Construction and applications of iterative methods for finding approximate solutions of nonlinear equations having unknown zeros of multiplicity with fractal geometry and dynamical behavior.

In this paper, we have applied two semi-analytical techniques: the Mohand variational iteration method (MVIM) and the q-homotopy Mohand transform method (q-HMTM) to derive approximate solutions of fractional-order nonlinear partial differential equations. Particularly, time-fractional FitzHugh Nagumo equation and Fisher equation based on Caputo derivative are explored. Both of them utilize well the features of Mohand transform and fractional calculus to represent the nonlocal and memory-dependent nature of the models. The validity and reliability of the suggested methods are justified by the comparison of the obtained solutions with the exact solutions known in the integer-order limit. The graphical and tabular analysis shows how the fractional order parameter ψ has a great impact on the solution profiles, and a smoothing effect and a delaying propagation effect occur as the fractional order parameter ψ reduces. In a close comparative analysis, both q-HMTM and MVIM provide highly precise results, where MVIM in some instances has a little better accuracy. The results affirm the effectiveness of these methods in addressing new complex systems of a fractional-order that occur in biological, physical, and engineering scenarios.

This paper examines two nonlinear time-fractional Navier-Stokes (NS) systems, in the context of the ϕ-Caputo fractional derivative that offers a generalized and versatile form of the memory and hereditary influence in fluid motion. In order to achieve the semi-analytical approximate solutions, two semi-analytical methods are designed and constructed, including the q-Homotopy ZZ Transform Method (q-HZZTM) and the ZZ Transform Variational Iteration Method (ZZ-VIM). The proposed strategies take the benefits of the ZZ transform, homotopy and variational iteration structures to deal with strong nonlinearities. Numerical tables are used to check and validate the accuracy and convergence of the obtained solutions against the exact solutions for various fractional orders. Moreover, the graphical illustrations are provided to examine the response behavior and the effects of the fractional-order parameter. It is found that both q-HZZTM and ZZ-VIM converge fast and are quite consistent with the exact solutions in the classical case and are stable in the case of fractional-order. The suggested framework is highly efficient, accurate, and flexible, and it is a potent analysis of nonlinear fractional fluid models emerging in applied science and engineering.

In this work, a time-fractional form of Richards' equation is considered to study the infiltration phenomenon in unsaturated porous media. The memory effects in the flow process are described using the Caputo-Fabrizio fractional operator. To obtain an approximate analytical solution of the governing nonlinear fractional partial differential equation, a hybrid analytical technique combining the Natural transform method and the variational iteration method is employed. The proposed Natural transform variational iteration method (NVIM) provides a rapidly convergent series solution and avoids complicated discretization or linearization procedures. A rigorous convergence and uniqueness analysis is carried out, which confirms the validity and accuracy of the proposed approach. The effectiveness and reliability of the method are demonstrated through the solution of the considered problem. The obtained solutions are illustrated through graphical representations using MATHEMATICA, where a comparison between the approximate analytical solution and the exact solution is carried out to validate the accuracy and effectiveness of the proposed method. In addition, the influence of the fractional parameter and other model parameters on the infiltration process is analyzed through graphical illustrations. The results demonstrate that the proposed approach provides accurate and efficient solutions and can serve as a useful analytical tool for solving nonlinear fractional differential equations arising in groundwater hydrology and related physical processes.

This paper aims to integrate the Laplace transformation method with the variational iteration method to deliver an analytical approximate solution for fractional-order integro-differential equations, where the fractional-order derivative and integration are defined in the conformable sense. The iterative solution sequence is obtained using the Laplace variational iteration method, and the convergence of this sequence of approximate solutions to the exact solution is established and demonstrated. First, we shall study the approximate solution of a linear fractional integro-differential equation, and secondly, solve the nonlinear fractional integro-differential equations modeled using conformable differointegration. Some illustrative examples are considered to verify the validity and accuracy of the proposed technique, in which approximate solutions are compared with the exact solutions if they exist. Through the comparison, we conclude that the present hybrid approach is very effective for solving this type of problem.

