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.

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.

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 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 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.

Homotopy perturbation method for MHD non-Newtonian Williamson fluid over exponentially stretching sheet with viscous dissipation and convective boundary condition

2019 ◽  
Vol 30 (11) ◽  
pp. 1950088 ◽  
Author(s):  
Khadijah M. Abualnaja
Keyword(s):  
Fluid Flow ◽  
Perturbation Method ◽  
Numerical Method ◽  
Homotopy Perturbation Method ◽  
Stretching Sheet ◽  
Fluid Temperature ◽  
Williamson Fluid ◽  
Homotopy Perturbation ◽  
Exponential Stretching ◽  
Special Cases

This research is aimed at presenting the two-dimensional steady fluid flow, represented by Williamson constitutive model past a nonlinear exponential stretching sheet theoretically. The system of ODEs describing the physical problem is successfully solved numerically with the help of the homotopy perturbation method (HPM). Special attention is given to study the convergence analysis of the proposed method. The influences of the physical governing parameters acting on the fluid velocity and the fluid temperature are explained with the help of the figures and tables. Further, the presented numerical method is employed to calculate both the rate of heat transfer and the drag force for the Williamson fluid flow. In particular, it is observed that both the Eckert number and the dimensionless convective parameter have the effect of enhancing the temperature of the stretching surface, while the inverse was noted for the dimensionless mixed convection parameter. Finally, the comparison with previous numerical investigations of other authors at some special cases which is reported here proves that the results obtained via homotopy perturbation method are accurate and the numerical method is reliable.

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 Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations

2017 ◽  
Vol 21 (4) ◽  
pp. 1843-1846 ◽  
Author(s):  
Zhen-Jiang Liu ◽  
Magaji Adamu ◽  
Enoch Suleiman ◽  
Ji-Huan He
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Analytical Solutions ◽  
Laplace Transformation ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Initial Approximation ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Linear Differential

Homotopy perturbation method is combined with Laplace transformation to obtain approximate analytical solutions of non-linear differential equations. An example is given to elucidate the solution process and confirm reliability of the method. The result indicates superiority of the method over the conventional homotopy perturbation method due its flexibility in choosing its initial approximation.

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.

Homotopy perturbation method with three expansions for Helmholtz-Fangzhu oscillator

2021 ◽  
Vol 35 (24) ◽  
Author(s):  
Ji-Huan He ◽  
Yusry O. El-Dib
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
High Accuracy ◽  
Singular Terms ◽  
Homotopy Perturbation ◽  
Damping Rate

This paper shows that the Toda oscillator can be converted to a new Helmholtz-Fangzhu equation, which is solved by the homotopy perturbation method with three expansions. The solution, the negative frequency and the damping rate are expanded into power forms in the homotopy parameter, the solution process is given step by step, and the results are of high accuracy. This paper gives a general approach to damp vibrations with singular terms,especially to the Helmholtz-Fangzhu oscillator.

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