Fractional Variational Iteration Method Research Articles

On the Solution of Local Fractional Differential Equations Using Local Fractional Laplace Variational Iteration Method

We present iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.

On the local fractional LWR model in fractal traffic flows in the entropy condition

Entropy plays an important role in the simulation of traffic flow distribution. This paper studies the entropy condition for the Lighthill–Whitham–Richards model of the fractal traffic flows described by local fractional calculus. We also discuss the solutions of non‐differentiability with graphs by using the local fractional variational iteration method. Copyright © 2015 John Wiley & Sons, Ltd.

Solving Fixed Final Time Fractional Optimal Control Problems Using the Parametric Optimization Method

AbstractIn this paper, the parametric optimization method is used to find optimal control laws for fractional systems. The proposed approach is based on the use for the fractional variational iteration method to convert the original optimal control problem into a nonlinear optimization one. The control variable is parameterized by unknown parameters to be determined, then its expression is substituted into the system state‐space model. The resulting fractional ordinary differential equations are solved by the fractional variational iteration method, which provides an approximate analytical expression of the closed‐form solution of the state equations. This solution is a function of time and the unknown parameters of the control law. By substituting this solution into the performance index, the original fractional optimal control problem reduces to a nonlinear optimization problem where the unknown parameters, introduced in the parameterization procedure, are the optimization variables. To solve the nonlinear optimization problem and find the optimal values of the control parameters, the Alienor global optimization method is used to achieve the global optimal values of the control law parameters. The proposed approach is illustrated by two application examples taken from the literature.

On the Exact Solution of Wave Equations on Cantor Sets

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples.

Numerical method for solving fractional coupled Burgers equations

In this paper, we use the fractional variational iteration method (FVIM) to solve a time- and space-fractional coupled Burgers equations. Some numerical examples are presented to show the efficiency of considered method. A comparison of the proposed method is made with the exact solution, adomain decomposition method (ADM), generalized differential transformation method (GDTM) and homotopy perturbation method (HPM).

An efficient computational technique for solving the Fokker–Planck equation with space and time fractional derivatives

This paper presents numerical solutions of the linear and nonlinear Fokker–Planck partial differential equations [FPPDEs] with space and time fractional derivatives through analytical solutions. These are treated by two analytical methods, namely, fractional reduced differential transform method [FRDTM] and fractional variational iteration method [FVIM] followed by some examples. Numerical results obtained by both FRDTM and FVIM are compared with some existing methods in the literature. This comparison shows the supremacy of FRDTM over FVIM and existing methods in terms of accuracy, simplicity and reliability.

Fractional Variational Iteration Method for Fractional Fornberg-Whitham Equation and Comparison with the Undetermined Coefficient Method

DOI: 10.9734/BJMCS/2015/15392 Editor(s): (1) Metin Basarir, Department of Mathematics, Sakarya University, Turkey. Reviewers: (1) Roshdi Khalil, Department of Mathematics, University of Jordan, Jordan. (2) Anonymous, Pakistan. (3) Anonymous, Turkey. (4) Jianfei Huang, College of Mathematics, Qingdao University, China. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=733&id=6&aid=7607

Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators

The local fractional Laplace variational iteration method (LFLVIM) is employed to handle the diffusion and wave equations on Cantor set. The operators are taken in the local sense. The nondifferentiable approximate solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

A Numeric?Analytic Method for Fractional Order Nonlinear PDE?s With Modified Riemann-Liouville Derivative by Means of Fractional Variational Iteration Method

In this article, an approximate analytical solution of fractional order nonlinear PDE’s with modified Riemann- Liouville derivative was obtained with the help of fractional variational iteration method (FVIM). It is showed that the solutions obtained by the FVIM are reliable and effective method for strongly nonlinear partial equations with modified Riemann-Liouville derivative. The solutions of our model equation can also be obtained from the known forms of the series solutions.

Laplace variational iteration method for the two-dimensional diffusion equation in homogeneous materials

In this paper, we suggest the local fractional Laplace variational iteration method to deal with the two-dimensional diffusion in homogeneous materials. The operator is considered in local fractional sense. The obtained solution shows the non-differentiable behavior of homogeneous materials with fractal characteristics.

A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer

In this work, the local fractional variational iteration method is employed to obtain approximate analytical solution of the two-dimensional diffusion equation in fractal heat transfer with help of local fractional derivative and integral operators.

Local Fractional Laplace Variational Iteration Method for Solving Nonlinear Partial Differential Equations on Cantor Sets within Local Fractional Operators

In this work, we discuss solutions of the nonlinear partial differential equations on
 Cantor sets within local fractional operators. The nondifferentiable approximate
 solutions are obtained by using the local fractional Laplace variational iteration method,
 which is the coupling method of local fractional variational iteration method and
 Laplace transform. The obtained results show the efficiency and accuracy of implements
 of the present method.

Analytical approximate solution of time-fractional Fornberg–Whitham equation by the fractional variational iteration method

The purpose of this paper was to obtain the analytical approximate solution of time-fractional Fornberg–Whitham, equation involving Jumarie’s modified Riemann–Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides the solution in the form of a convergent series with easily calculable terms. The obtained approximate solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency and accuracy of the method.

LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD FOR SOLVING VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITHIN LOCAL FRACTIONAL OPERATORS

The paper uses the Local fractional variational Iteration Method for solving the second kind Volterra integro-differential equations within the local fractional integral operators. The analytical solutions within the non-differential terms are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the integral equations.

Exact solutions of fractional heat-like and wave-like equations with variable coefficients

Purpose– This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.Design/methodology/approach– The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.Findings– The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.Originality/value– The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs.

Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow

We discuss the line partial differential equations arising in fractal vehicular traffic flow. The nondifferentiable approximate solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. The obtained results show the efficiency and accuracy of implements of the present method.

Fractional Variational Iteration Method for Fractional Cauchy Problems

The fractional variational iteration method is used to solve the fractional Cauchy problem. Some examples are given to elucidate the solution procedure and reliability of the obtained results. The variational iteration algorithm leads to exact solutions in the present study.

Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow

The local fractional Laplace variational iteration method is used for solving the nonhomogeneous heat equations arising in the fractal heat flow. The approximate solutions are nondifferentiable functions and their plots are also given to show the accuracy and efficiency to implement the previous method.