This work introduces a hybrid analytical technique for solving the Swift–Hohenberg equation by integrating the newly formulated Yasser–Jassim integral transform with the variational iteration method. The proposed framework is designed to efficiently handle both the classical and fractional forms of the equation, providing fast-convergent and highly accurate approximate solutions. To capture memory effects and nonlocal features more realistically, the Atangana–Baleanu fractional derivative in the Caputo sense is incorporated into the model. A rigorous convergence analysis is conducted, establishing sufficient conditions to ensure the stability, reliability, and accuracy of the iterative solutions. The performance of the method is assessed through a setiries of numerical experiments supported by graphical illustrations and tables of absolute error values, which collectively confirm the method’s superior accuracy and rapid convergence when compared with standard analytical approaches. Additionally, a detailed stability study of the fractional solutions is carried out and clearly verified through analytical arguments and numerical simulations. The results demonstrate that the combined Yasser–Jassim transform and variational iteration method offer a versatile and powerful tool for solving fractional-order partial differential equations. Beyond the Swift–Hohenberg equation, the proposed approach can be extended to a wide range of mathematical models, including nonlinear ordinary differential equations and integro-differential systems. Overall, the findings highlight the potential of this hybrid scheme to advance analytical methodologies within fractional calculus and nonlinear dynamical systems.

Retraction notice to "Variational iteration method along with intelligent computing system for the radiated flow of electrically conductive viscous fluid through porous medium" [Heliyon 9 (2023) e14365

In this research introduced a new framework to solve the fractional differential equations(FDEs) by using the Variation Iteration Method (HVIM) and Yasser Jassim Transform. Moreover the results obtained using this proposed method is very similar to that obtained using other strategies. .This methodology is employed to derive approximate analytical for solving Caputo fractional derivative. To establish the validity and efficacy of the current strategy, the solutions to three selected illustrative examples are presented and thoroughly analyzed. The findings obtained through the proposed method are comprehensively reviewed. The results consistently demonstrate that the (FHVIM) exhibits high efficiency, robust reliability, and ease of implementation, making it a promising and suitable tool for application across a wide spectrum of related problems in science and engineering disciplines.

ABSTRACT In this study, the drain spacing is computed using the variational iteration method (VIM) to the linearized Boussinesq equation. By applying at most two iterations of the VIM method under three different initial condition scenarios, three equations for drain spacing calculation were derived. These equations predict values of drain spacing that are generally closer to those of the van Schilfgaarde equation compared with the Glover–Dumm equation. Specifically, for values of the parameter λ , with (where m is the water table height at the midpoint of the drain spacing at time t and is the corresponding height at ) in the range of 0.42–0.9, the relative error (RE) for one of the three equations compared with the van Schilfgaarde equation is less than 10%, and less than 5% for λ between 0.51 and 0.75. The comparison with the Glover–Dumm equation showed that the same RE values are achieved when the ranges of λ values are considerably narrower. At large m values, the rate of water table recession is lower than that predicted by the van Schilfgaarde equation, whereas at later stages, the trend is reversed.

Variational Iteration Method for Fractal Differential Equations

Proportional delay differential equations (PDDEs) arise naturally in viscoelasticity, control theory, biology, population dynamics, and fractional-order physical models in which the future state depends on the value of the solution at a proportion of the current time, but their nonlinear nature and delay terms make analytic treatment challenging. This study develops a hybrid computational scheme that combines the Elzaki Transform (ET) and the Daftardar–Jafari Method (DJM) to obtain accurate analytical–approximate solutions for linear and nonlinear PDDEs. In the proposed approach, the Elzaki transform converts the PDDE into an algebraic functional equation, which is subsequently decomposed using DJM without the need for Adomian polynomials. The method is straightforward, computationally efficient, and capable of handling strong nonlinearities. Several illustrative examples are presented to demonstrate its efficiency, and the results confirm that the ET–DJM hybrid provides a powerful alternative to classical methods such as the Laplace transform, Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), and homotopy analysis methods.

