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

Abstract

We studied one essentially nonlinear two–point boundary value problem for a system of fractional differential equations. An original parametrization technique and a dichotomy-type approach led to investigation of solutions of two “model”-type fractional boundary value problems, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called “bifurcation” equations.

Non-local fractional boundary value problems with applications to predator-prey models

“…Similarly to the results in [9,10,12], where authors studied FDSs with the two-point linear and nonlinear boundary conditions, one can prove a uniform convergence of the sequence (4.6) to a parametrized limit function x ∞ (•, ξ, η) and its relation to the exact solution x(t) of the original FBVP (3.1), (3.2). (2) The sequence of functions (4.6) for t ∈ [0, T ] converges uniformly as m → ∞ to a parameter-dependent limit function…”

“…Since the proofs of Theorems 4.1-4.3 overlap with their analogues in [9,12] and do not contain any new techniques, we leave them to the reader. In the next section we modify iterations (4.6) by additionally interpolating them via the Lagrange polynomials, constructed at the Chebyshev nodes.…”

“…Recently a series of new results in approximation of solutions to the nonlinear fractional differential systems (FDS) of the Caputo type was published. Authors suggested a so-called "successive approximations technique" [19] to construct approximate solutions of the studied systems by incorporating also different types of linear [2,8,9,11] and nonlinear boundary conditions [10,12]. This approach demonstrates its effectiveness also when dealing with highly nonlinear systems, systems of a general order p ∈ (m, m + 1) or mixed order equations (see discussions in [3,4,8,9]).…”