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In this article, the fractional third‐order dispersive partial differential equations were investigated by using Shehu decomposition and variational iteration transform methods. The well known Riemann‐Liouville fraction integral, Caputo's fractional‐order derivative, Shehu transform for fractional‐order derivatives and Mittag‐Leffler function were used as the major basis of the methodology. The graphs and table show the solution behavior for various fractional order values. The comparison provided the signed agreement of the solutions to each other as well as with the exact result. The accurateness and efficiency of the suggested techniques are examined using two numerical experiments. The reliability and validity tests show that for various fractional‐order values, the newly proposed method is reliable, accurate, and efficient. [ABSTRACT FROM AUTHOR]

Numerical methods in the area of nonlinear systems are extensively implemented for computing their approximate solutions because these systems are very difficult to tackle analytically. There are various numerical techniques available in the literature to find the solutions of nonlinear oscillators. Variational iteration method (VIM) is one of these approaches which is convenient to implement for these kinds of problems. In this work, our study aims to identify the numerical solution of nonlinear oscillator by making use of variational iteration method associated with Formable transformation. For the smooth utilization of this approach, we have to formulate the Lagrange multiplier through variational theory. Furthermore, we develop a new unified iterative scheme for the correction functional of VIM, considering the Formable transformation. Several new schemes of correction functional can be deduced from the newly proposed method considering the duality relation of Formable transform. In support of our primary finding, we discuss numerical example as application. A number of Physical applications of nonlinear oscillators are available in the field of vibrations and oscillations but in recent times nonlinear oscillators are used to describe complicated systems or to address mechanical, electrical, and other engineering phenomenon. [ABSTRACT FROM AUTHOR]

In this investigation, we utilize two recent analytical schemes to unveil novel solitary wave solutions for the (1 + 1) -dimensional Mikhailov–Novikov–Wang integrable equation. The said equation serves as a mathematical model that captures specific physical phenomena, albeit lacking a direct physical interpretation. Nevertheless, it finds relevance in various systems within the realm of nonlinear waves in physics. Through the application of the aforementioned analytical schemes, we derive fresh solutions and evaluate their accuracy by employing the variational iteration method. The congruence observed between the analytical and numerical solutions of the investigated model serves as validation for the constructed solutions. Furthermore, we delve into exploring the implications of obtaining precise and ground breaking solitary wave solutions on the practical applications associated with the studied model. [ABSTRACT FROM AUTHOR]

In this article, the Laplace transformation method in connection with the variational iteration method will be used to solve approximately fuzzy random ordinary differential equations. After that, the sequence of approximated closed form iterated solutions is derived based on the general Lagrange multiplier evaluated using the well-known convolution theorem of the Laplace transformation method. In addition, two examples are given and solved to illustrate the reliability, efficiency and applicability of the proposed method, they are simulated using computer programs with two different generations of stochastic processes, namely the Wiener process or Brownian motion, which are 1000 and 10000, respectively. [ABSTRACT FROM AUTHOR]

In applicative biosensor technology, mathematical modeling plays an indispensable role in explaining the transport of electrical signals by analyzing the binding behavior of the biochemical enzyme inhibitors to the target molecule. The biosensors are extensively used in clinical diagnostics, drug detention, food analysis and environment monitoring because they are highly sensitive, reliable and relatively cheap. Dynamic mathematical models used for biological investigation serve the purpose efficiently with very reasonable outcomes. In this study, a time-independent mathematical model for biosensor enzyme–substrate–inhibitor system under uncompetitive inhibition based on the nonlinear diffusion equations taking into consideration the kinetic rate constants and the initial concentrations of enzyme, substrate and inhibitor has been formulated and solved analytically using variational iteration method (VIM). The reliability and accuracy has been proved by comparing our results with the solution obtained by standard VIM. Chosen biosensors showed desirable sensitivity, selectivity and potential for application on real samples. They are frequently made to prevent interference from undesirable components that are present in the monitored system. The VIM is effectively and easily used to obtain solution of nonlinear equations accurately. Further, the solution has been discussed exhaustively for different values of reaction parameters avoiding linearization and unrealistic assumptions and the results obtained significantly agree with existing literature. [ABSTRACT FROM AUTHOR]

