Upper and lower estimates for the separation of solutions to fractional differential equations

Abstract

Given a fractional differential equation of order $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions $$x_1$$ x 1 and $$x_2$$ x 2 , say, of the same differential equation, both of which are assumed to be defined on a common interval [0, T], and provide upper and lower bounds for the difference $$x_1(t) - x_2(t)$$ x 1 ( t ) - x 2 ( t ) for all $$t \in [0,T]$$ t ∈ [ 0 , T ] that are stronger than the bounds previously described in the literature.

Sign of the solutions of linear fractional differential equations and some applications

“…Remark 1.3. Theorem 1.1 was conjectured in 2008 by Diethelm [9] and solved partially therein. However, it was only in 2017 that a complete and correct proof of it was given (see [7] for the historical developments regarding this result).…”

A new approach to shooting methods for terminal value problems of fractional differential equations

“…First of all, we clarify under which conditions the problem is well posed. This question has been discussed and partially solved in [5,8]. A complete analysis is provided in [3] and additional aspects were presented in [9,14].…”

“…Proof. The upper bound is derived in [13,Theorem 5], the lower bound has been shown in [13,Theorem 4].…”

“…The major contribution of our algorithm is a new and advanced method for choosing the iterating guesses for initial values. Traditional approaches [5,7,15] have suggested to proceed by the classical bisection method, thus halving the size of the interval in which the "correct" choice of y(a) can be found in each step. Clearly this scheme is convergent, but it takes a large number of iterations to arrive in a sufficiently small neighbourhood of the exact solution because the size of the containment interval decreases rather slowly.…”

The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Affine Periodic Boundary Value Conditions

“…Due to the past effects of the phenomenon under consideration, fractional differential system can build more accurate and precise models than integer differential systems; therefore, it is widely used in many domains, for instance, physics, biology, chemistry, astronomy, economics, control theory, and ecology. For relevant research on this results, we refer the interested readers to see [1][2][3][4].…”