Abstract Fractional Monotone Approximation with Applications

George A Anastassiou

doi:10.3390/fractalfract5040158

Abstract

Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f∈Cp−1,1, p≥0 and let L be a linear abstract left or right fractional differential operator such that Lf≥0 over 0,1 or −1,0, respectively. We can find a sequence of polynomials Qn of degree ≤n such that LQn≥0 over 0,1 or −1,0, respectively. Additionally f is approximated quantitatively with rates uniformly by Qn with the use of first modulus of continuity of fp.

Highlights

Shisha continued this study by replacing the kth derivative with a linear differential operator of order k involving ordinary derivatives, again the approximation was with rates
We perform abstract fractional calculus, left and right monotone approximation theory of Caputo type, and we apply our results to Prabhakar fractional
There, the approximating polynomial Qn depends on f, ηn, h; which ηn depends on n, R p, n, k, s j, λ j ; which λ j depends on k j

Summary

Introduction

Shisha continued this study by replacing the kth derivative with a linear differential operator of order k involving ordinary derivatives, again the approximation was with rates. [4] (see chapters 1–8) went a step further, by starting the fractional monotone approximation, in that the linear differential operator is a fractional one, involving left or right side Caputo fractional derivatives. We define the left side Caputo fractional derivative of f of order α as follows: α. D∗−1 f stands for the left Caputo fractional derivative of f of order α j anchored at −1. Consider the linear left fractional differential operator k. We perform abstract fractional calculus, left and right monotone approximation theory of Caputo type, and we apply our results to Prabhakar fractional.

Abstract Fractional Calculus

About Prabhakar Fractional Calculus

From Generalized Non-Singular Fractional Calculus

Main Results

Conclusion