Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny–Lin equation

2017 ◽  
Vol 14 (12) ◽  
pp. 1750170 ◽  
Author(s):  
Saeede Rashidi ◽  
S. Reza Hejazi
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Group Analysis ◽  
Navier Stokes ◽  
Kawahara Equation ◽  
Navier Stokes Equation ◽  
Invariance Properties ◽  
Fractional Differential ◽  
Fifth Order ◽  
Sivashinsky Equation

This paper investigates the invariance properties of the time fractional Benny–Lin equation with Riemann–Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto–Sivashinsky equation and Navier–Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny–Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

Group formalism of Lie transformations, exact solutions and conservation laws of nonlinear time-fractional Kramers equation

2020 ◽  
Vol 17 (12) ◽  
pp. 2050190
Author(s):  
Zahra Momennezhad ◽  
Mehdi Nadjafikhah
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Group Analysis ◽  
Systematic Investigation ◽  
Analysis Method ◽  
Invariance Properties ◽  
Kramers Equation ◽  
Fractional Differential ◽  
Lie Transformations ◽  
Noether Operators

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear [Formula: see text]-dimensional time-fractional Kramers equation via Riemann–Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear [Formula: see text]-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov’s method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.

Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation

2016 ◽  
Vol 86 ◽  
pp. 8-15 ◽  
Author(s):  
Gangwei Wang ◽  
Abdul H. Kara ◽  
Kamran Fakhar ◽  
Jose Vega-Guzman ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Group Analysis ◽  
Fifth Order

Numerical and experimental study on the dynamic bearing properties of a four-pad and eight-pad tilting pad journal bearings in a vertical rotor

2021 ◽  
pp. 1-24
Author(s):  
Gudeta Berhanu Benti ◽  
David Jose Rondon ◽  
Rolf Gustavsson ◽  
Jan-Olov Aidanpää
Keyword(s):  
Dynamic Properties ◽  
Mean Value ◽  
Journal Bearings ◽  
Navier Stokes ◽  
Rigid Rotor ◽  
Navier Stokes Equation ◽  
Vertical Rotor ◽  
Tilting Pad ◽  
Fifth Order ◽  
Tilting Pad Journal Bearings

Abstract In this paper, the dynamics of tilting pad journal bearings with four and eight pads are studied and compared experimentally and numerically. The experiments are performed on a rigid vertical rotor supported by two identical bearings. Two sets of experiments are carried out under similar test setup. One set is performed on a rigid rotor with two four-pad bearings, while the other is on a rigid rotor with two eight-pad bearings. The dynamic properties of the two bearing types are compared with each other by studying the unbalance response of the system at different rotor speeds. Numerically, the test rig is modeled as a rigid rotor and the bearing coefficients are calculated based on Navier-Stokes equation. A nonlinear bearing model is developed and used in the steady state response simulation. The measured and simulated displacement and force orbits show similar patterns for both bearing types. Compared to the measurement, the simulated mean value and range (peak-to-peak amplitude) of the bearing force deviate with a maximum of 16 % and 38 %, respectively. It is concluded that, unlike the eight-pad TPJB, the four-pad TPJB excite the system at the third and fifth-order frequencies, which are due to the number of pads, and the amplitudes of these frequencies increase with the rotor speed.

Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques

2020 ◽  
Vol 34 (07) ◽  
pp. 2050045 ◽  
Author(s):  
Naila Nasreen ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Problems ◽  
Applied Physics ◽  
Kawahara Equation ◽  
Generalized Riccati Equation ◽  
All Solutions ◽  
Fifth Order

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

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