scholarly journals He's fractional derivative for the evolution equation

2020 ◽  
Vol 24 (4) ◽  
pp. 2507-2513
Author(s):  
Kang-Le Wang ◽  
Shao-Wen Yao
Keyword(s):  
Evolution Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Fractional Complex Transform ◽  
Fractional Evolution Equation ◽  
Homotopy Perturbation ◽  
Fractal Space

In this paper, He's fractional derivative is adopted to establish fractional evolution equations in a fractal space. He?s fractional complex transform is used to convent the fractional evolution equation into its traditional partner, and the homotopy perturbation method is used to solve the equations. Some illustrative examples are presented to show that the proposed technology is very excellent.

scholarly journals Application of he’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation

2020 ◽  
Vol 24 (5 Part A) ◽  
pp. 3023-3030 ◽  
Author(s):  
Naveed Anjum ◽  
Qura Ain
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Fractal Theory ◽  
Fractional Model ◽  
Fractional Complex Transform ◽  
Homotopy Perturbation ◽  
Holm Equation ◽  
Physical Understanding

In this article He?s fractional derivative is studied for time fractional Camassa-Holm equation. To transform the considered fractional model into a differential equation, the fractional complex transform is used and He?s homotopy perturbation method is adopted to solve the equation. Physical understanding of the fractional complex transform is elucidated by the two-scale fractal theory.

scholarly journals Numerical method for fractional Zakharov-Kuznetsov equations with He’s fractional derivative

2019 ◽  
Vol 23 (4) ◽  
pp. 2163-2170 ◽  
Author(s):  
Kang-Le Wang ◽  
Shao-Wen Yao
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Method ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Fractional Complex Transform ◽  
Homotopy Perturbation

In this paper, a fractional Zakharov-Kuznetsov equation with He's fractional derivative is studied by the fractional complex transform and He's homotopy perturbation method. The solution process is elucidated step by step to show its simplicity and effectiveness of the proposed method.

THE FRACTIONAL COMPLEX TRANSFORM: A NOVEL APPROACH TO THE TIME-FRACTIONAL SCHRÖDINGER EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050141
Author(s):  
QURA TUL AIN ◽  
JI-HUAN HE ◽  
NAVEED ANJUM ◽  
MUHAMMAD ALI
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Time Dependent ◽  
Nonlinear Part ◽  
Fractional Complex Transform ◽  
Homotopy Perturbation ◽  
Novel Approach ◽  
Time Fractional Derivative

This paper presents a thorough study of a time-dependent nonlinear Schrödinger (NLS) differential equation with a time-fractional derivative. The fractional time complex transform is used to convert the problem into its differential partner, and its nonlinear part is then discretized using He’s polynomials so that the homotopy perturbation method (HPM) can be applied powerfully. The two-scale concept is used to explain the substantial meaning of the fractional time complex transform and the solution.

scholarly journals He’s fractional derivative and its application for fractional Fornberg-Whitham equation

2017 ◽  
Vol 21 (5) ◽  
pp. 2049-2055 ◽  
Author(s):  
Kang-Le Wang ◽  
San-Yang Liu
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Fractional Complex Transform ◽  
Homotopy Perturbation ◽  
Fractal Process ◽  
Whitham Equation ◽  
Fractional Equation

Fractional Fornberg-Whitham equation with He?s fractional derivative is studied in a fractal process. The fractional complex transform is adopted to convert the studied fractional equation into a differential equation, and He's homotopy perturbation method is used to solve the equation.

scholarly journals Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Nonlinear Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Radial Symmetry ◽  
Space Time ◽  
Homotopy Perturbation ◽  
Initial Stage

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

scholarly journals A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets

2019 ◽  
Vol 3 (2) ◽  
pp. 30 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Cantor Sets ◽  
New Technique ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Fractional Derivative Operators ◽  
A New Technique

In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). 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.

Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

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