The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced.

In this paper, we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schrödinger‐Hirota equation. The spatio‐temporal dispersion term is included, in addition to group velocity dispersion Kerr law of nonlinearity are studied. A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method (IADM) and further sought for some analytical results for validation. The scheme is shown to be efficient and possessed high level of accuracy as demonstrated.

Classical Burgers’ equation is an indispensable dynamical evolution equation that is autonomously devised by Burgers and Harry Bateman in 1915 and 1948, respectively. This important model is featured through a nonlinear partial differential equation (NPDE). Furthermore, the model plays a crucial role in many areas of mathematical physics, including, for instance, fluid dynamics, traffic flow, nonlinear acoustics, turbulence phenomena, and linking convection and diffusion processes to state a few. Thus, in the present study, an efficient analytical approach by the name ‘generalized Riccati equation approach’ is adopted to securitize the class of (2+1)-dimensional Burgers’ equations by revealing yet another set of analytical structures to the governing single and vector-coupled Burgers’ equations. In fact, the besieged method of the solution has been proven to divulge various sets of hyperbolic, periodic, and other forms of exact solutions. Moreover, the method first begins by transforming the targeted NPDE to a nonlinear ordinary differential equation (NODE), and subsequently to a set of an algebraic system of equations; where the algebraic system is then solved simultaneously to obtain the solution possibilities. Lastly, certain graphical illustrations in 2- and 3-dimensional plots are set to be depicted - featuring the evolutional nature of the resulting structures, and thereafter, analyze the influence of the Reynolds number Ra on the respective wave profiles.

The current manuscript investigates by proposing new numerical schemes based on the Adomian's technique for the resolution of the dark and singular solutions of the Chen-Lee-Liu (CLL) equation. More precisely, the schemes are derived from the Wazwaz's modification of the Adomian's method and the improved Adomian's method for treating complex-valued evolution equations. The CLL model is applicable to a variety of applications including photonic and optical crystal fibers. The schemes which are implemented via the help of theMaplesoftware have many salient advantages as contained in the comparative analysis. Finally, we depict certain results graphically together with some supportive tables, in addition to some comprehensive remarks.

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.