Next Article in Journal
Classification of Two Dimensional Cellular Automata Rules for Symmetric Pattern Generation
Previous Article in Journal
SDAE-BP Based Octane Number Soft Sensor Using Near-infrared Spectroscopy in Gasoline Blending Process
Previous Article in Special Issue
New Stability Criterion for the Dissipative Linear System and Analysis of Bresse System
Article Menu
Issue 12 (December) cover image

Export Article

Open AccessArticle
Symmetry 2018, 10(12), 771; https://doi.org/10.3390/sym10120771

Nonlocal Symmetries for Time-Dependent Order Differential Equations

Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA
Received: 5 November 2018 / Revised: 12 December 2018 / Accepted: 17 December 2018 / Published: 19 December 2018
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
Full-Text   |   PDF [5988 KB, uploaded 19 December 2018]   |  

Abstract

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed. View Full-Text
Keywords: variable order derivative; fractional differential equation; Voltera equation; singular integrable kernel variable order derivative; fractional differential equation; Voltera equation; singular integrable kernel
Figures

Figure 1

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).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Ludu, A. Nonlocal Symmetries for Time-Dependent Order Differential Equations. Symmetry 2018, 10, 771.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top