Variational Iteration Method Solutions Research Articles

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

Multistage variational iteration method for a SEIQR COVID-19 epidemic model with isolation class

The SEIQR COVID-19 epidemic model is in the form of the system of first-order nonlinear differential equations. In this paper we propose multistage versions of the variational iteration method (VIM) to solve this COVID-19 epidemic model. The idea of multistage version is to divide the entire time domain into a finite number of subintervals and then implementing the VIM piecewisely on each subinterval. There are two kinds of multistage methods discussed in this paper, where the difference between the two methods lies in the number of restricted variations used in the correction functional. The multistage methods generally give more accurate solutions on longer time intervals than the classical versions. The multistage VIM with less number of restricted variations has the best performance among all types of variational iteration methods discussed in this paper. The accuracy of multistage VIM solution can be increased by using smaller size of subinterval or by implementing more iterations in each subinterval.

A comparative series solutions of Japanese encephalitis model using differential transform method and variational iteration method

AbstractIn this study, a deterministic mathematical model involving the transmission dynamics of Japanese encephalitis (JE) is presented and studied. The biologically feasible equilibria and their stability properties have been discussed. This study investigates a series of solutions to the system of ordinary differential equations (ODEs) in the transmission dynamics of JE. To get approximate series solutions of the JE model, we employed the differential transform method (DTM) and variational iteration method (VIM). DTM utilizes the transformed function of the original JE model, while VIM uses the general Lagrange multiplier to develop the correction functional for the JE model. The results show that the VIM solution is more accurate than the DTM solution for short intervals of time. In addition, the fractional compartmental model of JE is briefly discussed. We illustrated the profiles of the solutions of each of the compartments, from which we found that the fourth‐order Runge–Kutta method solutions are more accurate than the DTM and VIM solutions for long intervals of time.

The impacts of varying magnetic field and free convection heat transfer on an Eyring–Powell fluid flow with peristalsis: VIM solution

A mathematical model of a non-Newtonian fluid (Eyring–Powell fluid) flow with peristalsis under the effect of varying magnetic field and free convection heat transfer has been discussed. Most of the previous attempts of peristaltic flow problems for a non-Newtonian fluid have been solved by using the perturbation technique for a small non-Newtonian parameter, which gives some limits for the results. To avoid such restriction, the variational iteration method (VIM) is applied to solve the current model. A code is established by using the symbolic software MATHEMATICA to get the successive solutions. Semi-analytical solutions are found by VIM for the velocity, heat transfer, pressure gradient and stream function, which include Eyring–Powell fluid parameters. Moreover, a comparison between VIM and numerical solutions is discussed. The obtained results show that the variation of the heat transfer for the Newtonian fluid is large compared with the Eyring–Powell fluid. In addition, it is noteworthy that the peristaltic transport overcomes on Lorentz force nearby the peristaltic walls.

Variational Iteration Approach for Flexural Vibration of Rotating Timoshenko Cantilever Beams

This paper is concerned with the flexural vibration analysis of rotating Timoshenko beams by using the variational iteration method (VIM). Accurate natural frequencies and mode shapes of rotating Timoshenko beams under various rotation speeds and rotary inertia are obtained. The VIM solutions are verified by comparing with some existing results in the literature as well as validated from a comparison study with experimentally measured ones. High accuracy and efficiency of VIM are demonstrated by the use of only a small number of iteration steps required for convergence of the first to the tenth mode frequencies of rotating Timoshenko beam.

Dark and kink soliton solutions of the generalized ZK–BBM equation by iterative scheme

The generalized ZK–BBM equation is solved using iterative scheme of the Adomian decomposition method (ADM) and variational iteration method (VIM). A dark and a kink soliton solutions of the generalized ZK–BBM equation are obtained under initial conditions. The convergence analysis of the ADM and VIM solution shows that these solutions are convergent. The comparison of the ADM and VIM solutions with the exact solution shows that the solutions of the generalized ZK–BBM equation by the iterative methods are almost exact. The absolute errors show that the accuracy and efficiency of the ADM and VIM depend on the problem and its domain. It is found that the iterative scheme of Adomian decomposition method and variational iteration method are quite efficient for the soliton solution of the generalized ZK–BBM equation.

Analytical study for time and time-space fractional Burgers\u2019 equation

In this paper, the variational iteration method (VIM) is applied to solve the time and space-time fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the behavior of VIM solutions as the fractional derivative parameter is changed. The results obtained by VIM are compared with exact solutions and also with expansions of the exact solutions. VIM solutions are found to be in excellent agreement with these exact solutions.

VIM solution of squeezing MHD nanofluid flow in a rotating channel with lower stretching porous surface

A new analytical approach has been performed by using both similarity transformation and Variational Iteration Method (VIM) on the silver, copper, copper oxide, titanium oxide and aluminum oxide nanofluids flowing through a rotating channel with lower stretching porous wall under the squeezing magnetohydrodynamic (MHD) flow conditions. The effects of nanoparticles concentration, characteristic parameter of the flow, suction parameter and magnetic parameter are investigated on the velocity profiles and the shear stresses at lower and upper walls. The results show significant influence of nanoparticle additives on the shear stress and skin friction coefficient of the flow at the walls. The analytical results are compared with those obtained by numerical solution and remarkable agreement is achieved.