The work of this paper is motivated by the recently published article (Zeidan et al., Math Methods Appl Sci 43(5):2171–2188, 2020) in which the authors have discussed the Adomian decomposition method (ADM) to solve one dimensional Burgers' equation in viscous and inviscid forms. Here, we propose an effective and efficient semi-analytical method named variational iteration method (VIM) (He, Int J Non-linear Mech 34(4):699–708, 1999) to solve the Burgers' equations considered in Zeidan et al. (Math Methods Appl Sci 43(5):2171–2188, 2020). The novelty of the proposed scheme over ADM is proven by comparing the truncated series solutions and presented in the form of graphs and error tables. In addition to this, VIM is extended to solve 2D, 3D, and systems of Burgers' equations. Thanks to the scheme, closed-form solutions are obtained in most of the cases. The convergence analysis is also investigated for all the test problems. [ABSTRACT FROM AUTHOR]

In recent decades, numerous and varied numerical methods have been proposed and studied to approximate solutions for various classes of fractional differential equations, primarily those involving single-term or multipleorder equations. However, equations incorporating fractional iterated derivatives have not received widespread attention. In this work we describe a reliable strategy to approximate the solution of higher-order fractional differential equations where both the fractional derivative and the iterated derivatives are described in the Caputo sense. Specifically, we propose a fractional variational iteration method (FVIM) where the Lagrange multiplier associated with the correction term is explicitly determined by means of the Laplace transform. For the second-order case, we give a sufficient condition -involving the coefficients of the equation and the fractional order of the Caputo derivative-which guarantees the convergence of the sequence generated by the FVIM. Furthermore, this convergence is independent of the initial function considered for the iteration. Finally, some examples are presented in order to illustrate the applicability of the method and the reliability of the theoretical results obtained. In particular, for most of them we observe that the FVIM leads to the exact solution which shows the power of the method in practice. [ABSTRACT FROM AUTHOR]

In the present analysis, the natural transform decomposition and variational iteration transform methods have been employed to find an approximate analytical solution of the fractional order of Richards' equation. Some standard cases of Richards' equation have been discussed as an example to illustrate the high accuracy and reliability of proposed methods. The result obtained from the proposed method is very close to the exact solution of the problem. It is concluded that natural transform decomposition and variational iteration transform methods are better alternatives to some standard existing methods to solve some realistic problems arising in science and technology. [ABSTRACT FROM AUTHOR]

The heat transfer phenomenon in the flow of non-Newtonian fluids is important in the manufacturing processes of plastic products, cooling systems polymer solutions and extrusion and injection phenomena, etc. The motivation behind this study is to explore the thermal aspects of Burger non-Newtonian fluids by applying exponential heat source–sink, mixed convection phenomenon and thermal radiation. The analysis was done for different values of fluid viscosity. The numerical methodology is based on the implementation of a variational iteration technique. It is concluded that consideration of variable fluid viscosity is important to improve the heat transfer phenomenon more precisely. It is seen that the temperature profile increases with exponential heat source parameters. [ABSTRACT FROM AUTHOR]

Relevance. The study investigates the problem of stress-strain state and stability of porous functional-gradient size-dependent cylindrical panels. The composition and properties of alloys can differ and significantly affect the performance characteristics of products. Therefore, the research of material properties is relevant and contributes to the creation of new types of products demanded by the oil and gas industry. Aim. Development of a new model and creation of accurate methods for analyzing the stress-strain state of porous functional-gradient size-dependent micro/nano cylindrical panels taking into account geometrical nonlinearity. Methods. The method of variational iterations – the extended Kantorovich method is used to analyze the stress-strain state of cylindrical panels. The validity of the results is ensured by comparing the solutions obtained by the method of variational iterations in the first and second approximations with the solutions obtained by the authors, by the Bubnov–Galerkin method in higher approximations, by the finite difference method of the second order of accuracy, for which their convergence is investigated depending on a number of partitions of the integration area in the finite difference method and the number of series terms in the expansion of the basic functions in the Bubnov–Galerkin method. The results obtained by these methods are compared with the solutions obtained by other authors. It should be noted that the solutions obtained by the method of variational iterations for bending of functionally graded cylindrical panels under the action of transverse uniformly distributed load can be considered accurate. Results and conclusions. The authors have constructed the model of porous functional-gradient size-dependent cylindrical panels. Its use will allow studying the properties of alloys for producing drill pipes. The influence of material porosity type, porosity index, functional-gradient index, boundary conditions, size-dependent parameter, curvature parameters on the stress-strain state of cylindrical panels was analyzed using the developed method of variational iterations.

