New Iterative Method Research Articles

A Comparative Analysis of Two Semi Analytic Approaches in Solving Systems of First-Order Differential Equations

The resolution of systems of first-order ordinary differential equations (ODEs) stands as a pivotal pursuit with extensive implications across scientific and engineering domains. In tackling this fundamental task, this study undertakes a rigorous comparative assessment of two semi-analytic methodologies, the Variational Iterative Method (VIM) and the New Iterative Method (NIM). Motivated by the need to address a critical research gap, our investigation delves into these approaches' relative merits and demerits. Firstly, it conducts a comprehensive examination of VIM, a well-established method, juxtaposed with NIM, a relatively unexplored approach, to uncover their comparative strengths and limitations. Secondly, the study contributes to the existing knowledge in numerical methods for ODEs by shedding light on essential performance characteristics, including convergence properties, computational efficiency, and accuracy, across a diverse array of ODE systems. Through meticulous numerical experimentation, we not only reveal practical insights into the efficacy of VIM and NIM but also bridge a significant knowledge gap in the field of numerical ODE solvers. Our findings highlight VIM as the more effective method, thus advancing our understanding of semi-analytic approaches for solving ODE systems and furnishing valuable guidance for practitioners and researchers in selecting the most suitable method for their specific applications

Innovative Solutions for the Kadomtsev–Petviashvili Equation via the New Iterative Method

This research paper presents a new iterative method (NIM) for obtaining the solution to the potential Kadomtsev–Petviashvili (PKP) equation. NIM is a promising approach to solving complex mathematical problems, and its effectiveness and efficiency are highlighted through its application to the PKP equation. The results obtained through the use of NIM are compared to the exact solutions of the PKP equation, and it is found that the NIM approach provides results that are in close agreement with the exact solutions. This demonstrates the utility and accuracy of NIM and makes it a valuable tool for solving similar mathematical problems in the future. Furthermore, the lack of discretization in the NIM approach makes it a more convenient method for solving the PKP equation compared to traditional approaches that require discretization. Overall, the findings of this research paper suggest that NIM is a highly effective and convenient method for obtaining approximate analytical solutions to complex mathematical problems, such as the 2+1-dimensional PKP equation.

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis.

An evaluation of multispecies population dynamics models through numerical simulations using the new iterative method

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.

Solitary Pattern Solutions of gBBM Equation using New Iterative Method (NIM)

In this study, we focus on the solitary pattern solutions of the generalized Benjamin-Bona-Mahony equations (gBBM) and propose a new iterative method (NIM) for their numerical solution, given suitable initial conditions. Our proposed NIM approach generates numerical solutions in the form of a convergent power series with computationally simple components. Our results demonstrate that the NIM approach exhibits exceptional performance in terms of accuracy, efficiency, simplicity, stability, and reliability.

On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method

Abstract This investigation explores two numerical approaches: the optimal auxiliary function method (OAFM) and the new iterative method (NIM). These techniques address the physical fractional-order Klein-Gordon equations (FOKGEs), a class of partial differential equations (PDEs) that model various physical phenomena in engineering and diverse plasma models. The OAFM is a recently introduced method capable of efficiently solving several nonlinear differential equations (DEs), whereas the NIM is a well-established method specifically designed for solving fractional DEs. Both approaches are utilized to analyze different variations in FOKGE. By conducting numerous numerical experiments on the FOKGE, we compare the accuracy, efficiency, and convergence of these two proposed methods. This study is expected to yield significant findings that will help researchers study various nonlinear phenomena in fluids and plasma physics.

Comparative analysis of the fractional order Cahn-Allen equation

This current work presents a comparative study of the fractional-order Cahn-Allen (CA) equation, where the non-integer derivative is taken in the Caputo sense.The Cahn-Allen equation is an equation that assists in the comprehension of phase transitions and pattern formation in physical systems. This equation describes how different phases of matter, such as solids and liquids, change and interact throughout time. We employ two analytical methods: the Laplace Residual Power Series Method (LRPSM) and the New Iterative Method (NIM), to solve the proposed model. The LRPSM is a combination of the Laplace Transform and the Residual Power Series Method, while the New Iterative Method is a modified form of the Adomian Decomposition Method that does not require any type of polynomial or digitization. For the purpose of accuracy and reliability, we compare our findings with other methods and the exact solution used in the literature. Additionally, 2D and 3D plots are generated for various fractional order values denoted as p. These plots illustrate that as the fractional order p approaches 1, the graph of the approximate solution gradually coincides with the graph of the exact solution.

