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In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new generalized result to study the existence of solutions of nonlinear fractional boundary value problems (FBVPs). The proposed results unify the existence criteria of certain FBVPs including periodic and antiperiodic as special cases that have been previously studied separately in the literature. The method we employ is topological in its nature and manifests themselves in the forms of differential inequalities (lower and upper solutions, and coupled lower and upper solutions (CLUSs)). Two examples are given to demonstrate the applicability of the developed theoretical results. [ABSTRACT FROM AUTHOR]

In the present study, we introduce two new operational matrices of fractional Legendre function vectors in the sense of generalized Caputo-type fractional derivative and generalized Riemann–Liouville-type fractional integral operators. The derivative and integral operational matrices developed in the sense of Caputo and Riemann–Liouville operators are special cases of our proposed generalized operational matrices for β , η = 1. Then, we present a numerical method that is dependent on the generalized derivative and integral operational matrices. The applicability and accuracy of the presented method is tested by solving various problems and then comparing the results obtained otherwise by using various numerical methods including spectral collocation methods, spectral Tau method, stochastic approach, and Taylor matrix approach. Moreover, our presented method transforms the problems into Sylvester equations that are easily solvable by using MATLAB or MATHEMATICA. We believe that the newly derived generalized operational matrices and the presented method are expected to be further used to formulate and simulate many generalized Caputo-type fractional models. [ABSTRACT FROM AUTHOR]

In this article, the numerical method based on operational matrices of fractional order derivatives and integrals in the Caputo and Riemann-Liouville senses of two-parametric orthogonal shifted Jacobi polynomials is proposed for studying the approximate solutions for a generalized class of fractional order partial differential equations. The technique is extended herein to generalized classes of fractional order coupled systems having mixed partial derivatives terms. One salient aspect of this article is the development of a new operational matrix for mixed partial derivatives in the sense of Caputo. Validity of the method is established by comparing our simulated results with literature solutions obtained otherwise, yielding negligible errors. Furthermore, as a result of the comparative study, some results presented in the literature are extended and improved in the investigation herein. [ABSTRACT FROM AUTHOR]