Analysis and numerical methods for fractional differential equations with delay

A Modified Method for Solving Delay Differential Equations of Fractional Order

“…The Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator which is proposed by Caputo [9]. …”

“…First we convert the fractional order delay differential equation to fractional order non-delay differential equation by applying the method of steps [13], as …”

“…The most famous of these definitions that have been popularized in the world of fractional calculus are Riemann-Liouville and Grünwald-Letnikov definition. Also, Caputo, [9] reformulated the more "classic" definition of the Riemann-Liouville fractional derivative in order to use integer order initial conditions to solve his fractional order differential equations.…”

“…Although it seems natural to model certain processes and systems in engineering and other sciences with this kind of equations, only in the last few years has the attention of the scientific community been devoted to them [11], [12], [13]. This paper is organized as follows: In section 2 we recall the definitions of fractional derivatives and fractional integration, in section 3, a review of generalized Hat functions and their properties is described.…”

The controllability of nonlinear implicit fractional delay dynamical systems

“…Recently, Balachandran and Divya (2014) studied the controllability of nonlinear implicit fractional integrodifferential systems. Analysis and numerical methods for a fractional differential equation with delay were studied by Morgado et al (2013). The application of this equation was discussed by Bhalekar and Gejji (2010) or Bhalekar et al (2011).…”