scholarly journals Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ai-Min Yang ◽  
Zeng-Shun Chen ◽  
H. M. Srivastava ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Helmholtz Equation ◽  
Series Expansion ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Expansion Method ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Fractional Derivative Operators ◽  
Series Expansion Method

We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

scholarly journals Solving Helmholtz Equation with Local Fractional Derivative Operators

2019 ◽  
Vol 3 (3) ◽  
pp. 43 ◽  
Author(s):  
Baleanu ◽  
Jassim ◽  
Al Qurashi
Keyword(s):  
Fractional Derivative ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Nonlinear Pdes ◽  
Test Problems ◽  
Helmholtz Equations ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
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.

scholarly journals Local Fractional Variational Iteration Method for Inhomogeneous Helmholtz Equation within Local Fractional Derivative Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xian-Jin Wang ◽  
Yang Zhao ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Helmholtz Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Derivative Operator ◽  
Local Fractional Derivative ◽  
Variational Iteration

The inhomogeneous Helmholtz equation within the local fractional derivative operator conditions is investigated in this paper. The local fractional variational iteration method is applied to obtain the nondifferentiable solutions and the graphs of the illustrative examples are also shown.

scholarly journals A non-differentiable solution for the local fractional telegraph equation

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 225-231
Author(s):  
Jie Li ◽  
Ce Zhang ◽  
Weixing Liu ◽  
Yuzhu Zhang ◽  
Aimin Yang ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Series Expansion ◽  
Expansion Method ◽  
Telegraph Equation ◽  
Differentiable Solution ◽  
Local Fractional Derivative ◽  
Fractional Telegraph Equation ◽  
Telegraph Equations ◽  
Series Expansion Method

In this paper, we consider the linear telegraph equations with local fractional derivative. The local fractional Laplace series expansion method is used to handle the local fractional telegraph equation. The analytical solution with the non-differentiable graphs is discussed in detail. The proposed method is efficient and accurate.

scholarly journals Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yang Zhao ◽  
De-Fu Cheng ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Series Expansion ◽  
Present Method ◽  
Expansion Method ◽  
One Dimensional ◽  
Local Fractional Derivative ◽  
Simple Tool ◽  
Fractional Schrödinger Equations ◽  
The One ◽  
Series Expansion Method

The local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.

scholarly journals Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yong-Ju Yang ◽  
Liu-Qing Hua
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Transform Method ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Differential Transform ◽  
Differential Operation ◽  
Fractional Differential

We propose the variational iteration transform method in the sense of local fractional derivative, which is derived from the coupling method of local fractional variational iteration method and differential transform method. The method reduces the integral calculation of the usual variational iteration computations to more easily handled differential operation. And the technique is more orderly and easier to analyze computing result as compared with the local fractional variational iteration method. Some examples are illustrated to show the feature of the presented technique.

scholarly journals The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Hassan Kamil Jassim
Keyword(s):  
Fractional Derivative ◽  
Wave Equations ◽  
Three Dimensional ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Derivative Operator ◽  
Transform Method ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Dimensional Diffusion

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.

scholarly journals A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Xiaoqun Cao
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Variable Coefficients ◽  
Iteration Formula ◽  
Variational Iteration

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

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