Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

Delving Deeper: Growth Rates and the Marvelous Geometric Sequence

Students are often dazzled by the prodigious growth rate of the geometric sequence gn = 2n and the geometric series whose partial sums are Sn = 1 + 2 + 4 + 8 + … + 2n−1 = 2n − 1. Teachers sometimes note that the geometric sequence is the discrete “form” of an exponential function, which is characterized by very rapid growth. In particular, exponential functions grow faster than polynomial functions. A rigorous explanation of this claim is left to the calculus class in which students examine the relative growth rates of functions by using L'Hopital's rule. However, even by using tools developed in algebra and precalculus, teachers can explain to their students (or lead them to a justification) that the geometric sequence gn = 2n grows faster than any polynomial sequence. These tools are the binomial theorem, Pascal's triangle, mathematical induction, and an understanding of the end behavior of exponential and polynomial functions (i.e., what happens to the graphs as x approaches infinity).