Local Fractional Variational Iteration Research Articles

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

LOCAL FRACTIONAL VARIATIONAL ITERATION TRANSFORM METHOD: A TOOL FOR SOLVING LOCAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we use the local fractional variational iteration transform method LFVITM to solve a class of linear and nonlinear partial differential equations (PDEs), as well as a system of PDEs which are involving local fractional differential operators (LFDOs). The technique combines the variational iteration transform approach and the Yang–Laplace transform. To show how effective and precise the LFVITM four examples are given.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Analytical solution of local fractal continuum traffic flow model

This study developed a continuum traffic flow model with local fractional derivatives. This model overcomes the non-differentiable traffic parameters arising in vehicular traffic flow. The proposed model is solved using the local fractional Laplace variational iteration method (LFLVIM) and is well suited for analyzing the dynamical evolution of non-differentiable traffic density and speed function. Furthermore, the stability of the solution for the continuum model has also been discussed. Illustrative examples are also discussed to show the effectiveness of employing LFLVIM in the suggested model. Additionally, numerical simulations for each instance have been displayed. This research indicates that the utilized iterative approach is efficient and may be used to derive the non-differentiable solution to the proposed continuum traffic model.

A generalized local fractional LWR model of vehicular traffic flow and its solution

In this study, a generalized nonlinear local fractional Lighthill‐Whitham‐Richards (LFLWR) model has been developed. The local fractional variational iteration method (LFVIM) solves and analyzes the proposed model. Numerous works have been described in the past to address linear LWR and linear LFLWR models. This research highlighted on generalized nonlinear LFLWR model, and LFVIM is employed to derive non‐differentiable solutions of the suggested model. The existence and uniqueness of the solution of LFLWR model have also been established. Furthermore, several exemplary instances are discussed to demonstrate the success of implementing LFVIM to the proposed model. The numerical simulations for each of the cases have also been shown. Additionally, the obtained solutions of the suggested model have been compared with the solutions of the classical LWR model with non‐differentiable conditions in few examples. The study demonstrates that the employed iterative scheme is quite efficient and can be utilized to obtain the non‐differentiable solution to the proposed generalized nonlinear LFLWR traffic flow model.

Application of Local Fractional Variational Iteration Transform Method to Solve Nonlinear Wave-Like Equations within Local Fractional Derivative

Application of Local Fractional Variational Iteration Transform Method to Solve Nonlinear Wave-Like Equations within Local Fractional Derivative

Fractal Analytical Solutions for Nonlinear Two-Phase Flow in Discontinuous Shale Gas Reservoir

The paths of a two-phase flow are usually non-linear and discontinuous in the production of shale gas development. To research the influence mechanism between shale gas and water, several integer two-phase flow models have been studied but few analytical solutions have been obtained on shale gas and water pressure. This study first developed a local fractional mathematical model for gas and water two-phase flow in shale gas production. The model thus created considers the effects of capillary pressure, the fractal dimension of the flow pipe, and the discontinuity of the flow path. Second, the local fractional traveling wave method and variational iteration method were applied to this model for the development of iterative analytical solutions. Both shale gas and water pressure were analytically derived. Third, the depressurization process of the shale gas and water was analyzed, and a parametric study was conducted to explore the impacts of fractional dimension, entry capillary pressure, and travel wave velocity on shale gas pressure. Finally, our conclusions are drawn, based on the results of these studies.

A hybrid computational method for local fractional dissipative and damped wave equations in fractal media

In this paper, we propose a copulation of the local fractional variational iteration technique and local fractional natural transform (LFNT) for solving the dissipative and damped wave equations with local fractional derivatives. The convergence analysis for the solution obtained through this newly suggested method is also discussed. The numerical simulations for computed solutions are presented on the Cantor set. The computational procedure shows that the newly proposed local fractional method pertaining to the LFNT is very easy and effective to generate solutions for given local fractional wave equations (LFWEs). Moreover, the generated solutions are also in nice correspondence with the previous results acquired by other existing techniques.

