A new modified definition of Caputo–Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM)

2019 ◽  
Vol 346 ◽  
pp. 247-260 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Order Derivative ◽  
Analysis Method ◽  
Fractional Order Derivative ◽  
Homotopy Analysis ◽  
Definition Of

scholarly journals Homotopy Analysis Method for Three Types of Fractional Partial Differential Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Haidong Qu ◽  
Zihang She ◽  
Xuan Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Order Derivative ◽  
Analysis Method ◽  
Partial Differential ◽  
Homotopy Analysis ◽  
Differential Properties

In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractional-order derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.

scholarly journals Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1167
Author(s):  
Said Mesloub ◽  
Saleem Obaidat
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Initial Boundary Value

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

A Local Fractional Derivative

2003 ◽  
Author(s):  
Xiaorang Li ◽  
Christopher Essex ◽  
Matt Davison
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Local Property ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Local Fractional Derivative ◽  
Basic Properties ◽  
Differentiable Functions ◽  
Definition Of

A new definition of fractional order derivative is given and its basic properties are investigated. This definition is based on the Weyl derivative and is a local property of functions. It can be applied to non-differentiable functions and may be useful for studying fractal curves.

scholarly journals Davey-Stewartson Equation with Fractional Coordinate Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
H. Jafari ◽  
K. Sayevand ◽  
Yasir Khan ◽  
M. Nazari
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Good Agreement

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

Homotopy analysis method for solving Abel differential equation of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Khosro Sayevand ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Numerical Results ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Abel Differential Equation

AbstractIn this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

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