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In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

The present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Laplace decomposition and the conformable variational iteration methods to derive new numerical solutions of time and space-conformable fractional Wu-Zhang system. Some graphs for the obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology. [ABSTRACT FROM AUTHOR]

Abstract This study focuses on solving the Caudrey–Dodd–Gibbon ( $$\mathbb {CDG}$$ CDG ) equation using the Khater II ( $$\mathbb {KII}$$ KII ) method and the Variational Iteration ( $${\mathbb {V}}{\mathbb {I}}$$ V I ) method. The $$\mathbb {CDG}$$ CDG equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the $$\mathbb {KII}$$ KII and $${\mathbb {V}}{\mathbb {I}}$$ V I methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation’s behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

In this paper, a modified method is used to approximate the solution to the time-fractional n-dimensional Navier–Stokes equation. The modified method is the Variational Iteration Transform Method, which is implemented in the equation whose fractional order derivative is described in the Caputo sense. The proposed method’s findings are presented and examined using figures. It is demonstrated that the proposed method is efficient, dependable, and simple to apply to various science and engineering applications [ABSTRACT FROM AUTHOR]

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.

Abstract This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

In this article, the solution of a mathematical model of childhood diseases with the Jumarie fractional derivative is discussed by coupling the Laplace variational iteration method (LVIM). The solution of the proposed model not only exists but is also unique. The proposed model is dimensionally balanced as well as stable. This article aims to present an analytical approximation solution for non-linear fractional-order differential equations (FDEs) with the Jumarie derivative. The model has also demonstrated relevant numerical and graphical descriptions. [ABSTRACT FROM AUTHOR]

The objective of this paper is to examine the analytical properties of the nonlinear (1+1)-dimensional Camassa–Holm equation (ℂℍ), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III (핂hat.III) and improved Kudryashov (핀핂ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the ℂℍ issue. He’s variational iteration (ℍ핍핀) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model.The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the ℂℍ equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In the paper, the 1-D wave equation and non-linear diffusion equation are considered and the approximate solutions are obtained by using the variational iteration method. The obtained results show that the proposed method is efficient and simple. [ABSTRACT FROM AUTHOR]

This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

In this paper, a comparative study between two different methods for solving nonlinear timefractional coupled Boussinesq-Burger equation is conducted. The techniques are denoted as the Natural Transform Decomposition Method (NTDM) and the Variational Iteration Transform Method (VITM). To showcase the efficacy and precision of the proposed approaches, a pair of different numerical examples are presented. The outcomes garnered indicate that both methods exhibit robustness and efficiency, yielding approximations of heightened accuracy and the solutions in a closed form. Nevertheless, the VITM boasts a distinct advantage over the NTDM by addressing nonlinear predicaments without recourse to the application of Adomian polynomials. Furthermore, the VITM’s capacity to surmount challenges arising from the identification of the overarching Lagrange multiplier stands as an augmenting facet, amplifying its advantage over the NTDM technique. [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 paper, a new modified variational iteration method (MVIM) with a genetic algorithm has been applied for solving nonlinear partial differential equations. Therefore, a new correction function through an auxiliary parameter that makes sure the convergence of the standard method and improved results by using genetic techniques are introduced. The standard variational iteration method (VIM) is first applied to solve numerically the system of two-dimensional coupled Burgers' equations. Then an improvement on this method is done. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. The algorithm converges readily which yields correct solutions. Better accuracy in comparison with other previous methods has been noticed. Moreover, the method can be easily applied to a wide number of linear and nonlinear differential equations with better accuracy. [ABSTRACT FROM AUTHOR]

Physical systems frequently exhibit nonlinear behavior that remains unresolved in the majority of cases. In this study, we employ the Aboodh transform-based variational iteration method (ATVIM) to resolve the nonlinear model of a tapered beam. In order to solve the governing equation, the periodic motion is sought, and the explicit relationship between frequency and amplitude is revealed. The outcomes of the ATVIM approach are compared with those of other prevalent techniques, and a satisfactory concordance is observed between them. This study also provides an analytical approximation of the tapered beam for a detailed understanding of the effects of factors on the nonlinear frequency, which can be beneficial to researchers and engineers working on the analysis and design of structural projects [ABSTRACT FROM AUTHOR]

In this paper, a modified method is used to approximate the solution to the time-fractional n-dimensional Navier–Stokes equation. The modified method is the Variational Iteration Transform Method, which is implemented in the equation whose fractional order derivative is described in the Caputo sense. The proposed method's findings are presented and examined using figures. It is demonstrated that the proposed method is efficient, dependable, and simple to apply to various science and engineering applications.

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]

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.

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

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]

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]

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]

In this study, the numerical solution of the Volterra-integro differential equations was obtained by applying the variational iteration strategy with the shifted Vieta-Lucas polynomials. The proposed method builds the shifted Vieta-Lucas polynomials for the Volterra-integro differential equation which are then used as basis functions for the approximation. Numerical examples were given to establish the effectiveness and dependability of the recommended approach. Calculations were performed using Maple 2022 software.

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.

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.

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.

In the present investigation, Variational Iteration Method (VIM) is applied to solve the dynamic oscillation of a current-carrying wire in a magnetic field which is generated by a fixed current-carrying conductor parallel to the wire. Two linear springs are considered to restrict the wire to a rigid wall. In a special case, the periodic solution of the problem is obtained by VIM and compared with numerical solutions for different parameters. Results indicate high accuracy of this method which can be easily extended to solve other non-linear vibration equations, therefore can be applicable in other engineering problems. [ABSTRACT FROM AUTHOR]

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 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]

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 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.