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 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 Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Konuralp ◽  
H. Hilmi Sorkun
Keyword(s):  
Exact Solutions ◽  
Iteration Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Integrodifferential Equations ◽  
Test Problems ◽  
Application Process ◽  
Delay Function ◽  
Variational Iteration

Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay functionθ(t)vanishes inside the integral limits such thatθ(t)=qtfor0<q<1,t≥0. Either the approximate solutions that are converging to the exact solutions or the exact solutions of three test problems are obtained by using this presented process. The numerical solutions and the absolute errors are shown in figures and tables.

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 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 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 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.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

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