We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed.

In this study, we introduced the $\psi$-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, $\psi$, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the $\psi$-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various $\psi$-fractional differential equations.

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling.

This study develops a highly convergent computational method, the ψ-Laplace Adomian Decomposition Method (ψ-LADM), for solving coupled systems ofψ-Caputo Fractional Differential Equations (FDEs). The effectiveness of the proposed method has been assessed using various numerical test examples, including a real-world application for atmospheric convection models utilizing Lorenz chaotic dynamical systems. Notably, the method consistently produced solutions that matched the true solutions of the governing models. In the case of the Lorenz chaotic system, the obtained solutions accurately portrayed the characteristic phase portraits of a true chaotic system.

The main object of this paper is to investigate the existence of solutions for a self-adjoint coupled system of nonlinear second-order ordinary differential equations equipped with nonlocal multi-point coupled boundary conditions on an arbitrary domain. We apply the Leray–Schauder alternative, the Schauder fixed point theorem and the Banach contraction mapping principle in order to derive the main results, which are then well-illustrated with the aid of several examples. Some potential directions for related further researches are also indicated.

The Bagley-Torvik equation is an imperative differential equation that considerably arises in various branches of mathematical physics and mechanics. However, very few methods exist for the treatment of the model analytically; in fact, researchers frequently shop for semi-analytical and numerical methods in their studies. Therefore, the main goal of this research is to find theexact analytical solution for the fractional Bagley-Torvik equation fitted with Dirichlet boundary data, as well as a system of fractional Bagley-Torvik equations. Thus, this research aims to show that the modified Adomian decomposition method (MADM) via the proposed two algorithms is a very effective method for treating a class of Bagley-Torvik equations endowed with Dirichletboundary data. Certainly, MADM is a very powerful approach for solving dissimilar functional equations without the need for either linearization, discretization, perturbation, or even unnecessary restraining postulations. Additionally, the method reveals exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible. Lastly, some illustrative test problems of the governing model are examined to demonstrate the superiority of the proposed algorithms.