SYMMETRY ANALYSIS FOR A SEVENTH-ORDER GENERALIZED KdV EQUATION AND ITS FRACTIONAL VERSION IN FLUID MECHANICS

Abstract

KdV types of equations play an important role in many fields. In this paper, we study a seventh-order generalized KdV equation and its fractional version in fluid mechanics using symmetry. From symmetry, the corresponding vectors, symmetry reduction and conservation laws are derived. Potential equation is also analyzed with regard to the symmetry method. Based on the symmetry, similarity reductions and conservation laws are also presented. Subsequently, the fractional version of the seventh-order KdV equation is discussed. Finally, differential invariants are constructed for the special case.

Consistent Burgers equation expansion method and its applications to high- dimensional Burgers-type equations

“…Besides, new solutions with physical significance need to be further constructed and discovered. Through the continuous efforts of mathematicians and physicists, a large number of effective methods have been established and developed, for example, the symmetry method [5][6][7][8][9][10][11][12][13][14][15][16], inverse scattering transformation [1], ä B cklund transformation [17,18], Darboux transformations [19], the Hirotas bilinear direct method [20], auxiliary equation expansion method [21], generalized multi-symplectic method [22][23][24][25] homogeneous balance method [26], F-expansion method [27], homotopy perturbation method [28,29], CKs direct symmetry reduction method [30], consistent Riccati expansion method [31], consistent KdV expansion method [32] and so on.…”

Multiple Exp-Function Solutions, Group Invariant Solutions and Conservation Laws of a Generalized (2+1)-dimensional Hirota-Satsuma-Ito Equation

“…One of the most versatile methods for determining the exact solutions of NLEEs is the Lie symmetry method, also known as the Lie group analysis method. The Lie symmetry technique is an effective, dependable, and robust mathematical tool in symmetry theory, with some advantages over other enormously complicated mathematical techniques used to obtain closed-form invariant solutions of nonlinear evolution equations (NLEEs) [6,14,15].…”

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

“…The present work is related with the construction and implementation of a numerical technique to obtain the approximate solution of Kuramoto-Sivanshinsky equation (KSE) with initial-boundary conditions. The KSE is a great fundamental interest in the same way just as it is famous counterp-arts Korteweg-de Vries-Burger-Kuromato equation (Alimirzaluo & Nadjafikhah, 2019), fractional optical solitons of the space-time fractional nonlinear Schr€ odinger equations (Wu, Yu, & Wang, 2020), nonlinear wave-like equations stated by Kumar, Singh, Purohit, and Swroop (2019), symmetry breaking of infinite-dimensional dynamic system (Hu, Wang, Zhao, & Deng, 2020), vibration and elastic wave propagation in spatial flexible damping panel attached to four special spring (Hu, Zhang, & Deng, 2020), internal resonance of a flexible beam in a spatial tethered system (Hu, Ye, & Deng, 2020), minimum control energy of spatial beam with assumed attitude adjustment target (Hu, Yu, & Deng, 2020), symplectic analysis on orbit-attitude coupling dynamic problem of spatial rigid rod (Hu, Yin, Zheng, & Deng, 2020), interaction effects of DNA, RNA-polymerase, and cellular fluid on the local dynamic behaviours of DNA , different order of ordinary differential equations (Gebremedhin & Jena, 2019, 2020Jena, Mohanty, & Mishra, 2018;Mohanty, Jena, & Mishra, 2020;, B-spline collocation (Jena & Gebremedhin, 2021;Jena, Senapati, & Gebremedhin, 2020a, 2020b) symmetry analysis and rogue wave solutions for the (2 þ 1)dimensional nonlinear Schr€ odinger equation with variable coefficients (Wang, 2016), novel (3 þ 1)dimensional sine-Gorden and sinh-Gorden equation: Derivation symmetries and conservation laws (Wang, 2021), (2 þ 1)-dimensional KdV equation and mKdV equation: symmetries, group invariant solutions and conservation laws (Wang & Kara, 2019), symmetry analysis for a seventh-order generalized KdV equation and its fractional version in fluid mechanics (Wang, Liu, Wu, & Su, 2020), (2 þ 1)-dimensional Boiti-Leon-Pempinelli equation-Domail walls, invariance properties and conservation laws (Wang, Vega-Guzman, Biswas, Kamis Alzahrani, & Kara, 2020), (2 þ 1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solution (Wang, Yang, Gu, Gua...…”