Asymptotic integration of (1+α)-order fractional differential equations

2010/10/22 by Dumitru Baleanu, Baleanu, Dumitru, Octavian G. Mustafa +3
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1010.4675

16 pages

arxiv created 2010/10/22 · arxiv updated 2010/10/25

Abstract

\noindent\bf Abstract We establish the long-time asymptotic formula of solutions to the (1+α)--order fractional differential equation ₀^>i\cal O_t^1+αx+a(t)x=0, t>0, under some simple restrictions on the functional coefficient a(t), where ₀^>i\cal O_t^1+α is one of the fractional differential operators ₀D_t^α(x^′), (₀D_t^αx)^′=₀D_t^1+αx and ₀D_t^α(tx^′-x). Here, ₀D_t^α designates the Riemann-Liouville derivative of order α∈(0,1). The asymptotic formula reads as [a+O(1)]⋅ x_{\scriptstyle small}+b⋅ x_{\scriptstyle large} as t→+∞ for given a, b∈ℝ, where x_{\scriptstyle small} and x_{\scriptstyle large} represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ₀^>i\cal O_t^1+αx=0, t>0.

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