In this paper, the nonlinear shallow water equation appears in the mathematical modeling of tsunami wave propagation along the coastline of an ocean is solved numerically. The model investigates fractional derivatives and is presented through a system of nonlinear partial differential equations. The variational iteration method (VIM) and the residual power series method (RPSM) have been applied to the fractional system of equations for different parameters. In addition, tsunami wave velocity and run-up height with respect to coast slope and sea depth are analyzed for different time and order in detail. A detailed error analysis is presented for tsunami velocity and tsunami wave propagation. The results obtained are compared to other existing results to show the efficiency and reliability of the proposed methods.

The purpose of this work is to develop a semi-analytical numerical scheme for solving fractional order non-linear partial differential equations (FOPDEs), particularly inhomogeneous FOPDEs, expressed in terms of the Caputo-Fabrizio fractional order derivative operator. To achieve this goal, we examine several fractional versions of nonlinear model equations from the literature. We then present the proposed scheme, discussing its stability and convergence properties. We show that the proposed scheme is efficient and accurate, and we provide numerical examples to illustrate its performance. Our findings demonstrate that the scheme has significant potential for solving a wide range of complex FOPDEs. Overall, this work contributes to the advancement of numerical techniques for solving fractional order non-linear partial differential equations and lays a foundation for further research in this area.

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems. [ABSTRACT FROM AUTHOR]

The dynamic pull-in instability of a microstructure is a vast research field and its analysis is of great significance for ensuring the effective operation and reliability of micro-electromechanical systems (MEMS). A fractal modification for the traditional MEMS system is suggested to be closer to the real state as a practical application in the air with impurities or humidity. In this paper, we establish a fractal model for a class of electrostatically driven microstructure resonant sensors and find the phenomenon of pull-in instability caused by DC bias voltage and AC excitation voltage. The variational iteration method has been extended to obtain approximate analytical solutions and the pull-in threshold value for the fractal MEMS system. The result obtained from this method shows good agreement with the numerical solution. The simple and efficient operability is demonstrated through theoretical analysis and results comparisons. [ABSTRACT FROM AUTHOR]

This paper deals with fractal Van der Pol damped nonlinear oscillators equation having nonlinearity. By combining the techniques of the Laplace transform and the variational iteration method, we establish approximate periodic solutions for the fractal damped nonlinear systems. In this simple way, nonlinear differential equations can be easily converted into linear differential equations. Illustrative examples including the Van der Pol damped nonlinear oscillator reveal that this method is very effective and convenient for solving fractal nonlinear differential equations. Finally, comparison of the obtained results with those of the other achieved method, also reveals that this coupling method not only suggests an easier method due to the Lagrange multiplier but also can be easily extended to other nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper. the application of the natural transform decomposition method (NTDM) and variational iteration transform method (VITM) is applied to obtain the solution for the one-dimensional groundwater recharge through porous media in a fractional order with a singular kernel derivative. The mathematical formulation of the one-dimensional groundwater recharge problem has been derived with assuming that the average coefficient of diffusivity remains constant across the entire moisture content range, while the permeability is allowed to vary. The convergence and uniqueness of the solutions by the proposed methods have been analysed and discussed. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations. The obtained results demonstrate that the proposed methods are straightforward, highly accurate, and efficient. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce interval two-point fuzzy boundary value problems, in which the fuzziness course when the coefficients of the governing ordinary differential equation and/or the boundary conditions include fuzzy numbers of either triangular or trapezoidal types. Such equations will be solved by introducing the concept of α – level sets, α  [0,1] to treat the fuzzy ordinary differential equation into two nonfuzzy ordinary differential equations, which correspond to the lower and upper solutions of the interval fuzzy solutions. The well-known variational iteration method has been used to solve two-point fuzzy boundary value problems and linear equations have been examined. [ABSTRACT FROM AUTHOR]