Approximate analytic solution of the fractal Fisher’s equation via local fractional variational iteration method

The local fractional variational iteration method is applied to a modified Fisher?s equation defined on Cantor sets with the fractal conditions. The solution process is simple, and the accuracy of the approximate solution is high. The method provides an unrivaled tool for local differential equations. Key word: fractal Fisher?s equation, approximate analytical solutions, local fractional variational iteration method, local fractional derivative

Analytical method to solve the local fractional vehicular traffic flow model

This paper presents a new method to solve the local fractional partial differential equations (LFPDEs) describing fractal vehicular traffic flow. Firstly, the existence and uniqueness of solutions to LFPDEs were proved and then two schemes known as the basic method (BM) and modified local fractional variational iteration method (LFVIM) were developed to solve the local fractional PDEs. Multiple studies have been reported in the literature to solve these problems using iterative methods which are time‐consuming and prone to errors. For linear problems, basic method was found highly accurate and computationally sound. We derived a modified version of LFVIM to investigate and obtain the nondifferentiable solutions of linear, nonlinear, and nonhomogeneous LFPDEs arising in fractal vehicular traffic flow through illustrative examples. Study results show that both schemes are very effective and can be used successfully to solve fractal vehicular traffic flow problems.

An accurate method for nonlinear local fractional Wave-Like equations with variable coefficients

The basic motivation of the present study is to apply the local fractional Sumudu variational iteration method (LFSVIM) for solving nonlinear wave-like equations with variable coefficients and within local fractional derivatives. The derivatives operators are taken in the local fractional sense. The results show that the LFSVIM is an appropriate method to find non-differentiable solutions for similar problems. The results of the solved examples showed the flexibility of applying this method and its ability to reach accurate results even with these new differential equations.

ON THE APPROXIMATE SOLUTIONS FOR A SYSTEM OF COUPLED KORTEWEG–DE VRIES EQUATIONS WITH LOCAL FRACTIONAL DERIVATIVE

In this paper, we utilize local fractional reduced differential transform (LFRDTM) and local fractional Laplace variational iteration methods (LFLVIM) to obtain approximate solutions for coupled KdV equations. The obtained results by both presented methods (the LFRDTM and the LFLVIM) are compared together. The results clearly show that those suggested algorithms are suitable and effective to handle linear and as well as nonlinear problems in engineering and sciences.

Analytical solution to local fractional Landau-Ginzburg-Higgs equation on fractal media

The main objective of the present article is to introduce a new analytical solution of the local fractional Landau-Ginzburg-Higgs equation on fractal media by means of the local fractional variational iteration transform method, which is coupling of the variational iteration method and Yang-Laplace transform method.

The local fractional variational iteration method a promising technology for fractional calculus

In order to make the local variational iteration algorithm converge faster and more effective, the Sumudu transform is adopted and a proper initial solution is chosen. Some examples are given to show that the presented method is reliable, efficient and easy to implement from a computational viewpoint.

Solving Helmholtz Equation with Local Fractional Derivative Operators

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

The Approximate Solutions for Volterra Integro-Differential Equations within Local Fractional Integral Operators

In this paper, we use the Yang-Laplace transform on Volterra integro-differential equations of the second kind within the local fractional integral operators to obtain the nondifferentiable approximate solutions. The iteration procedure is based on local fractional derivative operators. This approach provides us with a convenient way to find solution with less computation as compared with local fractional variational iteration method. Some illustrative examples are discussed. The results show that the methodology is very efficient and simple tool for solving integral equations.

Application of Local Fractional Variational Iteration Method for Solving Fredholm integral equations Involving Local Fractional Operators

In this paper, the local fractional variational iteration method (LFVIM) is used for solving linear and nonlinear Fredholm integral equations of the second kind within local fractional derivative operators. To illustrate the ability and simplicity of the method, some examples are provided. The results reveal that the proposed method is very effective and simple and it leads to the exact solution.

A New Efficient Method for solving Helmholtz and Coupled Helmholtz Equations Involving Local Fractional Operators

In this manuscript, we apply a new technique, namely local fractional Laplace variational iteration method (LFVITM) on Helmholtz and coupled Helmholtz equations to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is the combined of the local fractional Laplace transform (LFLT) and variational iteration method (VIM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Fractional variational iteration method for solving the hyperbolic telegraph equation

In this article, an analytical solution procedure is described for solving one dimensional second order hyperbolic telegraph equation using a reliable semi-analytic method so called the local fractional variational iteration method (LFVIM) subject to the appropriate initial conditions. Various illustrate examples are carried out to check the accuracy, efficiency, and convergence of the described method.