Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

This article treats analytically and numerically to the three dimensional Rössler system. The governing equations of the problem are derived from traditional Lorentz system of fluid mechanics to nonlinear ordinary differential equations (ODEs) for modelling. A semi-analytic solution is developed by using New Iterative Method (NIM) whereas the numerical solution is presented by Runge-Kutta order four (RK4) scheme. A comparative study of the analytical and numerical solutions are made. The results confirm clearly that the two methods coincide closely for both the chaotic and non-chaotic cases of the determined system. This observation would be helpful to apply NIM on nonlinear problems of fluid flow with different fluid parameters in future.

This study explores the multispecies Lotka-Volterra population dynamics models, a captivating nonlinear mathematical framework with significant applications in natural sciences and environmental studies. The primary objective is to deliver precise solutions for these models using the New Iterative Method (NIM). Numerical simulations are conducted on three distinct types of nonlinear dynamic problems , comparing the accuracy of the NIM with that of the Perturbation Iteration Algorithm (PIA), existing exact solutions, and the traditional fourth-order Runge–Kutta method. A continuous step time of Δ = 0.001 was used for the Runge–Kutta method in all computations. Notably, the NIM's solutions for the nonlinear multispecies Lotka-Volterra models demonstrate very good accuracy, achieving convergence to the Runge–Kutta method's solutions within five iterations. The correctness of the NIM is found to be better than the other existing solutions. Its distinctive attribute lies in its computational efficiency, providing high accuracy without necessitating linearization, discretization, multipliers, or polynomials for nonlinear terms. This leads to simpler solution procedures while maintaining commendable accuracy. The findings underscore NIM's reliability and broad applicability in both linear and nonlinear models , highlighting its potential as an invaluable tool in numerical computation.

The human immunodeficiency virus (HIV) mainly attacks CD4⁺ T cells in the host. Chronic HIV infection gradually depletes the CD4⁺ T cell pool, compromising the host’s immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4⁺ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge–Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs.

In this paper, a numerical study has been undertaken on the susceptible-infected-recovered (SIR) epidemic model that encompasses the mechanisms of the evolution of disease transmission; a prophylactic vaccination strategy in the susceptible populations, depending on the infective individuals. We furnish numerical and graphical simulation combined with explicit series solutions of the proposed model using the New Iterative Method (NIM) and Modified New Iterative Method (MNIM). The analytic-numeric New Iterative Method failed to deliver accurate solution for the large time domain. A new reliable algorithm based on NIM, the coupling of the Laplace transforms, and the New Iterative method is called Modified New Iterative Method (MNIM) which is presented to enhance the validity domain of NIM techniques. The convergence analysis of the MNIM has also been illustrated. The simulation results show that the vaccination strategy can slow down the spread of the epidemic rapidly. Numerical results illustrate the excellent performance of the MNIM and show that the modified method is much more accurate than the NIM.