A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on some coupled systems of the governing model, and a speedier convergence rate was noted.

The present manuscript proposes a computational approach to efficiently tackle a class of two-point boundary value problems that features third-order nonlinear ordinary differential equations. Specifically, this approach is based upon a combination of the shooting method with a modification of the renowned Adomian decomposition method. The approach starts by transforming the governing BVP into two appropriate initial-value problems, and thereafter, solves the resulting IVPs recurrently. In addition, the application of this method to varied test models remains feasible—of course, this is supported by the competing Runge–Kutta method, among others, and reported through comparison plots and tables.

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.

In this paper, we propose a new numerical method for solving system of integro-differential equations featuring Volterra and Fredholm integrals. The proposed method depends on the successful application of the Discrete Adomian Decomposition Method (DADM) to solve highly complicated functional equations coupled with some numerical integration schemes of Trapezoidal and Simpson rules. The scheme is further simulated with the aid of symbolic computation and demonstrated on some test problems. We then carried out error analysis in comparison with some existing methods revealed that the present method is superior due to the high level of accuracy and less computational steps.

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable.

This research aims to present an efficient numerical method to find approximate numerical solutions for ordinary differential equations (ODEs), specifically, the class of two-point boundary value models. This approach is predicated on coupling the shooting method and the decomposition method, which the method is named as Efficient Decomposition Shooting Method (EDSM). Thus, we provide the complete outline of EDSM, and thereafter, apply it to some test models, comprising linear and nonlinear models. Further, the results are then validated by comparing them to the exact solution and other computational methods. Thus, the proposed method performs excellently over its peers, and is found to converge rapidly to the available exact solution with a good precision.