International Conference on Applications of Mathematics to Nonlinear Sciences. Electron. J. Diff. Eqns., Conference 24 (2017), pp. 47-61.

Differential equations of dynamical order

Andrei Ludu, Harihar Khanal

Abstract:
We introduce a special type of ordinary differential equations
$$ d^{\alpha(t)}x/dt^{\alpha(t)} = f(t,x(t)) $$
whose order of differentiation is a continuous function depending on the independent variable t. We show that such dynamical order of differentiation equations (DODE) can be solved as a Volterra integral equations of second kind with singular integrable kernel. We find the conditions for existence and uniqueness of solutions of such DODE. We present the numeric approach and solutions for particular cases for $\alpha(t) \in (0,2)$ and discuss the asymptotic approach of the DODE solutions towards the classical ODE solutions for $\alpha=1$ and 2.

Published November 15, 2017.
Math Subject Classifications: 34A08, 45G10, 65D30.
Key Words: Dynamical order differential equation; fractional differential equation; Voltera equationl; singular integrable kernel.

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Andrei Ludu
Department of Mathematics
Embry-Riddle Aeronautical University
Daytona Beach, FL, USA
email: ludua@erau.edu
Harihar Khanal
Department of Mathematics
Embry-Riddle Aeronautical University
Daytona Beach, FL, USA
email: harihar.khanal@erau.edu

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