Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

Abstract

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Carathéodory function and f (t, x, y) is singular at x = 0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.

Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

“…The unknown source term and the temperature distribution for the problem (1)- (3) are given by the series (17) and (16), where the unknowns u 0 (0), u 1n (0), u 2n (0) are calculated from (26), while f 0 , f 1n , f 2n are given by (28)-(30). In the next subsection we will show the uniqueness of the solution.…”

“…Indeed fractional calculus tools have numerous applications in nanotechnology, control theory, viscoplasticity flow, biology, signal and image processing etc, see the latest monographs, [9], [10], [11], [12], [13] articles [14], [15] and reference therein. The mathematical analysis of initial and boundary value problems (linear or nonlinear) of fractional differential equations has been studied extensively by many authors, we refer to [16], [17] , [18], [19], [20] and references therein.…”

Positive solutions for a semipositone fractional boundary value problem with a forcing term

“…The monographs [14,19] and the papers [13,16,18,20] are excellent sources for the theory and applications of fractional calculus. Among all the topics, the existence of positive solutions of BVPs of fractional differential equations has been extensively studied by many researchers in recent years; see, for example, [1,2,3,6,7,8,9,11,21,23] and the references therein. In particular, Goodrich [9] studied the BVP consisting of the equation [9], the author first obtained some properties of the Green's function associated with the problem.…”

Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

“…For the reader interested in such works, we refer to [2,4,7,8]. Of interest to the work presented, we point to research investigating the existence of solutions to fractional boundary value problems [1,6,[9][10][11][12].…”

“…In [1], the authors proved the existence of at least one positive solution to the Dirichlet boundary value problem…”