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

  • Authors:

  • Dumitru Bleanu;Octavian G. Mustafa;Ravi P. Agarwal

  • Affiliations:

  • Çankaya University, Department of Mathematics & Computer Science, ÖÖgretmenler Cad. 14 06530, Balgat-Ankara, Turkey;University of Craiova, DAL, Department of Mathematics & Computer Science, Tudor Vladimirescu 26, 200534 Craiova, Romania;Florida Institute of Technology, Department of Mathematical Sciences, Melbourne, FL 32901, USA

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2011

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Abstract

We establish the long-time asymptotic formula of solutions to the (1+@a)-order fractional differential equation "0^iO"t^1^+^@ax+a(t)x=0, t0, under some simple restrictions on the functional coefficient a(t), where "0^iO"t^1^+^@a is one of the fractional differential operators "0D"t^@a(x^'), ("0D"t^@ax)^'="0D"t^1^+^@ax and "0D"t^@a(tx^'-x). Here, "0D"t^@a designates the Riemann-Liouville derivative of order @a@?(0,1). The asymptotic formula reads as [b+O(1)]@?x"s"m"a"l"l+c@?x"l"a"r"g"e as t-+~ for given b, c@?R, where x"s"m"a"l"l and x"l"a"r"g"e represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation "0^iO"t^1^+^@ax=0, t0.