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 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 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 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 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 A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsaid ◽  
D. Hammad
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Gordon Equation ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Klein Gordon ◽  
Fractional Orders ◽  
Klein Gordon Equation

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

Dynamic Stability of an Axially Moving Yarn with Time-Dependent Tension

2014 ◽  
Vol 900 ◽  
pp. 753-756 ◽  
Author(s):  
You Guo Li
Keyword(s):  
Differential Equation ◽  
Homotopy Perturbation Method ◽  
Time Dependent ◽  
Second Law ◽  
Order Ordinary Differential Equation ◽  
Vibration Equation ◽  
Transversal Vibration ◽  
First Order ◽  
Homotopy Perturbation ◽  
Axially Moving

In this paper the nonlinear transversal vibration of axially moving yarn with time-dependent tension is investigated. Yarn material is modeled as Kelvin element. A partial differential equation governing the transversal vibration is derived from Newtons second law. Galerkin method is used to truncate the governing nonlinear differential equation, and thus first-order ordinary differential equation is obtained. The periodic vibration equation and the natural frequency of moving yarn are received by applying homotopy perturbation method. As a result, the condition which should be avoided in the weaving process for resonance is obtained.

Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

2016 ◽  
Vol 9 (1) ◽  
pp. 144-156 ◽  
Author(s):  
Majid Ghadiri ◽  
Mohsen Safi
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Simply Supported ◽  
Homotopy Perturbation ◽  
Functionally Graded Nanobeam

AbstractIn this paper, He's homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen's nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin's method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

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