Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

Christopher S. Goodrich; Benjamin Lyons; Mihaela T. Velcsov

doi:10.3934/cpaa.2020269

Article Contents

2021, Volume 20, Issue 1: 339-358. Doi: 10.3934/cpaa.2020269

This issue Previous Article Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional Next Article Isomorphism between one-dimensional and multidimensional finite difference operators

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

1.
School of Mathematics and Statistics, UNSW Sydney, NSW 2052 Australia
2.
Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114 USA
3.
Department of Mathematics, University of Nebraska-Omaha, Omaha, NE 68182-0243 USA

^* Corresponding author
^* Corresponding author

Received: March 31, 2020

Revised: July 31, 2020

Early access: November 2020

Published: January 2021

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Abstract

We investigate the relationship between the sign of the discrete fractional sequential difference $ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $ and the monotonicity of the function $ t\mapsto f(t) $. More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound

$ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $

for some $ \varepsilon>0 $. We prove that even though the fractional difference can be negative, the monotonicity of the function $ f $, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

Keywords:

Mathematics Subject Classification: Primary: 26A48, 33F05, 39A12, 65D15, 65Q20; Secondary: 39A99, 39B62.

Citation:

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Figure 1. Graphical representation of the set $ \mathscr{E}_{k, 0.1} $ for $ k \le 72 $

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Figure 2. Graphical representation of the set $ \mathscr{E}_{k, 0.01} $ for $ k \le 72 $

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Figure 3. Graphical representation of the set $ \mathscr{E}_{k, 0.001} $ for $ k \le 72 $

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Figure 4. Graphical representation of the set $ \mathscr{E}_{k, 0.0001} $ for $ k \le 72 $

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Figure 5. Graphical representation of the set $ \mathscr{E}_{k, 0.00001} $ for $ k \le 72 $

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Figure 6. Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 0.01, 0.001, 0.0001, 0.00001, 0.000001, 0.0000001 $. The cardinality increases from small (dark blue) to large (dark red) and the actual cardinalities are shown along the sidebar of each subplot

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Figure 7. Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 1/100, 1/150, 1/400, 1/650, 1/900, 1/1000 $. These correspond to the interval of $ \varepsilon $ reflected in the top two subplots of Figure 6. Notice the change of cardinality values as $ \varepsilon $ decreases

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