Symmetry group analysis of Benney system and an application for shallow-water equations

A Numerical Study for the Solution of Time Fractional Nonlinear Shallow Water Equation in Oceans

“…One form of the SWEs may be derived from Benney system. The Benney equations [7], which are derived from the two-dimensional and time-dependent motion of an inviscid homogeneous fluid in a gravitational field by assuming the depth of the fluid to be small compared to the horizontal wave lengths considered, are expressed as…”

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

“…The results presented here can be, in principle, extended by using the Lie-group symmetry analysis to obtain different kinds of similarity variables and corresponding reduced equations. Such an analysis may be performed along the lines given in references [13,3,10] and presented in specific cases in [17,14,15]. However, the presence of noninteger derivatives in equations treated here, makes this analysis nontrivial.…”

Lie symmetry solutions for two-phase mass flows

“…The Lie group theory has been extensively used to obtain solutions of differential equations arising from wide range of physical problems, including the reduction of differential equations, order reduction of ordinary differential equations, construction of invariant solutions, and mapping between the solutions in mechanics, applied mathematics and mathematical physics, and applied and theoretical physics [11,[19][20][21][22][23][24][25][26][27]. Lie algebra and symmetry group theory is an important and effective method for solving a system of non-linear partial differential equations via similarity forms and similarity solutions [19,23,[27][28][29][30][31][32].…”

“…Particularly related to our interest, Lie theory has been successively applied to construct and investigate self-similar solutions to gravity currents, and relatively simple shallow water or two-layer shallow water flows, impulse modified shallow water equations, dispersive shallow-water flows, Benney system, generalized Burgers equation, etc. [11,[25][26][27][33][34][35][36][37].…”