The purpose of this work is to develop a semi-analytical numerical scheme for solving fractional order non-linear partial differential equations (FOPDEs), particularly inhomogeneous FOPDEs, expressed in terms of the Caputo-Fabrizio fractional order derivative operator. To achieve this goal, we examine several fractional versions of nonlinear model equations from the literature. We then present the proposed scheme, discussing its stability and convergence properties. We show that the proposed scheme is efficient and accurate, and we provide numerical examples to illustrate its performance. Our findings demonstrate that the scheme has significant potential for solving a wide range of complex FOPDEs. Overall, this work contributes to the advancement of numerical techniques for solving fractional order non-linear partial differential equations and lays a foundation for further research in this area. [ABSTRACT FROM AUTHOR]

This study explores non-Newtonian fluid dynamics in an axisymmetric channel with a porous boundary, focusing on its relevance to turbine cooling systems in engineering and industry. Managing fluid flow in such systems is crucial. To address the complexity, mathematical tools like the Homotopy Perturbation Method, Variational Iteration Method, and Runge-Kutta 4th numerical method are used to solve nonlinear differential equations governing momentum and heat transfer. The investigation primarily aims to elucidate relationships in system parameters, including the Power Law index, Reynolds number, and Prandtl number. These relationships are compared with numerical techniques, assessing the precision and simplicity of the employed methods. The inquiry goes beyond conventional research paradigms, exploring constant parameters and trial function steps. The proposed solution reveals a theme of precision, simplicity, and efficient convergence. The correlation between the Reynolds number and the thermal boundary layer is a significant finding. Increasing the Reynolds number and adjusting the Power Law index results in a noticeable reduction in the thermal boundary layer. This has substantial implications for temperature profiles in coupled systems, potentially enhancing cooling effectiveness in various industrial applications.

The importance of differential equations of integer order and fractional order can be seen in many areas of engineering and applied sciences. The present work involves fractional order heat equations that arise in numerous applications of engineering and aims to find series solutions by the Laplace variational iteration method (LVIM). The method combines the Laplace transform and the variational iteration method. To show the efficiency and validity of LVIM, we have exemplarily considered 1-D, 2-D, and 3-D fractional heat equations and solve them by LVIM. Exact solutions are gained in expressions of the Mittag-Leffler function. The results are also explored through graphs and charts.

Simulation and numerical study for the blood ethanol concentration system (BECS) and the Lotka-Volterra system, i.e., predator-prey equations (PPEs) (both of fractional order in the Caputo sense) by employing a development accurate variational iteration method are presented in this work. By assessing the absolute error, and the residual error function, we can confirm the given procedure is effective and accurate. The outcomes demonstrate that the proposed technique is a suitable tool for simulating such models and can be extended to simulate other models.

In this paper, linear and nonlinear fourth-order Fractional Integro Differential Equations (FIDEs) with boundary value problems are solved by Laplace Transform with Modified Variational IterationMethod (LT-MVIM). A new technique based on the VIM is introduced to remove the random choice of initial guess by setting a specific rule depends on unknown parameters. These parameters contributed to the increase in the number of terms of the polynomial approximation and its degree, which, in turn, accelerates the convergence and increases the accuracy from one iteration compared to the standard method, where the initial approximation is still randomly chosen. Moreover, the standard method requires an infinite number of iterations, which need massive calculations in each iteration. Some examples are given in order to show the accuracy of the solutions obtained by the proposed method. Furthermore, comparisons are made between the solutions obtained by the proposed method and Laplace Transform Variational Iteration Method (LT-VIM) based on the exact solutions, revealing that the LT-MVIM contributes to accelerating the convergence of approximate solution to the exact solution by reducing the computational work to obtain the approximate solution using one iteration. Whereas, LT-VIM needs more iterations to obtain a suitable approximate solution, which results in an increase in the computational workload. [ABSTRACT FROM AUTHOR]

