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

• Published in: Computers & mathematics with applications (1987) 2011-08, Vol.62 (3), p.1492-1500
• Main Authors: Băleanu, Dumitru, Mustafa, Octavian G., Agarwal, Ravi P.
• Format: Article
• Language: English
• Subjects: Asymptotic integration; Asymptotic properties; Constrictions; Derivatives; Differential equations; Linear fractional differential equation; Mathematical analysis; Mathematical models; Operators; Solution space
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2011.03.021

We establish the long-time asymptotic formula of solutions to the ( 1 + α ) -order fractional differential equation 0 i O t 1 + α x + a ( t ) x = 0 , t > 0 , under some simple restrictions on the functional coefficient a ( t ) , where 0 i O t 1 + α is one of the fractional differential operators 0 D t α ( x ′ ) , ( 0 D t α x ) ′ = 0 D t 1 + α x and 0 D t α ( t x ′ − x ) . Here, 0 D t α designates the Riemann–Liouville derivative of order α ∈ ( 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 i O t 1 + α x = 0 , t > 0 .