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Symmetry 2018, 10(12), 771;

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)
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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

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

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