Numerical analytic method for solving a large class of generalized non-homogeneous variable coefficients KdV problems based on lie symmetries

2020 ◽  
pp. 2150115
Author(s):  
Azadeh R. Moghaddam ◽  
Mortaza Gachpazan
Keyword(s):  
Numerical Methods ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Analytic Method ◽  
Initial Value ◽  
Lie Symmetry Method ◽  
Generalized Kdv Equation ◽  
Symmetry Method ◽  
Reduced Problem

In this study, we use Lie symmetry method to reduce a generalized KdV equation with initial and boundary conditions with non-homogeneous variable coefficients to initial value problem. In particular, we concentrate on the cases that the reduced IVP cannot be solved analytically. We also compare the approximated solutions of IVP using numerical methods with IBVP, and note that they are more efficient than doing the same procedure for IBVP. In fact, it is shown that reducing IBVP to IVP and solving the reduced problem numerically will lead to more accurate solutions.

Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics

2018 ◽  
Vol 28 (3) ◽  
pp. 518-531 ◽  
Author(s):  
Sudao Bilige ◽  
Yanqing Han
Keyword(s):  
Fluid Mechanics ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Lie Symmetry ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Content Type ◽  
Lie Symmetry Method ◽  
Homotopy Perturbation ◽  
Symmetry Method

Purpose The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics. Design/methodology/approach The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods. Findings First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions. Originality/value The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

scholarly journals Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gang-Wei Wang ◽  
Tian-Zhou Xu
Keyword(s):  
Lie Algebra ◽  
Group Analysis ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Optimal System ◽  
Explicit Solutions ◽  
Kawahara Equation ◽  
Lie Symmetry Method ◽  
Symmetry Method ◽  
Similarity Reductions

The simplified modified Kawahara equation with variable coefficients is studied by using Lie symmetry method. Then we obtain the corresponding Lie algebra, optimal system, and the similarity reductions. At last, we also give some new explicit solutions for some special forms of the equations.

Homoclinic breather-wave and singular periodic wave for a (2 + 1)D GSWW equation

2019 ◽  
Vol 29 (3) ◽  
pp. 1000-1009 ◽  
Author(s):  
Kang Xiaorong ◽  
Xian Daquan
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Lie Symmetry ◽  
Bilinear Method ◽  
Content Type ◽  
Lie Symmetry Method ◽  
Dynamical Behaviors ◽  
Higher Dimensional ◽  
Symmetry Method

Purpose The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation. Design/methodology/approach The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation. Findings A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented. Research limitations/implications As a typical nonlinear evolution equation, some dynamical behaviors are also discussed. Originality/value These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

scholarly journals Construction of invariant solutions and conservation laws to the $(2+1)$-dimensional integrable coupling of the KdV equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ben Gao ◽  
Qinglian Yin
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Tanh Method ◽  
The Kdv Equation ◽  
Integrable Coupling ◽  
Symmetry Method

AbstractUnder investigation in this paper is the $(2+1)$ ( 2 + 1 ) -dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Maria Ihsane El Bahi ◽  
Khalid Hilal
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Nonlinear Partial Differential Equation ◽  
Lie Symmetry ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Symmetry Operators ◽  
Generalized Time ◽  
Symmetry Method ◽  
Conservation Theorem

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

New soliton solutions of the CH–DP equation using Lie symmetry method

2018 ◽  
Vol 32 (20) ◽  
pp. 1850234 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Symmetry Method

In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

scholarly journals Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Mostafa Hesamiarshad
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Symbolic Computation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Reduced Equations ◽  
Symmetry Generators ◽  
Symmetry Method

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

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