APPROXIMATE SOLUTION OF VACCINATION, SUSCEPTIBLE, EXPOSED, INFECTED, RECOVERED (VSEIR) MODEL OF TUBERCULOSIS IN NORTH SUMATERA INDONESIA

In this paper, a vaccination susceptible exposed infected recovered (VSEIR) model f o rtuberculosis in North Sumatera is discussed. The VSEIR model is formed by a system of nonlinear differentialequations. The approximate solution of this model is obtained by using the step variational iteration method(SVIM) and variational iteration method (VIM). VIM used the general Lagrange multiplier in the correctionfunctional and runs iteratively. SVIM also used the general Lagrange multipliers for construction of the correctionfunctional for the problems, and runs by step approach, where the computation is done b y d i v i d i n g theinterval into subintervals with time steps. The two methods are alternative methods to obtain the approximatesolutions of the VSEIR model. Comparison is made with the conventional numerical method, t h e fourth-orderRunge–Kutta method (RK4). From the results, the SVIM solution is more accurate than the VIM solution for longtime interval when compared to RK4. Keywords: VSEIR model, General Lagrange Multiplier, Variational Iteration Method, Step Variational Iteration Method,Fourth Order Runge Kutta

Investigation on vibration of single-walled carbon nanotubes by variational iteration method

In this paper, the variational iteration method (VIM) has been used to investigate the non-linear vibration of single-walled carbon nanotubes (SWCNTs) based on the nonlocal Timoshenko beam theory. The accuracy of results is examined by the fourth-order Runge–Kutta numerical method. Comparison between VIM solutions with numerical results leads to highly accurate solutions. Also, the behavior of deflection and frequency in vibrations of SWCNTs are studied. The results show that frequency of single walled carbon nanotube versus amplitude increases by increasing the values of B.

Analytical solution for a suspension bridge by applying HPM and VIM

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are used to solve the large amplitude torsional oscillations equations in a nonlinearly suspension bridge. This paper compares the HPM and VIM in order to solve the equations of nonlinearly suspension bridge. A comparative study between the HPM and VIM is presented in this work. The achieved results reveal that the HPM and VIM are very effective, convenient and quite accurate to nonlinear partial differential equations. These methods can be easily extended to other strongly nonlinear oscillations and can be found widely applicable in engineering and science. The Laplace transform method is applied to obtaining the Lagrange multiplier in the VIM solution.

Numerical Analytic Solution of SIR Model of Dengue Fever Disease in South Sulawesi using Homotopy Perturbation Method and Variational Iteration Method

In this research, the susceptible–infected–recovered (SIR) model of dengue fever is considered. We have implemented two analytical techniques, namely the variational iteration method (VIM) and the homotopy perturbation method (HPM) for solving the SIR model. The Lagrange multiplier was investigated for the VIM and He’s polynomial approach for the HPM was used. In these schemes, the solution takes the form of a convergent series with easily computable components. The resultsshow thatthe VIM solution is more accurate than the HPM solution for short time intervals, whereasthe HPM is more accurate than the VIM for long time intervalswhencompared with the fourth-orderRunge-Kutta method (RK4).We found that the HPM and the RK4 were in excellent conformance.

Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

He’s Variational Iteration Method for Nonlinear Jerk Equations: Simple but Effective

The variational iteration method (VIM) for determining periodic solutions of non-linear jerk equations involving third-order time-derivative is presented. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. We propose first-order approximate VIM solution for this equation and we compare this solution with the solution obtained by harmonic balance method.

He's variational iteration method for nonlinear jerk equations: Simple but effective

The variational iteration method (VIM) for determining periodic solutions of non-linear jerk equations involving third-order time-derivative is presented. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. We propose first-order approximate VIM solution for this equation and we compare this solution with the solution obtained by harmonic balance method.

Stability Analysis of Two-Segment Stepped Columns with Different End Conditions and Internal Axial Loads

Members with varying geometrical and/or material properties are commonly used in many engineering applications. Stepped columns with internal axial loads constitute a special case of such nonuniform columns. Crane columns in industrial buildings or structural columns supporting intermediate floors are important applications of stepped members in civil engineering. Since neither axial load nor stiffness is constant along the column height, the stability analysis of a stepped column is usually more complicated than that of a uniform column. Determination of exact buckling loads for stepped columns with different end conditions is not always practical. This paper shows that variational iteration method (VIM), a kind of analytical technique recently proposed for solution of nonlinear differential equations, can satisfactorily be used to obtain approximate solutions for buckling loads of stepped columns with internal axial loads. VIM solutions perfectly match with the exact solutions available in the literature for some special cases of two-segment stepped columns. For many other cases, that is, for various values of three design parameters, namely, (i) load ratio, (ii) stiffness ratio, and (iii) length ratio, approximate buckling loads for two-segment stepped columns are determined using VIM and presented in tabular form which can easily be used by design engineers.

Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach

The present work deals with the problem of the nonlinear vibrations of multi-walled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer force. The variational iteration method (VIM) is adopted to obtain the amplitude–frequency curves for large-amplitude vibrations of single-, double- and triple-walled carbon nanotubes. The influences of changes in material constants of the surrounding elastic medium and the geometric parameters on the vibration characteristics of multi-walled carbon nanotubes are investigated. The results from the VIM solution are compared and shown to be in excellent agreement with the available solutions from the open literature. The capability of the present analytical technique is clarified in terms of numerical accuracy as well as computational efficiency.