Simulation and numerical study for the blood ethanol concentration system (BECS) and the Lotka-Volterra system, i.e., predator-prey equations (PPEs) (both of fractional order in the Caputo sense) by employing a development accurate variational iteration method are presented in this work. By assessing the absolute error, and the residual error function, we can confirm the given procedure is effective and accurate. The outcomes demonstrate that the proposed technique is a suitable tool for simulating such models and can be extended to simulate other models. [ABSTRACT FROM AUTHOR]

This study unveils a novel approach, integrating the variational iteration method and numerical integration, to address the n-th order fuzzy random ordinary differential equations' linear fuzzy initial value problems. The robustness of the variational iteration method, a proven and reliable technique, ensures the effectiveness of the proposed approach. The sequence of approximations generated by this method is scrutinized to confirm its convergence towards the exact solution, demonstrating the method's precision. Two distinct examples, each with a different number of Brownian motion generations, are simulated to elucidate the practical application of the proposed approach. The outcomes affirm the method's reliability and efficiency in tackling such complex mathematical problems. [ABSTRACT FROM AUTHOR]

The amplitudes of the forced oscillations of a nano-structure conveying fluid are the solutions of an inhomogeneous integral-differential system. This is solved by an easily accessible scheme based on the variational iteration method (VIM), Galerkin's method and the Laplace transform techniques. The presented method is accompanied by the study of the convergence of the iterative process and of the errors. In the literature, the dynamic response of a viscoelastic nanotube conveying fluid is frequently obtained by an iterative method. This leads to the double convolution products, whose presence will be avoided in the new method proposed in this paper. Thus, the numerical results will be obtained much faster and more accurately. [ABSTRACT FROM AUTHOR]

In this investigation, the fractional Fornberg–Whitham equation (FFWE) is solved and analyzed via the variational iteration method (VIM) and Adomian decomposition method (ADM) with the help of the Aboodh transformation (AT). The FFWE is an important model for describing several nonlinear wave propagations in various fields of science and plasma physics. The AT provides a powerful tool for transforming fractional-order differential equations (DEs) into integer-order ones, making them more amenable to analytical solutions. Accordingly, the main objective of this investigation is to demonstrate the effectiveness and accuracy of ADM and VIM in deriving some approximations for the FFWE. Furthermore, we highlight the advantages and potential applications of these methods in solving other fractional-order nonlinear problems in several scientific fields, especially in plasma physics and some engineering problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractional-order arms race model. The model has emerged as an important tool for the investigation of international conflict and arms races. The variational iteration method, the homotopy perturbation method, and the adomian decomposition method are used to solve the mathematical model with Caputo’s fractional derivative. Several numerical computations have been provided to establish the validity and accuracy of the acquired results. It is shown that the fractional-order model can be solved easily using semi-analytical methods. The results obtained by all methods are compared. [ABSTRACT FROM AUTHOR]

The dynamics of micro/nanoelectromechanical systems (M/NEMS) is a core research area in micromechanics. Due to the nonlinearities and the singular nature of actuation forces that emerge in these systems, it has become a promising and challenging research area. The foremost objective of this manuscript is to examine the dynamics of M/NEMS by approximating rational terms involved in M/NEMS structures. An M/NEMS switch under electromagnetic force is adopted to reveal the effectiveness of the expansion of rational terms. Taylor series is employed to approximate the rational function into the summation of simple terms. The well-known variational iteration method is engaged to obtain the dynamic pull-in threshold value, the nonlinear frequency, and the analytical solution of the objective system. The solution obtained from the proposed strategy exhibits good agreement with observations obtained numerically. As opposed to the existing approaches, the suggested scheme achieves a high level of accuracy. [ABSTRACT FROM AUTHOR]

