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 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 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 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 modification fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operators

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 165-175 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Jassim ◽  
Hasib Khan
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
New Technique ◽  
Transform Method ◽  
Variational Iteration ◽  
Kdv Equations ◽  
Gas Dynamic ◽  
Coupled Kdv Equations ◽  
A New Technique

In this paper, we apply a new technique, namely local fractional variational iteration transform method on homogeneous/non-homogeneous non-linear gas dynamic and coupled KdV equations to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative and integral operators. This method is the combination of the local fractional Laplace transform and variational iteration method. 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.

scholarly journals Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

2020 ◽  
Vol 9 (1) ◽  
pp. 470-482
Author(s):  
Aniqa Zulfiqar ◽  
Jamshad Ahmad
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Conformable Derivative ◽  
Transform Method ◽  
Fractional Equations ◽  
Differential Transform ◽  
Foam Drainage Equation

AbstractIn this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Hassan Kamil Jassim ◽  
Canan Ünlü ◽  
Seithuti Philemon Moshokoa ◽  
Chaudry Masood Khalique
Keyword(s):  
Laplace Transform ◽  
Iteration Method ◽  
Wave Equations ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Cantor Set ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Variational Iteration

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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ai-Min Yang ◽  
Jie Li ◽  
H. M. Srivastava ◽  
Gong-Nan Xie ◽  
Xiao-Jun Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Variational Iteration ◽  
Linear Partial Differential Equations

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.

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 Variational iteration method for studying perihelion precession and deflection of light in General Relativity

2016 ◽  
Vol 4 (2) ◽  
pp. 52 ◽  
Author(s):  
V.K. Shchigolev
Keyword(s):  
General Relativity ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Simple Problem ◽  
Geodesic Equations ◽  
Variational Iteration ◽  
Deflection Of Light ◽  
Perihelion Shift ◽  
Main Ideas

A new approach in studying the planetary orbits and deflection of light in General Relativity (GR) by means of the Variational iteration method (VIM) is proposed in this paper. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of VIM are given. The appropriate correct functionals are constructed for the geodesics in the spacetime of Schwarzschild, Reissner-Nordström and Kiselev black holes. In these cases, the Lagrange multiplier is obtained from the stationary conditions for the correct functionals. Then, VIM leads to the simple problem of computation of the integrals in order to obtain the approximate solutions of the geodesic equations. On the basis of these approximate solutions, the perihelion shift and the light deflection have been obtained for the metrics mentioned above.

Sign in / Sign up

Export Citation Format

Share Document