scholarly journals Exact Solutions and Conservation Laws of a Generalized (1 + 1) Dimensional System of Equations via Symbolic Computation

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2916
Author(s):  
Sivenathi Oscar Mbusi ◽  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution Equations ◽  
Theoretical Physics ◽  
Dimensional System ◽  
Ansatz Method ◽  
New Exact Solutions ◽  
Time Space ◽  
First Time

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

scholarly journals Some Methods about Finding the Exact Solutions of Nonlinear Modified BBM Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
First Time ◽  
Bbm Equation

Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.

Application of the Subequation Method to Some Differential Equations of Time-Fractional Order

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Ahmet Bekir ◽  
Esin Aksoy
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
New Exact Solutions

The main goal of this paper is to develop subequation method for solving nonlinear evolution equations of time-fractional order. We use the subequation method to calculate the exact solutions of the time-fractional Burgers, Sharma–Tasso–Olver, and Fisher's equations. Consequently, we establish some new exact solutions for these equations.

Application of the Exp-Functionmethod to the Riccati Equation and New Exact Solutions with Three Arbitrary Functions of Quantum Zakharov Equations

2008 ◽  
Vol 63 (10-11) ◽  
pp. 646-652 ◽  
Author(s):  
Mohamed A Abdou ◽  
Essam M. Abulwafa
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
New Exact Solutions ◽  
Generalized Riccati Equation ◽  
Solitary Solutions

The Exp-function method with the aid of the symbolic computational system is used for constructing generalized solitary solutions of the generalized Riccati equation. Based on the Riccati equation and its generalized solitary solutions, new exact solutions with three arbitrary functions of quantum Zakharov equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.

scholarly journals Bright and dark 1– soliton solutions to perturbed Schrodinger – hirota equation with power law nonlinearity via semi – inverse variation method and ansatz method

2017 ◽  
Vol 5 (2) ◽  
pp. 39 ◽  
Author(s):  
S. Subhaschandra Singh
Keyword(s):  
Exact Solutions ◽  
Power Law ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Variation Method ◽  
Science And Engineering ◽  
Ansatz Method ◽  
Hirota Equation

This paper studies perturbed Schrodinger Hirota equation with power law nonlinearity by obtaining its 1 – soliton solutions via He’s semi – inverse variation method and the Ansatz method and the results reveal that these methods are very effective ones for obtaining exact solutions to various types of nonlinear evolution equations appearing in the studies of science and engineering.

New Exact Solutions to a Class of Nonlinear Evolution Equations with Symbolic Computations

2013 ◽  
Vol 645 ◽  
pp. 312-315
Author(s):  
Jian Ya Ge ◽  
Tie Cheng Xia
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Method ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Searching for exact solutions to nonlinear evolution equations is an important topic in mathematical physics and engineering. Many methods of finding exact solutions have been presented such as the inverse scattering method, algebraic method and so on. In this paper, by using Fan sub-equation method with the help of Maple, several meaningful solutions are obtained including bell shape solutions, trigonometric function solutions, twist shape solutions and Jacobi elliptic function solutions for a class of nonlinear evolution equation. This method can be applied to other nonlinear partial differential equations.

scholarly journals New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jun Su ◽  
Genjiu Xu
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Wronskian Determinant ◽  
New Exact Solutions ◽  
Positon Solutions ◽  
Bkp Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

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