The efficiency of fuel consumptions and improved performances is real challenge in the vehicle engines. Recent advances in thermal engineering suggested various tools to improve the capacitance of engine oil for which the interaction of nanoparticles is more dynamical. The aim of current research is to enhance the thermal aspects of engine oil with suspension of Williamson hybrid nanofluid. The hybrid nanofluid is the suspension of molybdenum disulfide MoS2 and zinc oxide ZnO. In order to increase the heating capability of system, external heat source and viscous dissipation features are contributed. The flow analysis is based on moving surface contains porous medium. After modelling the problem, the analytical outcomes of attributed with variational iteration method (VIM). The computational simulations are performed with help of Lagrange Multiplier technique. The impact of numerous physical parameters on skin fraction, Nusselt numbers, temperature distribution and velocity profile is exhibited.

The amplitudes of the forced oscillations of a nano-structure conveying fluid are the solutions of an inhomogeneous integral-differential system. This is solved by an easily accessible scheme based on the variational iteration method (VIM), Galerkin’s method and the Laplace transform techniques. The presented method is accompanied by the study of the convergence of the iterative process and of the errors. In the literature, the dynamic response of a viscoelastic nanotube conveying fluid is frequently obtained by an iterative method. This leads to the double convolution products, whose presence will be avoided in the new method proposed in this paper. Thus, the numerical results will be obtained much faster and more accurately.

The focus of this study is to investigate novel and precise solitary wave solutions of the modified Kawahara (MK) equation using recent and accurate computational techniques. The MK equation, a nonlinear partial differential equation, has significant applications in various fields, including fluid dynamics and plasma physics The comprehension of nonlinear wave phenomena relies heavily on solitary wave solutions, making it necessary to develop accurate and efficient computational techniques for their identification. In this study, the Khater II method is employed as a computational technique, while the variational iteration method is used as a numerical scheme.The Khater II method is a powerful computational technique that has exhibited promising outcomes in solving nonlinear wave equations. It uses an analytical framework to transform the partial differential equation into a set of ordinary differential equations that are amenable to straightforward solutions, thereby enabling the construction of exact solutions. Conversely, the variational iteration method is a numerical scheme that enhances solution accuracy through iterative approximations.By using the Khater II method as a computational technique and the variational iteration method as a numerical scheme, this study identifies novel and precise solitary wave solutions of the MK equation. These solutions provide valuable insights into the dynamics and behavior of nonlinear waves across various applications. The effectiveness of the employed techniques is demonstrated by comparing them with other commonly utilized computational methods for solving the MK equation. The results indicate that the combined application of the Khater II method and the variational iteration method yields more accurate and efficient solutions. This heightened accuracy and efficiency are critical for comprehending and predicting the behavior of the studied model in real-world applications. Thus, this study underscores the significance of employing recent and accurate computational techniques, such as the Khater II method and the variational iteration method, to identify novel and precise solitary wave solutions of the MK equation. The obtained solutions contribute to an improved understanding of the model’s behavior and hold practical implications across diverse scientific and engineering domains.

Owing to motivated applications of nanofluids in industrial and technological processes, many novel attempts have been reported by investigators in recent century. The interesting applications subject to the nanomaterials is noted in hybrid-powered engines, solar systems, thermal management, heat exchanger, energy generation, microelectronics etc. This framework presents a Cattaneo-Christov heat flux model for nonlinear convective transport of Walter-B nanofluid due to extending surface. The model is further supported with the non-uniform heat source and activation energy applications. The solutal thermal and mass flux constraints are utilized to inspect the thermal outcomes. The transformation of partial differential system to nonlinear ordinary differential system is carried out with proper conversions. The nonlinear system is analytically tackled with implementation of variational iteration method (VIM). The role of physical parameters appeared in the flow modelling are physical justified.