In this research paper, we have proposed a new technique for resolving the Benjamin-Ono and Buckmaster equations that come up in many engineering and science applications. The double Elzaki transform and the Adomian polynomials are coupled in the suggested hybrid approach. Experiments have been carried out to verify the correctness and simplicity of the suggested scheme. To assess the effectiveness of the suggested scheme, the outcomes so obtained are compared with the results obtained by the variational iteration method.

In this research work, a novel non-linear mathematical model has been proposed considering susceptible, quarantined, infected, recovered, and removed compartments before and after the 1st dose and 2nd dose of vaccination. For this dynamics model, the novel coronavirus COVID-19, a contagious disease, is taken as a case study in which its transmission, impact of vaccination, and mitigation have been discussed. This model may be helpful in numerous fields of epidemiology and dynamical systems; moreover, Mason Graph has been used to describe the mathematical model. The stability analysis and disease-free equilibrium points have been deliberated for the model. In this work, the semi-analytical technique Variational Iteration Method has been employed, which will assist researchers in the future by showing that if the rate of immunized personnel rises, then the infection rate decreases. It has been observed that the non-vaccinated personnel decrease with the passage of time due to the awareness campaign programs of the governments. Furthermore, it was observed that the removed rate also decreases with the passage of time as the immunized personnel rises. Mathematical software MAPLE has been used to calculate the analytical solutions of the aforementioned mathematical model.

This article investigates mathematical simulations of Michaelis–Menten kinetics in differential biochemical reactions by implementing fractional derivatives. It establishes numerical computations for the concentrations of enzymes, substrates, inhibitors, products, and several complex intermediates using the homotopy perturbation method (HPM), homotopy analysis method (HAM), and variational iteration method (VIM). The focus is on Caputo fractional derivatives. Numerical examples illustrate HPM, HAM, and VIM comparisons to enhance accuracy and understanding. The conclusion recaps the key findings of this biochemical reaction model involving fractional derivatives, including the relevant numerical results and graphical representations.

The Benjamin–Bona–Mahony–Burgers (BBMB) equation is an important nonlinear partial differential equation in fluid mechanics, used to describe the propagation of long waves. Due to its nonlinearity, obtaining exact solutions is difficult, which motivates the use of efficient approximate methods.In this paper, the BBMB equation is solved using the Variational–Homotopy Perturbation Method (VHPM), which combines the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM), providing a rapidly convergent series solution without requiring a small perturbation parameter. The results obtained by VHPM are compared with those of VIM and HPM. The solution is expressed as a series and truncated at the second-order term due to the method’s fast convergence. Numerical examples are presented for both homogeneous and non-homogeneous cases, and the accuracy is evaluated using the error norms L_2 and L_∞.The results show that VHPM produces more accurate and stable solutions than the other methods. The reported error norms and graphical comparisons confirm the effectiveness and reliability of the proposed technique. Consequently, the VHPM is shown to be a powerful and efficient tool for solving the BBMB equation and similar nonlinear partial differential equations. Keywords: Benjamin, Bona, Mahony, Berger’s Equation, Variational, Homotopy Perturbation Method.

This paper presents a comprehensive review and significant theoretical extension of the Variational Iteration Method (VIM), a powerful semi-analytical technique for solving differential equations. While traditional VIM has demonstrated success for various linear and nonlinear problems, several fundamental limitations have constrained its broader application. This work introduces: (1) A systematic operator-theoretic framework for determining Lagrange multipliers for variable-coefficient and composite operators, extending VIM's applicability beyond canonical forms; (2) Novel adaptive convergence acceleration algorithms that guarantee improved convergence rates through parameter optimization; (3) Multi-scale formulations capable of handling problems with disparate temporal/spatial scales without numerical stiffness; and (4) Demonstrations in cutting-edge application domains including climate dynamics, epidemiology, and nonlinear elasticity. Through rigorous mathematical analysis, comprehensive numerical validation against contemporary methods, and practical implementation guidelines, we establish enhanced VIM as not merely an alternative technique, but as a robust computational framework offering unique advantages in accuracy, efficiency, and physical insight for 21st-century scientific computing challenges.

