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.

Based on the application of the standard Adomian method, the current paper proposes two modification approaches for the classes of the fractional differential equations and the system of fractional differential equations, which are featured through initial-value problems. Certainly, the constructed iterative schemes for the two classes are shown to be reliable, considering a number of test problems for demonstration, and upon deploying other existing numerical approaches for contrasting. In fact, the proposed schemes are found to portray less error, rapidity, accuracy and consume less computational time among others.

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.

In this work, we investigate the existence of solution for some nonlinear singular problem of Kirchhoff type involving Riemann-Liouville Fractional Derivative and the p-Laplacian operator. The main tools are based on the variational method, precisely, we use the minimisation of the corresponding functional in a suitable fractional spaces. Our main result significantly complement and improves the previous ones due to [6] and [31].

In this work, we establish the existence of three solutions (one of them is positive, one is negative and the other one has non-constant sing) for some fractional boundary value problems involving the $ \psi $-Riemann-Liouville derivative and the $ p $-Laplace operator. More precisely, we use the variational method to prove that the associated functional energy satisfies the Palais-smale condition, and has a critical point in three-manifold sets. In my opinion, this is the first time that the signs of three existing solutions have been studied for a problem involving the $ \psi $-Riemann-Liouville derivative.

This work studies a novel class of nonlinear multi-term fractional q-difference equations with nonlocal boundary conditions. To utilize the fixed-point tools, the given problem is transformed into an equivalent fixed-point problem. The Leray–Schauder alternative is used to demonstrate the existence of solutions to the considered problem, and Banach’s fixed-point theorem is used to establish the uniqueness of the solutions.