A New Extension of Optimal Auxiliary Function Method to Fractional Non-Linear Coupled ITO System and Time Fractional Non-Linear KDV System

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions.

Numerical Analysis of Nonlinear Coupled Schrödinger–KdV System with Fractional Derivative

In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and using residual correction to improve the accuracy of the solution. The second method is a new iterative method (NIM) that simplifies the problem and obtains a recursive formula for the solution. Both methods are applied to the Schrödinger–KdV system with fractional derivatives, which arises in many physical applications. Numerical experiments are performed to compare the accuracy and efficiency of the two methods. The results show that both methods can produce highly accurate solutions for the fractional Schrödinger–KdV system. However, the new iterative method is more efficient in terms of computational time and memory usage. Overall, our study demonstrates the effectiveness of the residual power series method and the new iterative method in solving fractional-order Schrödinger–KdV systems and provides a valuable tool for researchers and practitioners in applied mathematics and physics.

Solving One Species Lotka–volterra Equation by the New Iterative Method (NIM)

In this paper, we investigate the use of the new iterative method, referred to as the NIM, for solving the one species LotkaVolterra equation. This equation, which describes the dynamics of populations in ecological systems, has been widely studied in the field of mathematical biology. However, finding an analytical solution to this equation can be challenging. To overcome this, we propose using the NIM as an alternative method for solving the equation. To demonstrate the effectiveness of the NIM, we conduct a comparative study between it and other wellestablished techniques such as the differential transformation method (DTM), the variational iteration method (VIM), and the Adomian decomposition method (ADM). Through numerical simulations, we show that the NIM is able to accurately and efficiently solve the one species LotkaVolterra equation, making it a promising tool for researchers in the field of mathematical biology.

Closed-Form Solutions for Cauchy-Euler Differential Equations through the New Iterative Method (NIM)

Closed-Form Solutions for Cauchy-Euler Differential Equations through the New Iterative Method (NIM)

Improving Numerical Solutions for the Generalized Huxley Equation: The New Iterative Method (NIM)

Improving Numerical Solutions for the Generalized Huxley Equation: The New Iterative Method (NIM)

Application of the New Iterative Method (NIM) to the Generalized Burgers–Huxley Equation

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, such as Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), and the exact solution, and found that it is efficient and easy to use. NIM has the advantage of quick convergence, easy implementation, and handling of a wide range of initial conditions. The comparison of the present symmetrical results with the existing literature is satisfactory.

Accurate Numerical Treatment on a Stochastic SIR Epidemic Model with Optimal Control Strategy

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.

Numerical Treatment on a Chaos Model of Fluid Flow Using New Iterative Method

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.

A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

This article presents a new iterative method (NIM) for the investigation of the approximate solution of the Klein-Gordon and sine-Gordon equations. This approach is formulated on the combination of the Mohand transform and the homotopy perturbation method. Mohand transform (MT) is capable to handle the linear terms only, thus we introduce homotopy perturbation method (HPM) to tackle the nonlinear terms. This NIM derives the results in the form of a series solution. The proposed method emphasizes the stability of the derived solutions without any linearization, discretization, or hypothesis. Graphical representation and absolute error demonstrate the efficiency and authenticity of this scheme. Some numerical models are illustrated to show the compactness and reliability of this strategy.

The Integral Iterative Method for Approximate Solution of Newell-Whitehead-Segel Equation

In this paper, the Newell-Whitehead-Segel (NWS) equation is solved using the integral iterative method (IIM) to determine the accuracy and effectiveness of the method. Comparison of results obtained by IIM with the exact solution and other existing results obtained by other methods such as new iterative method (NIM), Adomian decomposition method (ADM) and Laplace Adomian decomposition method (LADM) revealed the accuracy and effectiveness of the method. The approximation results obtained by IIM is comparable with the others. IIM is reliable and easier in solving the nonlinear problems since this method is simple, straightforward and does not require calculating multiple integral and demand less computational work.

A Modified Iterative Algorithm for Numerical Investigation of HIV Infection Dynamics

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.

Applications of New Iterative Method to fractional non linear coupled ITO system

In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The results obtained by the proposed method are compared with that of Homotopy Perturbation Method (HPM). It is shown that the proposed method is accurate for strongly nonlinear fractional coupled system of PDEs.

Numerical Investigation of Fractional‐Order Kawahara and Modified Kawahara Equations by a Semianalytical Method

In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractional‐order Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractional‐order and the well‐known approach of least squares. Fractional‐order derivatives are taken in the Caputo sense with numerical values in the closed interval [0, 1]. The suggested method is directly applied to fractional‐order Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature.