scholarly journals On varitional iteration method for fractional calculus

2017 ◽  
Vol 21 (4) ◽  
pp. 1707-1712 ◽  
Author(s):  
Hai-Gen Wu ◽  
Yue Hu
Keyword(s):  
Power Series ◽  
Fractional Calculus ◽  
Iteration Method ◽  
Solution Process ◽  
Fractional Power ◽  
Variational Iteration Method ◽  
Computational Procedure ◽  
Fractional Differential ◽  
Fractional Power Series ◽  
Main Shortcoming

Modification of the Das? variational iteration method for fractional differential equations is discussed, and its main shortcoming involved in the solution process is pointed out and overcome by using fractional power series. The suggested computational procedure is simple and reliable for fractional calculus.

scholarly journals The fractional power series method an efficient candidate for solving fractional systems

2018 ◽  
Vol 22 (4) ◽  
pp. 1745-1751 ◽  
Author(s):  
Feng Ren ◽  
Yue Hu
Keyword(s):  
Power Series ◽  
Solution Process ◽  
Fractional Power ◽  
Initial Solution ◽  
Great Success ◽  
Series Method ◽  
Power Series Method ◽  
Fractional Systems ◽  
Fractional Differential ◽  
Fractional Power Series

The fractional power series method was originally proposed to solve a fractional differential equation. This paper extends the method to a system of fractional differential equations with great success. How to construct an initial solution, plays an important role in the solution process and an example is given to elucidate the choice of the initial solution.

scholarly journals Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Fei Xu ◽  
Yixian Gao ◽  
Xue Yang ◽  
He Zhang
Keyword(s):  
Power Series ◽  
Fractional Power ◽  
Boussinesq Equations ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Fractional Power Series ◽  
Residual Power

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

scholarly journals Method for Evaluating Fractional Derivatives of Fractional Functions

2020 ◽  
pp. 286-290
Author(s):  
Chii-Huei Yu
Keyword(s):  
Power Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Differential Problem ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Fractional Power Series

This paper studies the fractional differential problem of fractional functions, regarding the modified Riemann-Liouvellie (R-L) fractional derivatives. A new multiplication and the fractional power series method are used to obtain any order fractional derivatives of some elementary fractional functions.

scholarly journals Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1070
Author(s):  
Pshtiwan Othman Mohammed ◽  
José António Tenreiro Machado ◽  
Juan L. G. Guirao ◽  
Ravi P. Agarwal
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Power ◽  
Power Series Solution ◽  
True Nature ◽  
Nonlinear Fractional Differential Equations ◽  
Adomian Decomposition ◽  
Fractional Differential ◽  
Fractional Power Series

Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples.

scholarly journals An analytical solution for a fractional heat-like equation with variable coefficients

2017 ◽  
Vol 21 (4) ◽  
pp. 1759-1764 ◽  
Author(s):  
Run-qing Cui ◽  
Tao Ban
Keyword(s):  
Analytical Solution ◽  
Power Series ◽  
Solution Process ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Series Method ◽  
Power Series Method ◽  
Fractional Power Series

The fractional power series method is used to solve a fractional heat-like equations with variable coefficients. The solution process is elucidated, and the results show that the method is simple but effective.

scholarly journals Fractional power series method for solving fractional differemtial equation

2016 ◽  
Vol 12 (4) ◽  
pp. 6156-6159 ◽  
Author(s):  
Runqing Cui ◽  
Yue Hu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Power ◽  
The Other ◽  
Series Method ◽  
Power Series Method ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Fractional Power Series

we use fractional power series method (FPSM) to solve some linear or nonlinear fractional differential equations . Compared to the other method, the FPSM is more simple, derect and effective.

Fractional calculus for nanoscale flow and heat transfer

2014 ◽  
Vol 24 (6) ◽  
pp. 1227-1250 ◽  
Author(s):  
Hong-Yan Liu ◽  
Ji-Huan He ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Content Type ◽  
Variational Iteration ◽  
Fractal Media ◽  
Fractional Differential ◽  
Analytical Approaches

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings – Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value – Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

scholarly journals FRACTIONAL POWER SERIES APPROACH FOR THE SOLUTION OF FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
MUHAMMAD AKBAR ◽  
RASHID NAWAZ ◽  
SUMBAL AHSAN ◽  
KOTTAKKARAN SOOPPY NISAR ◽  
KAMAL SHAH ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Power ◽  
Power Series Method ◽  
Transform Method ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Optimal Homotopy Asymptotic Method ◽  
Trigonometric Transform ◽  
Fractional Power Series

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable.

scholarly journals Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

2019 ◽  
Author(s):  
Josef Rebenda
Keyword(s):  
Power Series ◽  
Initial Value Problem ◽  
Fractional Power ◽  
Order System ◽  
Initial Value ◽  
Caputo Fractional Derivatives ◽  
Differential Transform ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Fractional Power Series

The differential transformation, an approach based on Taylor's theorem, is proposed as convenient for finding exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem results in a reliable and expected outcome.

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