In this paper, we discuss fractional differential equations, including the Fokker-Planck equation and fractional diffusion differential equations, which are closely related to chemistry and engineering. To solve these equations, we employ the Laplace Variational Iteration Method (LVIM), which combines the Laplace transform with He’s Variational Iteration Method. To demonstrate the efficiency and validity of LVIM, we consider two 1-D Fokker-Planck equations and three fractional diffusion equations in 1-D, 2-D, and 3-D. We solve these equations using LVIM, and the results are presented analytically in tables and graphically using MATLAB for different values of the fractional order and these results are then compared with those obtained by existing methods. The solutions obtain as infinite series, and for certain values of the fractional order, they are found to be similar to the exact results. GANIT J. Bangladesh Math. Soc. 45.2 (2025) 031–044

Understanding how tumour cells interact with the immune system is a key focus in modern mathematical oncology, especially given cancer’s continuing global impact. This study presents a mathematical framework designed to explore how white blood cell (WBC) activity influences tumour progression. The model uses a coupled nonlinear reaction diffusion system to capture both the spatial and temporal dynamics of tumour immune interactions. To solve this complex system, the Variational Iteration Method (VIM) is applied, providing efficient semi-analytical approximations, while stability analysis identifies conditions for tumour elimination or persistence. Numerical simulations show that WBC concentration plays a pivotal role: low immune cell counts lead to rapid tumour growth, whereas higher levels slow tumour expansion. The model also reveals a two-phase tumour behaviour, with an initial period of immune-mediated suppression followed by gradual mass increase, reflecting the transition from immune control to tumour dominance. Overall, this approach highlights the delicate balance between tumour proliferation, immune response, and microenvironmental influences. These findings underscore the potential of mathematical modelling to inform treatment strategies and contribute to the advancement of precision oncology.

Complex systems of ordinary and partial differential equations govern the great majority of phenomena in mathematical physics and engineering. By using sophisticated analytical frameworks to solve these governing equations, this study offers a thorough investigation into the dynamics of fluid flow and heat transfer. In particular, the study investigates the coupled effects of mass and heat transfer over an inclined stretching sheet in the magneto-hydrodynamic (MHD) flow of a non-Newtonian Burgers' fluid through a porous medium. Two reliable analytical techniques, the Variational Iteration Method (VIM) and the Variational of Parameter Method (VPM), are used to preserve high accuracy while guaranteeing computational efficiency. These techniques offer sophisticated, closed-form approximations that avoid the high computational costs of conventional numerical grid-based simulations. The impact of different physical parameters on the temperature and velocity distributions is examined in detail, and graphical representations are offered to confirm the dependability and consistency of the suggested analytical models. The results provide a theoretical foundation for understanding the sensitivity of flow parameters, which is essential for the future design and optimization of transport mechanisms in MHD-based industrial systems.

The task of studying the processes of heat conduction in objects located in nonlinear environments is reduced to solving boundary value problems for a nonlinear heat conduction equation, in which the coefficients and/or the power function of heat sources vary with temperature according to certain rules. Among the numerical approaches applicable to solving such problems for nonlinear equations of mathematical physics, one can distinguish the finite difference method, finite element method, variational, projection and manifold iterative methods. Given those options, we can consider the method of two-sided approximations as the most useful since it allows the researcher to obtain a convenient estimate for the error of the approximate solution and justify the existence of a solution to the original problem. The aim of the article is to study the applicability of the method of two-sided approximations based on usage of Green's functions to solve the first boundary value problem for a nonlinear one-dimensional heat conduction equation with a power-dependent temperature coefficient of heat conduction and an exponentially temperature-dependent power function of internal heat sources. To achieve the set goal, the original problem was replaced, and the new obtained boundary value problem was reduced to the equivalent Hammerstein integral equation, which was considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a unique positive solution to the problem and the conditions for two-sided convergence of approximations to it were formulated. The developed method was implemented programmatically and investigated when solving a test problem. The results of the computational experiment are given via graphical and tabular information