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study, we apply the fractional Laplace variational iteration method (FLVIM), a computer methodology for exploring fractional Navier–Stokes equation solutions. In light of the theory of fixed points and Banach spaces, this paper also explores the uniqueness and convergence of the solution of general fractional differential equations obtained by the suggested method. In addition, the fractional Laplace variational iteration method solution's error analysis is covered. The computational technique also clearly demonstrates the validity and dependability of the suggested method for solving fractional Navier‐Stokes equations. Furthermore, the obtained solutions are a perfect fit with previously established solutions. [ABSTRACT FROM AUTHOR]

Solving uncertain differential equations is a critical subject in the field of uncertainty theory, where uncertain differential equations are a sort of differential equations that involve Liu processes. Currently, considerable effort has been put into addressing this issue. Regrettably, analytic solutions to uncertain differential equations are not always accessible. As a result, several numerical methods have been investigated. However, numerical methods have certain limitations in terms of providing a continuous representation of the solution as well as more information about the solution. This paper will propose a novel algorithm based on the variational iteration method (VIM) for solving uncertain differential equations analytically or approximately analytically. The associated numerical experiments show that the proposed method is an efficient tool for solving uncertain differential equations. [ABSTRACT FROM AUTHOR]

We have proposed a development of the variational iteration method (VIM), or extended Kantorovich method, by studying physically nonlinear (FN) or geometrically nonlinear (GN) Kirchhoff nanoplates as an example. The modified couple stress theory was used for modeling size-dependent factors of the Kirchhoff nanoplates. Nested one into the other iteration procedures of the Birger method of variable elasticity parameters, of the variational iteration method (VIM), and of the Newton–Raphson method for physically nonlinear (FN) Kirchhoff nanoplates were constructed. The solution of problems for geometrically nonlinear (GN) Kirchhoff nanoplates was carried out on the basis of the variational iteration method and the Newton–Raphson method. The validity of the results was ensured by the coincidence of the results obtained via several methods of reducing partial differential equations to ordinary differential equations and via the finite difference method. The computational effectiveness of the proposed iterative procedure was demonstrated in terms of both accuracy and performance. A comparison of the results obtained showed that the variational iteration method (VIM) is the most efficient and fastest of all the methods considered both for problems with physical nonlinearity and for geometrically nonlinear problems. [ABSTRACT FROM AUTHOR]

In this paper, a generalized Korteweg-de Vries equation involving a temporal fractional derivative and a spatial fractal derivative is studied. The temporal fractional derivative can describe the non-local property and memory property, while the spatial fractal derivative can model the space discontinuity. Its approximate analytical solution is presented using He's variational iteration method, which is extremely effective for the fractal-fractional differential equations. [ABSTRACT FROM AUTHOR]

The goal of the work is to solve the nonlinear convection-diffusion-reaction problem using the variational iteration method with the combination of the Chebyshev wavelet. This work developed a hybrid iterative technique named as Variational iteration method with the Chebyshev wavelet for the solutions of nonlinear convection-diffusion-reaction problems. The aim of applying the derived algorithm is to achieve fast convergence. During the solution of the given problem, the restricted variations will be mathematically justified. The effects of the scaling and other parameters like diffusion parameter, convection parameter, and reaction parameter on the solution are also focused on by their suitable selection. The approximate results include the error profiles and the simulations. The results of variational iteration with the Chebyshev wavelet are compared with variational iteration method, the Modified variational iteration method, and the Variational iteration method with Legendre wavelet. The error profiles allow us to compare the results with well-known existing schemes. [ABSTRACT FROM AUTHOR]

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method's ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option. [ABSTRACT FROM AUTHOR]

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order η = (0 , 1 ] with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable l < χ < L and l , L ∈ R . To confirm the effectiveness of the proposed methods of non-integer order η , the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods. [ABSTRACT FROM AUTHOR]

Approximate solution of fuzzy initial value problem composed of interval fuzzy ordinary differential equation subject to initial conditions given as fuzzy intervals is the main objective of this work. The considered fuzzy initial value problem was given with Hukuhara definition of the difference and derivatives for intervalvalued functions, in which two cases are considered depending on the comparison between the lower and upper functions of the solution derivative. The followed definition for presenting the fuzzy interval is generalized using trapezoidal fuzzy number, while the followed approximated method used in this work is the variational iteration method and its modification to increase the accuracy and the rate of convergence to the exact solution. [ABSTRACT FROM AUTHOR]

