scholarly journals Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation

2019 ◽  
Vol 65 (5 Sept-Oct) ◽  
pp. 529 ◽  
Author(s):  
M. S. Hasheim ◽  
M. Inc ◽  
M. Bayram
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Conformable Fractional Derivative ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Symmetry Properties ◽  
Lie Symmetry Approach

In this paper, the time fractional Kolmogorov-Petrovskii-Piskunov (FKP) equation is analyzed by means of Lie symmetry approach. The FKP is reduced to ordinary differential equation of fractional order via the attained point symmetries. Moreover, the simplest equation method is used in construct the exact solutions of underlying equation with recently introduced conformable fractional derivative.

scholarly journals Application of Lie Symmetry Analysis and Simplest Equation Method for Finding Exact Solutions of Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Hossein Jafari ◽  
Nematollah Kadkhoda ◽  
Chaudry Massod Khalique
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Equation Method ◽  
Lie Symmetry Analysis ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Lie Symmetry Approach

The Lie symmetry approach with simplest equation method is used to construct exact solutions of the bad Boussinesq and good Boussinesq equations. As the simplest equation, we have used the equation of Riccati.

scholarly journals Exact Solutions ofϕ4Equation Using Lie Symmetry Approach along with the Simplest Equation and Exp-Function Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Hossein Jafari ◽  
Nematollah Kadkhoda ◽  
Chaudry Masood Khalique
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Travelling Wave ◽  
Lie Symmetry ◽  
Equation Method ◽  
Travelling Wave Solutions ◽  
Symmetry Approach ◽  
Wave Solutions ◽  
Simplest Equation Method ◽  
Lie Symmetry Approach

This paper obtains the exact solutions of theϕ4equation. The Lie symmetry approach along with the simplest equation method and the Exp-function method are used to obtain these solutions. As a simplest equation we have used the equation of Riccati in the simplest equation method. Exact solutions obtained are travelling wave solutions.

Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

2020 ◽  
Vol 30 (01) ◽  
pp. 2050004 ◽  
Author(s):  
Jianli Liang ◽  
Longkun Tang ◽  
Yonghui Xia ◽  
Yi Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Leibniz Rule ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics

2021 ◽  
pp. 2150163
Author(s):  
Vinita ◽  
S. Saha Ray
Keyword(s):  
Conservation Laws ◽  
Quantum Physics ◽  
Physical Structure ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Similarity Reduction ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Lie Symmetry Approach ◽  
Conservation Theorem

A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.

scholarly journals A solution method for integro-differential equations of conformable fractional derivative

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 7-14 ◽  
Author(s):  
Mustafa Bayram ◽  
Veysel Hatipoglu ◽  
Sertan Alkan ◽  
Sebahat Das
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Solution Method ◽  
Approximate Solutions ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Numerical Examples

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

Exact solutions to the space-time fraction Whitham–Broer–Kaup equation

2020 ◽  
Vol 34 (16) ◽  
pp. 2050178
Author(s):  
Damin Cao ◽  
Cheng Li ◽  
Fajiang He
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Space Time ◽  
Polynomial Discriminant

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective.

Regarding Rational Exponential and Hyperbolic Functions Solutions of Conformable Time-Fractional Phi-four Equation

2018 ◽  
Author(s):  
Asim Zafar ◽  
Alper Korkmaz ◽  
Bushra Khalid ◽  
Hadi Rezazadeh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Function Method ◽  
Wave Transformation ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Hyperbolic Function Method ◽  
Projected Methods

In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.

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