scholarly journals Lie Symmetry Analysis and Explicit Solutions for the Time-Fractional Regularized Long-Wave Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nisrine Maarouf ◽  
Hicham Maadan ◽  
Khalid Hilal
Keyword(s):  
Vector Fields ◽  
Group Analysis ◽  
Power Series Expansion ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Visual Interpretation ◽  
Long Wave ◽  
Rlw Equation ◽  
Regularized Long Wave Equation ◽  
Liouville Fractional Derivative

This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdèlyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.

scholarly journals Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nisrine Maarouf ◽  
Khalid Hilal
Keyword(s):  
Power Series Expansion ◽  
Planck Equation ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Liouville Fractional Derivative ◽  
Detailed Derivation ◽  
Conservation Theorem ◽  
Fokker Planck

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

Solitons, Lie Group Analysis and Conservation Laws of a (3+1)-Dimensional Modified Zakharov-Kuznetsov Equation in a Multicomponent Magnetised Plasma

2017 ◽  
Vol 72 (12) ◽  
pp. 1159-1171 ◽  
Author(s):  
Xia-Xia Du ◽  
Bo Tian ◽  
Jun Chai ◽  
Yan Sun ◽  
Yu-Qiang Yuan
Keyword(s):  
Conservation Laws ◽  
Group Analysis ◽  
Power Series Expansion ◽  
Expansion Method ◽  
Negative Ion ◽  
Point Symmetry ◽  
Higher Temperature ◽  
Lie Point Symmetry ◽  
Symmetry Generators ◽  
Positive Ion

AbstractIn this paper, we investigate a (3+1)-dimensional modified Zakharov-Kuznetsov equation, which describes the nonlinear plasma-acoustic waves in a multicomponent magnetised plasma. With the aid of the Hirota method and symbolic computation, bilinear forms and one-, two- and three-soliton solutions are derived. The characteristics and interaction of the solitons are discussed graphically. We present the effects on the soliton’s amplitude by the nonlinear coefficients which are related to the ratio of the positive-ion mass to negative-ion mass, number densities, initial densities of the lower- and higher-temperature electrons and ratio of the lower temperature to the higher temperature for electrons, as well as by the dispersion coefficient, which is related to the ratio of the positive-ion mass to the negative-ion mass and number densities. Moreover, using the Lie symmetry group theory, we derive the Lie point symmetry generators and the corresponding symmetry reductions, through which certain analytic solutions are obtained via the power series expansion method and the (G′/G) expansion method. We demonstrate that such an equation is strictly self-adjoint, and the conservation laws associated with the Lie point symmetry generators are derived.

The Solution of the Regularized Long Wave Equation Using the Fourier Leap-Frog Method

2010 ◽  
Vol 65 (4) ◽  
pp. 268-276 ◽  
Author(s):  
Hany N. Hassan ◽  
Hassan K. Saleh
Keyword(s):  
Fourier Transform ◽  
Solitary Waves ◽  
Wave Equations ◽  
Space Dimension ◽  
Nonlinear Wave Equations ◽  
Long Wave ◽  
Rlw Equation ◽  
Regularized Long Wave Equation ◽  
Waves Interaction ◽  
Space Dependence

An efficient numerical method is developed for solving nonlinear wave equations by studying the propagation and stability properties of solitary waves (solitons) of the regularized long wave (RLW) equation in one space dimension using a combination of leap frog for time dependence and a pseudospectral (Fourier transform) treatment of the space dependence. Our schemes follow very accurately these solutions, which are given by simple closed formulas. Studying the interaction of two such solitons and three solitary waves interaction for the RLW equation. Our implementation employs the fast Fourier transform (FFT) algorithm.

scholarly journals Application of (G′G)-expansion method to Regularized Long Wave (RLW) equation

2011 ◽  
Vol 61 (8) ◽  
pp. 2044-2047 ◽  
Author(s):  
M.M. Kabir ◽  
A. Borhanifar ◽  
R. Abazari
Keyword(s):  
Expansion Method ◽  
Long Wave ◽  
Rlw Equation

scholarly journals Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Özkan Güner ◽  
Dursun Eser
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Long Wave ◽  
Functional Variable ◽  
Liouville Derivative ◽  
Functional Variable Method ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

We apply the functional variable method, exp-function method, and(G′/G)-expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE) in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.

Rosenau-KdV Equation Coupling with the Rosenau-RLW Equation: Conservation Laws and Exact Solutions

2017 ◽  
Vol 18 (6) ◽  
pp. 451-456 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Long Wave ◽  
Rlw Equation ◽  
Kudryashov Method ◽  
Regularized Long Wave Equation ◽  
De Vries

AbstractWe compute the conservation laws for the Rosenau-Kortweg de Vries equation coupling with the Regularized Long-Wave equation using Noether’s approach through a remarkable method of increasing the order of the Rosenau-KdV-RLW equation. Furthermore, exact solutions for the Rosenau- KdV-RLW equation are acquired by employing the Kudryashov method.

Analytic solutions of the scattering by two multilayered dielectric spheres

1992 ◽  
Vol 70 (9) ◽  
pp. 696-705 ◽  
Author(s):  
A-K. Hamid ◽  
I. R. Ciric ◽  
M. Hamid
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Plane Electromagnetic Wave ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Addition Theorem ◽  
Modal Expansion ◽  
Modal Expansion Method ◽  
Dielectric Spheres ◽  
Scattered Fields

The problem of plane electromagnetic wave scattering by two concentrically layered dielectric spheres is investigated analytically using the modal expansion method. Two different solutions to this problem are obtained. In the first solution the boundary conditions are satisfied simultaneously at all spherical interfaces, while in the second solution an iterative approach is used and the boundary conditions are satisfied successively for each iteration. To impose the boundary conditions at the outer surface of the spheres, the translation addition theorem of the spherical vector wave functions is employed to express the scattered fields by one sphere in the coordiante system of the other sphere. Numerical results for the bistatic and back-scattering cross sections are presented graphically for various sphere sizes, layer thicknesses and permittivities, and angles of incidence.

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