Copyright of Journal of the College Of Basic Education is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A system of nonlinear partial differential equations was solved using a modified variational iteration method (MVIM) combined with a genetic algorithm. The modified method introduced an auxiliary parameter (p) in the correction functional to ensure convergence and improve the outcomes. Before applying the modification, the traditional variational iteration method (VIM) was used firstly. The method was applied to numerically solve the system of Schrödinger-KdV equations. By comparing the two methods in addition to some of the previous approaches, it turns out the new algorithm converges quickly, generates accurate solutions and shows improved accuracy. Additionally, the method can be easily applied to various linear and nonlinear differential equations.

In this paper, a mathematical model is constructed based on the deformation theory of plasticity for studying the stress-strain state of Kirchhoff nanoplates (nanoeffects are taken into account according to the modified moment theory of elasticity). An economical and correct iterative method for calculating the stress-strain state of nanoplates has been developed — the method of variational iterations (the extended Kantorovich method). The method of variational iterations (the extended Kantorovich method) has the advantage over the Bubnov – Galerkin or Ritz method in that it does not require specifying a system of approximating functions satisfying boundary conditions, because the method of variational iterations builds a system of approximating functions at each iteration, which follows from solving an ordinary differential equation after applying the Kantorovich procedure. The correctness of the method is ensured by the convergence theorems of the method of variable elasticity parameters by I. I. Vorovich, Yu. P. Krasovsky and the convergence theorems of the method of variational iterations by V. A. Krysko, V. F. Kirichenko. In addition, the reliability of the solutions for elastic Kirchhoff nanoplates obtained using the variational iteration method is ensured by comparison with the exact Navier solution and solutions using Bubnov – Galerkin methods in higher approximations, finite differences and finite elements. The developed method and the methodology for calculating elastic-plastic deformation of Kirchhoff nanoplates, which is based on this method, are effective in terms of machine time costs compared with the methods of Bubnov – Galerkin in higher approximations, finite differences, Kantorovich – Vlasov, Weindiner and especially finite elements. The influence of the nano coefficient, the types of dependences of strain intensity (stress intensity on the elastic-plastic behavior of the nanoplates) has been studied.

To solve fractional-order differential equations (FODEs) with Antagana-Baleanu fractional operator (ABFO), an efficient strategy based on variational iteration method (VIM) and natural transform(NT) is given. Natural variational iteration technique is the name of this method (NVIM). This work also investigates the convergence of the solution of general FODEs obtained by the suggested method given the theory of fixed point and Banach spaces. Furthermore, the error analysis of the NVIM solution is also discussed. Two problems are solved to validate and efficacy demonstrate the of the present. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution. The NVIMs numerical results reveal that the technique is simple to implement and computationally appealing.

In this paper, we developed a new variational iteration method using the quasilinearization method and Adomian polynomial to solve nonlinear differential equations. The convergence analysis of our new method is also discussed under the Lipschitz continuity condition in Banach space. Some application problems are included to test the efficacy of our proposed method. The behavior of the method is investigated for different values of parameter t. This is a powerful technique for solving a large number of nonlinear problems. Comparisons of our technique were made with the available exact solution and existing methods to examine the applicability and efficiency of our approach. The outcome revealed that the proposed method is easy to apply and converges to the solution very fast. [ABSTRACT FROM AUTHOR]

This work aims at a new semi-analytical method called the variational iteration transform method for investigating fractional-order Emden–Fowler equations. The Shehu transformation and the iterative method are applied to achieve the solution of the given problems. The proposed method has the edge over other techniques as it does not required extra calculations. Some numerical problems are used to test the validity of the suggested method. The solution obtained has demonstrated that the proposed technique has a higher level of accuracy. The proposed method is capable of tackling various nonlinear fractional-order problems due to its simple implementation. [ABSTRACT FROM AUTHOR]