Fractal differential equations and fractal-time dynamical systems |
| |
Authors: | Abhay Parvate A D Gangal |
| |
Institution: | (1) Department of Physics, University of Pune, 411 007 Pune, India;(2) Centre for Modeling and Simulation, University of Pune, 411 007 Pune, India |
| |
Abstract: | Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous
and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal
subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate
framework. A new calculus calledF
α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF
α-integral andF
α-derivative respectively. TheF
α-integral is suitable for integrating functions with fractal support of dimension α, while theF
α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function
occur naturally as solutions ofF
α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.
We discuss construction and solutions of some fractal differential equations of the form whereh is a vector field andD
F,t
α
is a fractal differential operator of order α in timet. We also consider some equations of the form whereL is an ordinary differential operator in the real variablex, and(t,x) ∈F × Rn whereF is a Cantor-like set of dimension α.
Further, we discuss a method of finding solutions toF
α-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed
back to get those of the former. This is illustrated with a couple of examples. |
| |
Keywords: | Fractal-time dynamical systems fractal differential equations fractal calculus Cantor functions subdiffusion fractal-time relaxations |
本文献已被 SpringerLink 等数据库收录! |
|