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Mathematics 2019, 7(3), 286; https://doi.org/10.3390/math7030286

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

1,* and 2
1
Department of Mathematics, China University of Mining and Technology Beijing, Ding No. 11 Xueyuan Road, Haidian District, Beijing 100083, China
2
School of Science, Shandong Jiaotong University, Jinan 250023, China
*
Author to whom correspondence should be addressed.
Received: 29 December 2018 / Revised: 4 March 2019 / Accepted: 15 March 2019 / Published: 20 March 2019
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PDF [802 KB, uploaded 20 March 2019]

Abstract

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results. View Full-Text
Keywords: variable order fractional derivative; initial value problem; fractional differential equations; piecewise constant functions; approximate solution variable order fractional derivative; initial value problem; fractional differential equations; piecewise constant functions; approximate solution
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Zhang, S.; Hu, L. Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis. Mathematics 2019, 7, 286.

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