Symmetry reductions, exact solutions, stability analysis and conservation laws of time-fractional Sharma–Tasso–Olever equation

2019 ◽  
Vol 16 (06) ◽  
pp. 1950087
Author(s):  
Youwei Zhang
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Group Analysis ◽  
Lie Group Analysis ◽  
Series Solutions ◽  
Analysis Method ◽  
Symmetry Reductions ◽  
Wave Solutions ◽  
Power Series Solutions ◽  
Detailed Derivation

In this paper, Lie group analysis method is applied to consider a vector field and symmetry reductions for time-fractional Sharma–Tasso–Olver equation, exact hyperbolic wave solutions, power series solutions and its convergence are investigated. Stability and boundedness analysis of trivial solution to the reduced ordinary differential equation is shown by constructing appropriate Lyapunov function. Conservation laws of the equation are well constructed with a detailed derivation making use of Noether’s operator.

Symmetry reductions, explicit solutions, convergence analysis and conservation laws via multipliers approach to the Chen–Lee–Liu model in nonlinear optics

2019 ◽  
Vol 33 (04) ◽  
pp. 1950035 ◽  
Author(s):  
Aliyu Isa Aliyu ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Mustafa Bayram ◽  
Dumitru Baleanu
Keyword(s):  
Nonlinear Optics ◽  
Power Series ◽  
Conservation Laws ◽  
Convergence Analysis ◽  
Symmetry Analysis ◽  
Explicit Solutions ◽  
Series Solutions ◽  
Symmetry Reductions ◽  
Power Series Solutions

In this paper, symmetry analysis is performed for the nonlinear Chen–Lee–Liu equation (NCLE) arising in temporal pulses. New forms of explicit solutions of the equation are constructed using the optimal systems by applying the power series solutions (PSS) technique and the convergence of the PSS is investigated. Finally, the conservation laws (Cls) of the model is studied using the multiplier approach.

scholarly journals Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 874
Author(s):  
Selahattin Gülşen ◽  
Shao-Wen Yao ◽  
Mustafa Inc
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Group Analysis ◽  
Nonlinear Ordinary Differential Equation ◽  
Series Solutions ◽  
Nonlinear Odes ◽  
Fractional Differential Operator ◽  
Exact Power ◽  
Power Series Solutions ◽  
Fractional Differential

In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.

Bright soliton solutions, power series solutions and travelling wave solutions of a (3+1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (06) ◽  
pp. 1850082
Author(s):  
Ding Guo ◽  
Shou-Fu Tian ◽  
Li Zou ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Bright Soliton ◽  
Travelling Wave Solutions ◽  
Series Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Power Series Solutions ◽  
De Vries

In this paper, we consider the (3[Formula: see text]+[Formula: see text]1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the nonlinear waves in plasma physics and fluid dynamics. By using solitary wave ansatz in the form of sech[Formula: see text] function and a direct integrating way, we construct the exact bright soliton solutions and the travelling wave solutions of the equation, respectively. Moreover, we obtain its power series solutions with the convergence analysis. It is hoped that our results can provide the richer dynamical behavior of the KdV-type and KP-type equations.

LIE SYMMETRY ANALYSIS TO THE WEAKLY COUPLED KAUP–KUPERSHMIDT EQUATION WITH TIME FRACTIONAL ORDER

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950052 ◽  
Author(s):  
ZHENLI WANG ◽  
LIHUA ZHANG ◽  
CHUANZHONG LI
Keyword(s):  
Fractional Order ◽  
Group Analysis ◽  
Nonlinear Ordinary Differential Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Explicit Solutions ◽  
Lie Group Analysis ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Weakly Coupled

The aim of this paper is to apply the Lie group analysis method to the weakly coupled Kaup–Kupershmidt (KK) equation with time fractional order. We considered the symmetry analysis, explicit solutions to the weakly coupled time fractional KK (TF-KK) equation with Riemann–Liouville (RL) derivative. The weakly coupled TF-KK equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. We solve the reduced fractional ODE using the sub-equation method.

Lie group analysis, Hamiltonian equations and conservation laws of Born–Infeld equation

2014 ◽  
Vol 07 (03) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Reza Hejazi
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Lie Group ◽  
Group Analysis ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Group Method ◽  
Lie Group Analysis ◽  
Hamiltonian Equations

Lie symmetry group method is applied to study the Born–Infeld equation. The symmetry group is given, and similarity solutions associated to the symmetries are obtained. Finally the Hamiltonian equations including Hamiltonian symmetry group and conservation laws are determined.

scholarly journals Exact Solutions and Conservation Laws of the Time-Fractional Gardner Equation with Time-Dependent Coefficients

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2434
Author(s):  
Ruixin Li ◽  
Lianzhong Li
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Time Dependent ◽  
Series Solutions ◽  
Lie Symmetry Analysis ◽  
Gardner Equation ◽  
Power Series Solutions ◽  
Conservation Theorem

In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some specific cases are obtained. By virtue of the fractional conservation theorem, the conservation laws are constructed.

scholarly journals Improved Numerical Computation to Solve Lane-Emden type Equations

2021 ◽  
Vol 10 (3) ◽  
pp. 1980-1984
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Present Method ◽  
Mathematical Physics ◽  
Equilibrium Density ◽  
Series Solutions ◽  
Maclaurin Series ◽  
Emden Equation ◽  
Power Series Solutions

Lane-Emden equation is also of fundamental importance in mathematical physics, celestial mechanics,and computer science. It can be used to describe stellar structures, equilibrium density distribution in polytrophicisothermal gas, thermal behavior in mutual attraction of its molecules. An improved numerical method is developed for solving Lane-Emden type differential equations. The method is based on power series solutions of differential equations and Maclaurin series expansion. A python program is written to carry out numerical calculations. Five examples are solved and shown in this paper, the solutions obtained by the program are compared with the exact solutions of differential equation, an excellent agreement is found between them. The present method improves runtime.

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