In this research study, we establish some necessary conditions to check the uniqueness-existence of solutions for a general multi-term $\psi$-fractional differential equation via generalized $\psi$-integral boundary conditions with respect to the generalized asymmetric operators. To arrive at such purpose, we utilize a procedure based on the fixed-point theory. We follow our study by suggesting two numerical algorithms called the Dafterdar-Gejji and Jafari method (DGJIM) and the Adomian decomposition method (ADM) techniques in which a series of approximate solutions converge to the exact ones of the given $\psi$-RLFBVP and the equivalent $\psi$-integral equation. To emphasize for the compatibility and the effectiveness of these numerical algorithms, we end this investigation by providing some examples showing the behavior of the exact solution of the existing $\psi$-RLFBVP compared with the approximate ones caused by DGJIM and ADM techniques graphically.

This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term fractional coupled system of Navier equations under the fractional boundary conditions. The existence theory is studied regarding solutions of the given coupled sequential Navier boundary problems via the Krasnoselskii’s fixed-point theorem on two nonlinear operators. Moreover, the Banach contraction principle is applied to investigate the uniqueness of solution. We then focus on the Hyers–Ulam-type stability of its solution. Furthermore, the approximate solutions of the proposed coupled fractional sequential Navier system are obtained via the generalized differential transform method. Lastly, the results of this research are supported by giving simulated examples.

In this paper, a new kind of mathematical modeling is studied by providing a five-compartmental system of differential equations with respect to new hybrid generalized fractal-fractional derivatives. For the first time, we design a model of giving up smoking to analyze its dynamical behaviors by considering two parameters of such generalized operators; i.e., fractal dimension and fractional order. We apply a special sub-category of increasing functions to investigate the existence of solutions. Uniqueness property is derived by a standard method based on the Lipschitz rule. After proving stability property, the equilibrium points are obtained and asymptotically stable solutions are studied. Finally, we illustrate all analytical results and findings via numerical algorithms and graphs obtained by Lagrangian piece-wise interpolation, and discuss all behaviors of the relevant solutions in the fractal-fractional system.

This paper deals with the stability results for solution of a fractional integro-differential problem with integral conditions. Using the Krasnoselskii’s, Banach fixed point theorems, we proof the existence and uniqueness results. Based on the results obtained, conditions are provided that ensure the generalized Ulam stability of the original system. The results are illustrated by an example.

This paper deals with the existence and uniqueness results for the solution of a Neutral fractional integro-differential problem with nonlocal conditions. Using the Nonlinear alternative for single valued maps, Krasnoselskii's and Banach fixed point theorems to proof our main results. An example is given to illustrate our main results.

This work is reserved for the study of a special category of boundary value problems ( BVP s) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. This fractional model and its boundary value conditions (BVCs) involve different simple BVP s, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. The Banach principle gives a unique solution for this Liouville–Caputo BVP . Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view.