scholarly journals Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Hossein Jafari ◽  
Hassan Kamil Jassim
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Laplace Equation ◽  
Decomposition Methods ◽  
Fractional Operators ◽  
Local Fractional Derivative ◽  
Function Decomposition ◽  
Adomian Decomposition ◽  
And Function ◽  
Fractional Function

We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

scholarly journals Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Dumitru Baleanu ◽  
J. A. Tenreiro Machado ◽  
Carlo Cattani ◽  
Mihaela Cristina Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Decomposition Methods ◽  
Cantor Set ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Local Fractional Derivative ◽  
Variational Iteration

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

scholarly journals Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shun-Qin Wang ◽  
Yong-Ju Yang ◽  
Hassan Kamil Jassim
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Decomposition Method ◽  
Wave Equations ◽  
Inhomogeneous Wave ◽  
Function Decomposition ◽  
Fractional Differential ◽  
Fractional Function

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

2018 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Fractional Operators ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Derivative Order

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

scholarly journals Application of Natural Transform Method to Fractional Pantograph Delay Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
M. Valizadeh ◽  
Y. Mahmoudi ◽  
F. Dastmalchi Saei
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
New Method ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Delay Differential

In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.

scholarly journals Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shao-Hong Yan ◽  
Xiao-Hong Chen ◽  
Gong-Nan Xie ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Method ◽  
Decomposition Method ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Fokker Planck Equations ◽  
Fokker Planck

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

scholarly journals The need for the fractional operators

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
E. A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Time ◽  
Space Time ◽  
Fractional Operators ◽  
Partial Differential ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

AbstractIn this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

scholarly journals The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Chun-Guang Zhao ◽  
Ai-Min Yang ◽  
Hossein Jafari ◽  
Ahmad Haghbin
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Local Fractional Derivative ◽  
Fractional Differential

The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.

scholarly journals Numerical Solution of Coupled System of Nonlinear Partial Differential Equations Using Laplace-Adomian Decomposition Method

2016 ◽  
Vol 12 (8) ◽  
pp. 6530-6544
Author(s):  
Mohamed S M. Bahgat
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Coupled System ◽  
Decomposition Methods ◽  
Partial Differential ◽  
Adomian Decomposition

Aim of the paper is to investigate applications of Laplace Adomian Decomposition Method (LADM) on nonlinear physical problems. Some coupled system of non-linear partial differential equations (NLPDEs) are considered and solved numerically using LADM. The results obtained by LADM are compared with those obtained by standard and modified Adomian Decomposition Methods. The behavior of the numerical solution is shown through graphs. It is observed that LADM is an effective method with high accuracy with less number of components.

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