Stable monotone iterative solutions to a class of boundary value problems of nonlinear fractional order differential equations

Abstract

We construct sufficient conditions for existence of extremal solutions to boundary value problem (BVP) of nonlinear fractional order differential equations (NFDEs). By combing the method of lower and upper solution with the monotone iterative technique, we construct sufficient conditions for the iterative solutions to the problem under consideration. Some proper results related to Hyers-Ulam type stability are investigated. Base on the proposed method, we construct minimal and maximal solutions for the proposed problem. We also construct and provide maximum error estimates and test the obtain results by two examples.

Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

“…The procedure of AHPM is effective and accurate, as compared with existing analytical techniques. All the figures show that the symmetry or asymmetry of the solutions of the original problems is invariant by using the homotopy deformation Equation (10). All the computation work, tables, and figures are handled by Matlab software.…”

“…Furthermore, when we substitute Equations ( 14) and (15) in Equation (10) and equate like power of p, the obtained series of simpler linear problems are…”

“…In fact, in fractional calculus, many researchers have focused on the various schemes and aspects of partial differential equations and fractional order partial differential equations (FPDEs) as well, see [4][5][6][7][8][9][10][11]. In this regard, various types of techniques have been developed for numerical solutions of non-linear and linear differential equations of integer order.…”

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

“…In our case, due to the definition of the Riemann-Liouville fractional derivative and the lack of regularity of the solutions we are looking for, the arguments used on those references are not suitable for Problem (1). So, we will center our efforts in the construction of the Green's function and in to define operators in suitable spaces where to save the lack of the regularity of the obtained integral kernel.…”

S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations

“…In the recent years, notable contributions have been made to both the theory and applications of fractional differential equations (FDEs) [7,9,44,14,39,40,41,42,45,33,35,2]. For some recent and potential applications of FDEs, we refer to reader the monographs [10,22,3,4,13,15,39,36,37,29,6]. In order to investigate the behaviour and properties of different FDEs, number of researchers are working on their solutions.…”