1
  • Definition 2: See PowerDerivative
  • Remark 2
  • Definition 3: See PowerDerivative
  • Remark 3
  • Lemma 1
  • proof
  • Remark 4
  • Proposition 1
  • ...and 18 more