Analysis of Generalised Boussinesq Coupled Equations Using Lie Symmetry

Abstract

In the last decades, Nonlinear partial differential equations (NPDEs) have become essential tools to model complex phenomena that arise in different aspects of science and engineering such as hydrodynamics. Therefore, constructing exact and approximate solutions of NLPDEs is of great importance in mathematical sciences. Previously authors have done similar work with restriction of K and L to be one. In this paper we solve the generalised Boussinesq coupled equations:

𝒖𝒕 + 𝑲𝒗𝒙 + 𝑳𝒖𝒖𝒙 = 𝟎; 𝑲 > 𝟎; 𝑳 > 𝟎 𝒗𝒕 + 𝒖𝒗𝒙 + 𝒖𝒙𝒙𝒙 = 𝟎

using Lie symmetry of differential equations where u = u (x; t) is the velocity of water and v = v (x; t) the total depth of water and subscripts denote partial derivatives. The positive constants K, L would enable further analysis of optimal water depth and velocity be determined.