On One Interpolation Type Fractional Boundary-Value Problem

Abstract

We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called “interpolation” type boundary restrictions. Under certain conditions, the aforementioned problem is simplified via a proper parametrization technique, and with the help of the numerical-analytic method, the approximate solutions are constructed.

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

“…Using relation (9), we reduce the study of the original problem ( 6) and (7) to an investigation of solutions of two decomposed "model" problems with separated linear boundary conditions, dependent on parameters:…”

“…with z ∈ D a , λ ∈ D a+b 2 , and η ∈ D b defined by (9). In addition, we connect to the parametrized FBVPs (10)-( 13) the correspondent sets:…”

“…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]).…”

“…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.…”