On the Cauchy–Nicoletti Type Two-Point Boundary-Value Problem for Fractional Differential Systems

Abstract

We deal with a system of quasilinear fractional differential equations, subjected to the Cauchy-Nicoletti type boundary conditions. The task of explicit solution of such problems is difficult and not always solvable. Thus we suggest a suitable numerical-analytic technique that allows to construct an approximate solution of the studied fractional boundary value problem with high precision.

Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

“…The parametrization (9) reduces the study of the original FBVP (6) and (7) with nonlinear boundary conditions on the full interval [a, b] to the investigation of two decomposed problems (10), ( 11) and (12), ( 13), defined on the half intervals a, a+b 2 and a+b 2 , b , respectively. This approach allows to diminish some values in the qualitative analysis of the given FBVP and to essentially improve the estimates of the iteration schemes, presented in the coming sections (see also discussions in [6][7][8][9][10]).…”

“…In this section, we formulate some auxiliary lemmas, needed later on. In terms of fractional integrals, they were first proven in [6] for the interval [0, T] and later generalized over the interval [a, b] (see discussion in [10]). Then, for all t ∈ [a, b], the following estimate is true:…”

“…Lemma 2. [10] Let {α m (•, τ, I)} m∈N be a sequence of continuous functions at the interval [a, b] given by:…”

“…The novel technique suggested in this paper has never been applied in study of FBVPs. It allowed us to improve and essentially sharpen the estimates obtained in [6][7][8][9][10]. Along with the other well-known approximate methods, dealing with the fractional differential equations and their systems (see discussions in [17][18][19][20][21]), the aforementioned approach complements the fundamental study of such essentially nonlinear problems.…”

“…In the current paper, we provide a new view on the successive approximations approach [5], recently used for the study of FBVPs for periodic, Cauchy-Nicoletti-type, and interpolation boundary conditions (see [6][7][8][9][10]). An original parametrization technique, initially suggested for the reduction of nonlinearities in boundary restrictions (see discussions [11,12]), and a dichotomy-type approach (see results in [13][14][15][16]) led to the investigation of solutions of two "model"-type FBVPs, containing some artificially introduced parameters.…”